# On the Parametric Stokes Phenomenon for Solutions of Singularly Perturbed Linear Partial Differential Equations

Authors: Stéphane Malek
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study a family of singularly perturbed linear partial differential equations with irregular typeϵt2∂t∂zSXi(t,z,ϵ)+(ϵt+1)∂zSXi(t,z,ϵ)=∑(s,k0,k1)∈𝒮bs,k0,k1(z,ϵ)ts∂tk0∂zk1Xi(t,z,ϵ) in the complex domain. In a previous work, Malek (2012), we have given sufficient conditions under which the Borel transform of a formal solution to the above mentioned equation with respect to the perturbation parameter ϵ converges near the
origin in ℂ and can be extended on a finite number of unbounded sectors with small opening and bisecting directions, say κi∈[0,2π), 0≤i≤ν−1 for some integer ν≥2. The proof rests on the construction of neighboring sectorial holomorphic solutions to the first mentioned equation whose differences have exponentially small bounds in the perturbation parameter (Stokes phenomenon) for which the classical Ramis-Sibuya theorem can be applied. In this paper, we introduce new conditions for the Borel transform to be analytically continued
in the larger sectors {ϵ∈ℂ∗/arg(ϵ)∈(κi,κi+1)}, where it develops isolated singularities of logarithmic type lying on some half lattice. In the proof, we use a criterion of analytic continuation of the Borel transform described by Fruchard and Schäfke (2011) and is based on a more accurate description of the Stokes phenomenon for the sectorial solutions mentioned above.


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## Body

## 1. Introduction

We consider a family of singularly perturbed linear partial differential equations of the form(1.1)ϵt2∂t∂zSXi(t,z,ϵ)+(ϵt+1)∂zSXi(t,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)ts∂tk0∂zk1Xi(t,z,ϵ)
for given initial conditions
(1.2)(∂zjXi)(t,0,ϵ)=φi,j(t,ϵ),0≤i≤ν-1,0≤j≤S-1,
where ϵ is a complex perturbation parameter, S is some positive integer, ν is some positive integer larger than 2, and 𝒮 is a finite subset of ℕ3 with the property that there exists an integer b>1 with
(1.3)S≥b(s-k0+2)+k1,s≥2k0
for all (s,k0,k1)∈𝒮, and the coefficients bs,k0,k1(z,ϵ) belong to 𝒪{z,ϵ} where 𝒪{z,ϵ} denotes the space of holomorphic functions in (z,ϵ) near the origin in ℂ2. In this work, we make the assumption that the coefficients of (1.1) factorize in the form bs,k0,k1(z,ϵ)=ϵk0b~s,k0,k1(z,ϵ) where b~s,k0,k1(z,ϵ) belong to 𝒪{z,ϵ}. The initial data φi,j(t,ϵ) are assumed to be holomorphic functions on a product of two sectors 𝒯×ℰi, where 𝒯 is a fixed bounded sector centered at 0 and ℰi, 0≤i≤ν-1, are sectors with opening larger than π centered at the origin whose union form a covering of 𝒱∖{0}, where 𝒱 is some neighborhood of 0. For all ϵ≠0, this family belongs to a class of partial differential equations which have a so-called irregular singularity at t=0 (in the sense of [1]).In the previous work [2], we have given sufficient conditions on the initial data φi,j(t,ϵ), for the existence of a formal series
(1.4)X^(t,z,ϵ)=∑k≥0‍Hk(t,z)ϵkk!∈𝒪(𝒯){z}[[ϵ]]
solution of (1.1), with holomorphic coefficients Hk(t,z) on 𝒯×D(0,δ) for some disc D(0,δ), with δ>0, such that, for all 0≤i≤ν-1, the solution Xi(t,z,ϵ) of the problem (1.1), (1.2) defines a holomorphic function on 𝒯×D(0,δ)×ℰi which is the 1-sum of X^ on ℰi. In other words, for all fixed (t,z)∈𝒯×D(0,δ), the Borel transform of X^ with respect to ϵ defined as ℬ(X^)(s)=∑k≥0‍Hk(t,z)sk/(k!2) is holomorphic on some disc D(0,s0) and can be analytically continued (with exponential growth) to sectors 𝒢κi, centered at 0, with infinite radius and with the bisecting direction κi∈[0,2π) of the sector ℰi. But in general, due to the fact that the functions Xi do not coincide on the intersections ℰi∩ℰi+1 (known as the Stokes phenomenon), the Borel transform cannot be analytically extended to the whole sectors Sκi,κi+1={s∈ℂ*/arg(s)∈(κi,κi+1)} for all 0≤i≤ν-1, where by convention κν=κ0, ℰν=ℰ0, and Xν=X0.In this work, we address the question of the possibility of analytic continuation, location of singularities, and behaviour near these singularities of the Borel transform within the sectorSκi,κi+1. More precisely, our goal is to give stronger conditions on the initial data φi,j(t,ϵ) under which the Borel transform ℬ(X^)(s) can be analytically continued to the full-punctured sector Sκi,κi+1 except a half lattice of points λk/t, k∈ℕ∖{0}, depending on t and some well-chosen complex number λ∈ℂ* and moreover develop logarithmic singularities at λk/t (Theorem 5.8).In a recent paper of Fruchard and Schäfke, see [3], an analogous study has been performed for formal WKB solutions y(x,ϵ)=exp((x2/2-x3/3)/ϵ)x-1/2(x-1)-1/2v^(x,ϵ) to the singularly perturbed Schrödinger equation
(1.5)ϵ2y′′(x,ϵ)=x2(x-1)2y(x,ϵ),
where v^(x,ϵ)=∑n≥0‍yn(x)ϵn is a formal series with holomorphic coefficients yn on some domain avoiding 0 and 1. The authors show that the Borel transform of v^ with respect to ϵ converges near the origin and can be analytically continued along any path avoiding some lattices of points depending on (x2/2-x3/3). We also mention that formal parametric Stokes phenomenon for 1-dimensional stationary linear Schrödinger equation ϵ2y′′(z)=Q(z)y(z), where Q(z) is a polynomial, has been investigated by several other authors using WKB analysis, see [4–6]. In a more general framework, analytic continuation properties related with the Stokes phenomenon have been studied by several authors in different contexts. For nonlinear systems of ODEs with irregular singularity at ∞ of the form y′(z)=f(z,y(z)) and for nonlinear systems of difference equations y(z+1)=g(z,y(z)), under nonresonance conditions, we refer to [7, 8]. For linearizations procedures for holomorphic germs of (ℂ,0) in the resonant case, we make mention to [9, 10]. For analytic conjugation of vector fields in ℂ2 to normal forms, we indicate [11, 12]. For Hamiltonian nonlinear first-order partial differential equations, we notice [13].In the proof of our main result, we will use a criterion for the analytic continuation of the Borel transform described by Fruchard and Schäfke in [3] (Theorem (FS) in Theorem 5.8). Following this criterion, in order to prove the analytic continuation of the Borel transform ℬ(X^)(s), say, on the sector Sκ0,κ1, for any fixed (t,z)∈𝒯×D(0,δ), we need to have a complete description of the Stokes relation between the solutions X0 and X1 of the form
(1.6)X1(t,z,ϵ)-X0(t,z,ϵ)=∑h=1m‍e-ah/ϵXh,0(t,z,ϵ)+𝒪(e-Ceiα/ϵ)
for all ϵ∈ℰ0∩ℰ1, for some integer m≥1, where {ah}1≤h≤m is a set of aligned complex numbers such that arg(ah)=α∈(κ0,κ1) with |ak|<C (for some C>0) and Xh,0(t,z,ϵ), h≥1, are the 1-sums of some formal series G^h(ϵ)∈𝒪(𝒯×D(0,δ))[[ϵ]] on ℰ0. If the relation (1.6) holds, then ℬ(X^)(s) can be analytically continued along any path in the punctured sector (Sκ0,κ1∩D(0,C))∖{ah}1≤h≤m and has logarithmic growth as s tends to ah in a sector. Actually, under suitable conditions on the initial data φi,j(t,ϵ), we have shown that such a relation holds for ak=λk/t, for some well-chosen λ∈ℂ* and for all k≥1, see (5.145) in Theorem 5.8. In order to establish such a Stokes relation (1.6), we proceed in several steps.In the first step, following the same strategy as in [2], using the linear map T↦T/ϵ=t, we transform the problem (1.1) into an auxiliary regularly perturbed singular linear partial differential equation which has an irregular singularity at T=0 and whose coefficients have poles with respect to ϵ at the origin, see (4.9). Then, for λ∈ℂ*, we construct a formal  transseries expansion of the form
(1.7)Y^(T,z,ϵ)=∑h≥0‍exp(-λh/T)h!Y^h(T,z,ϵ)
solution of the problem (4.9), (4.10), where each Y^h(T,z,ϵ)=∑m≥0‍Yh,m(z,ϵ)Tm/m! is a formal series in T with coefficients Yh,m(z,ϵ), which are holomorphic on a punctured polydisc D(0,δ)×(D(0,ϵ0)∖{0}). We show that the Borel transform of each Y^h(T,z,ϵ) with respect to T, defined by Vh(τ,z,ϵ)=∑m≥0‍Yh,m(z,ϵ)τm/(m!2), satisfies an integrodifferential Cauchy problem with rational coefficients in τ, holomorphic with respect to (τ,z) near the origin and meromorphic in ϵ with a pole at zero, see (4.20), (4.21). For well-chosen λ and suitable initial data, we show that each Vh(τ,z,ϵ) defines a holomorphic function near the origin with respect to (τ,z) and on a punctured disc with respect to ϵ and can be analytically continued to functions Vh,i(τ,z,ϵ) defined on the products Si×D(0,δ)×ℰi, where Si, 0≤i≤ν-1 are suitable open sectors with small opening and infinite radius. Moreover, the functions Vh,i(τ,z,ϵ) have exponential growth rate with respect to (τ,ϵ), namely, there exist A,B,K>0 such that
(1.8)supz∈D(0,δ)|Vh,i(τ,z,ϵ)|≤Ah!BheK|τ|/|ϵ|
for all (τ,z,ϵ) in their domain of definition and all h≥0 (Proposition 4.12). In order to get these estimates, we use the Banach spaces depending on two parameters β∈ℕ and ϵ with norms ||·||β,ϵ of functions v(τ) bounded by exp(Kβ|τ|/|ϵ|) for some bounded sequence Kβ already introduced in [2]. If one expands the functions Vh,i(τ,z,ϵ)=∑β≥0‍vh,i,β(τ,ϵ)zβ/β! with respect to z, we show that the generating function ∑h≥0,β≥0‍||vh,i,β(τ,ϵ)||β,ϵuhxβ/(h!β!) can be majorized by a series Wi(u,x) which satisfies a Cauchy problem of Kowalevski type (4.47), (4.48) and is therefore convergent near the origin in ℂ2.We construct a sequence of actual functionsYh,i(T,z,ϵ), h≥0, 0≤i≤ν-1 as Laplace transform of the functions Vh,i(τ,z,ϵ) with respect to τ along a halfline Li=ℝ+e-1γ⊂Si∪{0}. We show that the functions Xh,i(t,z,ϵ)=Yh,i(ϵt,z,ϵ) are holomorphic functions on the domain 𝒯×D(0,δ)×ℰi and that the functions Gh,i(ϵ):=Xh,i+1(t,z,ϵ)-Xh,i(t,z,ϵ) are exponentially flat as ϵ tends to 0 on ℰi+1∩ℰi as 𝒪(𝒯×D(0,δ))-valued functions. In the proof, we use, as in [2], a deformation of the integration's path in Xh,i and the estimates (1.8). Using the Ramis-Sibuya theorem (Theorem (RS) in Proposition 4.15), we deduce that each Xh,i(t,z,ϵ) is the 1-sum of a formal series G^h(ϵ)∈𝒪(𝒯×D(0,δ))[[ϵ]] on ℰi, for 0≤i≤ν-1 (Proposition 4.15). We notice that the functions X0,i(t,z,ϵ) actually coincide with the functions Xi(t,z,ϵ) mentioned above solving the problem (1.1), (1.2). We deduce that, for a suitable choice of λ, the function
(1.9)Z0(t,z,ϵ)=∑h≥0‍exp(-λh/ϵt)h!Xh,0(t,z,ϵ)
solves (1.1) on the domain 𝒯×D(0,δ)×(ℰ0∩ℰ1).In the second part of the proof, we establish the connection formulaX0,1(t,z,ϵ)=Z0(t,z,ϵ) which is exactly the Stokes relation (1.6) on 𝒯×D(0,δ)×(ℰ0∩ℰ1) (Proposition 5.2). The strategy we follow consists in expressing both functions X0,1 and Z0 as Laplace transforms of objects that are no longer functions in general but distributions supported on ℝ+ which are called staircase distributions in the terminology of [8]. We stress the fact that such representations of transseries expansions as generalized Laplace transforms were introduced for the first time by Costin in the paper [8]. Notice that similar arguments have been used in the work [14] to study the Stokes phenomenon for sectorial holomorphic solutions to linear integro-differential equations with irregular singularity.In Lemma5.5, we show that Z0 can be written as a generalized Laplace transform in the direction arg(λ) of a staircase distribution 𝕍(r,z,ϵ)=∑β≥0‍𝕍β(r,ϵ)zβ/β!∈𝒟'(σ,ϵ,δ), which is a convergent series in z on D(0,δ) with coefficients 𝕍β(r,ϵ) in some Banach spaces of staircase distributions 𝒟β,σ,ϵ′ on ℝ+ depending on the parameters β and ϵ (see Definition 2.3). We observe that the distribution 𝕍(r,z,ϵ) solves moreover an integro-differential Cauchy problem with rational coefficients in r, holomorphic with respect to z near the origin and meromorphic with respect to ϵ at zero, see (5.80), (5.81). The idea of proof consists in showing that each function Xh,0(t,z,ϵ) can be expressed as a Laplace transform in a sequence of directions ζn tending to arg(λ) of a sequence of staircase distributions 𝕍h,n(r,z,ϵ) (which are actually convergent series in z with coefficients that are C∞ functions in r on ℝ+ with exponential growth). Moreover, each distribution 𝕍h,n(r,z,ϵ) solves an integro-differential Cauchy problem (5.37), (5.38), whose coefficients tend to the coefficients of an integro-differential equation (5.39), (5.40), as n tends to ∞, having a unique staircase distribution solution 𝕍h,∞(r,z,ϵ). Under the hypothesis that the initial data (5.38) converge to (5.40) as n→+∞, we show that the sequence 𝕍h,n(r,z,ϵ) converges to 𝕍h,∞(r,z,ϵ) in the Banach space 𝒟′(σ,ϵ,δ) with precise norm estimates with respect to h and n (Lemma 5.3). In order to show this convergence, we use a majorazing series method together with a version of the classical Cauchy-Kowalevski theorem (Proposition 2.22) in some spaces of analytic functions near the origin in ℂ2 with dependence on initial conditions and coefficients applied to the auxiliary problem (5.66), (5.68). Using a continuity property of the Laplace transform (3.5), we show that each function Xh,0(t,z,ϵ) can be actually expressed as the Laplace transform of 𝕍h,∞(r,z,ϵ) in the direction arg(λ) and finally that Z0 itself is the Laplace transform of some staircase distribution 𝕍(r,z,ϵ) solving (5.80), (5.81).On the other hand, in Lemma5.7, under suitable conditions on φ1,j(t,ϵ), 0≤j≤S-1, we can also write X0,1(t,z,ϵ) as a generalized Laplace transform in the direction arg(λ) of the staircase distribution mentioned above 𝕍(r,z,ϵ) solving (5.80), (5.81). Therefore, the equality X0,1(t,z,ϵ)=Z0(t,z,ϵ) holds on 𝒯×D(0,δ)×(ℰ0∩ℰ1). The method of proof consists again in showing that X0,1(t,z,ϵ) can be written as Laplace transform in a sequence of directions ξn tending to arg(λ) of a sequence of staircase distributions 𝕍n(r,z,ϵ) (which are actually convergent series in z with coefficients that are C∞ functions in r on ℝ+ with exponential growth). Moreover, each distribution 𝕍n(r,z,ϵ) solves an integro-differential Cauchy problem (5.98), (5.99), whose coefficients tend to the coefficients of the integro-differential equation (5.80). Under the assumption that the initial data (5.99) converge to the initial data (5.81), we show that the sequence 𝕍n(r,z,ϵ) converges to the solution of (5.80), (5.81) (i.e., 𝕍(r,z,ϵ)) in the Banach space 𝒟′(σ,ϵ,δ), as n→+∞, see Lemma 5.6. This convergence result is obtained again by using a majorazing series technique which reduces the problem to the study of some linear differential equation (5.106), (5.109), whose coefficients and initial data tend to zero as n→+∞. Finally, by continuity of the Laplace transform, X0,1(t,z,ϵ) can be written as the Laplace transform of 𝕍(r,z,ϵ) in direction arg(λ).After Theorem5.8, we give an application to the construction of solutions to some specific singular linear partial differential equations in ℂ3 having logarithmic singularities at the points (λk/t,t,z), for k∈ℕ∖{0}. We show that under the hypothesis that the coefficients bs,k0,k1 are polynomials in ϵ, the Borel transform ℬ(X^)(s) turns out to solve the linear partial differential equation (5.149). We would like to mention that there exists a huge literature on the study of complex singularities and analytic continuation of solutions to linear partial differential equations starting from the fundamental contributions of Leray in [15]. Several authors have considered Cauchy problems a(x,D)u(x)=0, where a(x,D) is a differential operator of some order m≥1, for initial data ∂x0hu|x0=0=wh, 0≤h<m. Under specific hypotheses on the symbol a(x,ξ), precise descriptions of the solutions of these problems are given near the singular locus of the initial data wh. For meromorphic initial data, we may refer to [16–18] and for more general ramified multivalued initial data, we may cite [19–23].The layout of this work is as follows.In Section2, we introduce Banach spaces of formal series whose coefficients belong to spaces of staircase distributions and we study continuity properties for the actions of multiplication by C∞ functions and integro-differential operators on these spaces. In this section, we also exhibit a Cauchy Kowalevski theorem for linear partial differential problems in some space of analytic functions near the origin in ℂ2 with dependence of their solutions on the coefficients and initial data which will be useful to show the connection formula (5.28) stated in Section 5.In Section3, we recall the definition of a Laplace transform of a staircase distribution as introduced in [8] and we give useful commutation formulas with respect to multiplication by polynomials, exponential functions, and derivation.In Section4, we construct formal and analytic transseries solutions to the singularly perturbed partial differential equation with irregular singularity (1.1).In Section5, we establish the crucial connection formula relying on the analytic transseries solution Z0(t,z,ϵ) and the solution X0,1(t,z,ϵ) of (1.1). Finally, we state the main result of the paper which asserts that the Borel transform ℬ(X^)(s) in the perturbation parameter ϵ of the formal solution X^(t,z,ϵ) of (1.1) can be analytically continued along any path in the punctured sector Sκ0,κ1∖∪h≥1{λh/t} and has logarithmic growth as s tends to λh/t in a sector for all h≥1.


## 2. Banach Spaces of Formal Series with Coefficients in Spaces of Staircase Distributions: A Cauchy Problem in Spaces of Analytic Functions

### 2.1. Weighted Banach Spaces of Distributions

We define𝒟(ℝ+) to be the space of complex valued C∞-functions with compact support in ℝ+, where ℝ+ is the set of the positive real numbers x>0. We also denote by 𝒟'(ℝ+) the space of distributions on ℝ+. For f∈𝒟'(ℝ+), we write f(k) the k-derivative of f in the sense of distribution, for k≥0, with the convention f(0)=f.Definition 2.1.
A distributionf∈𝒟'(ℝ+) is called staircase if f can be written in the form
(2.1)f=∑k=0∞‍(Δk(f))(k),
for unique integrable functions Δk(f)∈L1(ℝ+) such that the support supp(Δk(f)) of Δk(f) is in [k,k+1] for all k≥0.Remark 2.2.
Given a compact setK∈ℝ+, a general distribution Λ∈𝒟′(ℝ+) can always be written as a k-derivative of a continuous function on ℝ+ restricted to the test functions with support in K, where k depends on K, see [24].Definition 2.3.
Letσ>0 be a real number, b>1 an integer and let rb(β)=∑n=0β‍1/(n+1)b for all integers β≥0. Let ℰ be an open sector centered at 0 and let ϵ∈ℰ. We denote by Lβ,σ,ϵ the vector space of all locally integrable functions f∈Lloc1(ℝ+) such that
(2.2)∥f(r)∥β,σ,ϵ∶=∫0∞‍|f(τ)|exp(-σ|ϵ|rb(β)τ)dτ
is finite. We denote by 𝒟β,σ,ϵ′ the vector space of staircase distributions f=∑k=0∞‍(Δk(f))(k) such that
(2.3)∥f∥β,σ,ϵ,d=∑k=0+∞‍(σ|ϵ|rb(β))k∥Δk(f)∥β,σ,ϵ
is finite.Remark 2.4.
Letϵ,σ,β such that |ϵ|<σrb(β). If f∈𝒟β,σ,ϵ′, then f∈𝒟β′,σ,ϵ′ for all β′≥β and we have that h↦||f||h,σ,ϵ,d is a decreasing sequence on [β,+∞). Likewise, if f∈𝒟β,σ~,ϵ′, then f∈𝒟β,σ,ϵ′ for all σ≥σ~ and we have that σ↦||f||β,σ,ϵ,d is a decreasing sequence on [σ~,+∞).Letℋ be the Heaviside one-step function defined by ℋ(r)=1, if r≥0 and ℋ(r)=0, if r<0. Let 𝒫 the operator defined on distributions T∈𝒟'(ℝ+) by 𝒫T=ℋ*T. For a subset A⊂ℝ, we denote by 1A the function which is equal to 1 on A and 0 elsewhere.The proofs of the following Lemmas2.5 and 2.6, Propositions 2.7, 2.8, 2.9, and Corollary 2.10 are given in the appendix of [25], see also [8].Lemma 2.5.
Letk≥0 and f=F(k)∈𝒟′(ℝ+), where F∈L1(ℝ+) and supp(F)⊂[k,+∞). Then f is a staircase distribution and the decomposition of f has the following terms Δ0=Δ1=⋯=Δk-1=0, Δk=F1[k,k+1] and for n≥1, Δk+n=Gn1[k+n,k+n+1] where Gn=𝒫(Gn-11[k+n,+∞)) and G0=F.Lemma 2.6.
Letf be as in Lemma 2.5 and ϵ,σ,β such that |ϵ|<σrb(β). Then, one has
(2.4)∥Δk+n∥β,σ,ϵ≤(σ|ϵ|rb(β))-n∥F∥β,σ,ϵ,
if n=0,1,2 and for n≥3,
(2.5)∥Δk+n∥β,σ,ϵ≤e(2-n)(σ/|ϵ|)rb(β)nn-1(n-1)!∥F∥β,σ,ϵ.Proposition 2.7.
Letf∈Lβ,σ/2,ϵ and ϵ,σ,β such that |ϵ|<σrb(β)/2. Then f belongs to 𝒟′β,σ,ϵ and the decomposition (2.1) of f has the following terms Δn=Gn1[n,n+1] with Gn=𝒫(Gn-11[n,+∞)) and G0=f, for n≥0. Moreover, there exists a universal constant C1>0 such that ||f||β,σ,ϵ,d≤C1||f||β,σ/2,ϵ.Proposition 2.8.
The set𝒟(ℝ+) of C∞-functions with compact support in ℝ+ is dense in 𝒟β,σ,ϵ′ for all β≥0, σ>0 and ϵ∈ℰ.Proposition 2.9.
Letϵ,σ,β such that |ϵ|<σrb(β). For all f,f~∈𝒟β,σ,ϵ′, we have f*f~∈𝒟β,σ,ϵ′. Moreover, there exists a universal constant C2>0 such that
(2.6)∥f*f~∥β,σ,ϵ,d≤C2∥f∥β,σ,ϵ,d∥f~∥β,σ,ϵ,d
for all f,f~∈𝒟β,σ,ϵ′.In this paper, for all integersk≥1, we will denote ∂r-kf(r) the convolution ℋ*k*f for all f∈𝒟β,σ,ϵ′ where ℋ*k stands for the convolution product of ℋ with itself k-1 times for k≥2 and with the convention that ℋ*1=ℋ. From Propositions 2.7 and 2.9, we deduce the following.Corollary 2.10.
Letϵ,σ,β be such that |ϵ|<σrb(β) and let k≥1 be an integer. For all f∈𝒟β,σ,ϵ′, one has ∂r-kf(r)∈𝒟β,σ,ϵ′. Moreover, there exists a universal constant C3>0 such that
(2.7)∥∂r-kf(r)∥β,σ,ϵ,d≤C3(|ϵ|σrb(β))k∥f(r)∥β,σ,ϵ,d
for all f∈𝒟β,σ,ϵ′.In the next proposition, we study norm estimates for the multiplication by bounded analytic functions.Proposition 2.11.
Letσ and β≥0 such that
(2.8)32σ|ϵ|rb(β)e1-(σ/|ϵ|)rb(β)<1,|ϵ|<σrb(β),
and let h be a C∞-function on ℝ+ such that there exist constants Ch>0,μ>0 and ρ>|ϵ|/(σrb(β)) such that
(2.9)|h(q)(r)|≤Chq!(ρ(r+μ))(q+1)
for all r∈ℝ+. Then, for all f∈𝒟β,σ,ϵ′, we have h(r)f(r)∈𝒟β,σ,ϵ′. Moreover, there exists a constant C4>0 (depending on μ,ρ) such that
(2.10)∥h(r)f(r)∥β,σ,ϵ,d≤C4Ch∥f(r)∥β,σ,ϵ,d
for all f∈𝒟β,σ,ϵ′.Proof.
The proof can be found in [14] and is inspired from [25, Lemma 2.9.1], but for the sake of completeness, we sketch it below. Without loss of generality, we can assume that f has the following form f(t)=Δk(k)(t), where Δk∈L1(ℝ+) with supp(Δk)∈[k,k+1], for k≥1. Put gk,j(t)=h(k-j)(t)Δk(t). Then, supp(gk,j(t))⊂[k,k+1].
From the Leibniz formula, we get the identity(2.11)h(t)Δk(k)(t)=∑j=0k‍k!j!(k-j)!gk,j(j)(t).
On the other hand, one can rewrite gk,j(j)(t)=(𝒫[k-j]gk,j)(k), where supp(𝒫[k-j]gk,j)∈[k,+∞) and 𝒫[q] denotes the qth iteration of 𝒫.
Due to Lemma2.5, gk,j(j) can be written gk,j(j)=∑l=k+∞‍(Δ~l,j)(l), with Δ~l,j=Gl,j1[l,l+1],Gl,j=𝒫(Gl-1,j1[l,+∞)) and Gk,j=𝒫[k-j]gk,j.
Therefore, we get the following identity(2.12)h(t)Δk(k)(t)=(h(t)Δk(t))(k)+∑j=0k-1‍k!j!(k-j)!Δ~k,j(k)+∑j=0k-1‍k!j!(k-j)!∑l=k+1+∞‍Δ~l,j(l).
First of all, we have(2.13)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d=(σrb(β)|ϵ|)k∫0+∞‍|h(t)Δk(t)|e-σrb(β)t/|ϵ|dt≤Chρμ(σrb(β)|ϵ|)k∥Δk(t)∥β,σ,ϵ,
where Ch>0 is given in (2.9). From Lemma 2.6, we have the estimates
(2.14)∥Δ~k+n,j∥β,σ,ϵ≤(|ϵ|σrb(β))n∥𝒫[k-j]gk,j∥β,σ,ϵ,∥Δ~l,j∥β,σ,ϵ≤e(2-(l-k))(σ/|ϵ|)rb(β)(l-k)l-k-1(l-k-1)!∥𝒫[k-j]gk,j∥β,σ,ϵ,
for n=0,1,2 and all l≥k+3. Now, we give estimates for ||𝒫[k-j]gk,j||β,σ,ϵ.
Using the Taylor formula with integral remainder and the hypothesis (2.9), we get
(2.15)|𝒫[k-j]gk,j(t)|≤Ch(k-j)!(k-j-1)!∫kt‍(t-s)k-j-1(ρ(s+μ))1+(k-j)|Δk(s)|ds.
Hence, from the Fubini theorem and the identity
(2.16)∫s+∞‍e-(σrb(β)/|ϵ|)t(t-s)k-j-1dt=e-(σrb(β)/|ϵ|)s(k-j)!(|ϵ|σrb(β))(k-j),
we deduce
(2.17)∫k+∞‍e-(σrb(β)/|ϵ|)t|𝒫[k-j]gk,j(t)|dt≤Ch(k-j)(k-j)!(|ϵ|σrb(β))(k-j)∫k+∞‍e-(σrb(β)/|ϵ|)s|Δk(s)|(ρ(s+μ))1+(k-j)ds
and hence
(2.18)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ≤Ch(k-j)(k-j)!(ρ(k+μ))1+k-j(|ϵ|σrb(β))(k-j)∥Δk(s)∥β,σ,ϵ.
From (2.14) and (2.18), we obtain
(2.19)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)(t)∥β,σ,ϵ,d≤ChAk(σrb(β)|ϵ|)k∥Δk(s)∥β,σ,ϵ,
for n=0,1,2, all k≥1, where
(2.20)Ak=∑j=0k-1‍k!(k-j)j!(ρ(k+μ))1+k-j(|ϵ|σrb(β))(k-j).
Now, we need to estimate Ak. Due to the Stirling formula, k!~kke-k(2πk)1/2 as k tends to infinity, there exists a universal constant C4,1>0 such that
(2.21)Ak≤C4,1kk(k+μ)k1ρ(k+μ)(2πk)1/2e-k∑j=0k-1‍(k-j)((k+μ)(σrb(β)/|ϵ|)ρ)jj!((σrb(β)/|ϵ|)ρ)k
for all k≥1. Using the hypothesis σrb(β)ρ/|ϵ|≥1, we have
(2.22)∑j=0k-1‍(k-j)(k+μ)jj!((σrb(β)/|ϵ|)ρ)j((σrb(β)/|ϵ|)ρ)k≤∑j=0k-1‍(k-j)(k+μ)jj!=k(k+μ)k-1(k-1)!-μ∑j=0k-2‍(k+μ)jj!.
Using again the Stirling formula, we get a constant C4,μ>0 (depending on μ) such that
(2.23)k(k+μ)k-1(k-1)!≤C4,μk1/2ek
for all k≥1. Moreover,
(2.24)μ∑j=0k-2‍(k+μ)jj!≤μ∑j=0+∞‍(k+μ)jj!=μek+μ.
Hence,
(2.25)∑j=0k-1‍(k-j)(k+μ)jj!((σrb(β)/|ϵ|)ρ)j((σrb(β)/|ϵ|)ρ)k≤(C4,μk1/2+μeμ)ek
for all k≥1. Finally, we obtain a constant C4,μ,ρ>0 depending only on ρ,μ such that
(2.26)Ak≤C4,μ,ρ
for all k≥1. From (2.14) and (2.18), we have
(2.27)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤ChAkA~k(σrb(β)|ϵ|)k∥Δk(s)∥β,σ,ϵ,
where
(2.28)A~k=∑l=k+3+∞‍(σrb(β)|ϵ|)l-ke(2-(l-k))(σrb(β)/|ϵ|)(l-k)(l-k-1)(l-k-1)!=∑h=3∞‍(σrb(β)|ϵ|)he(2-h)(σrb(β)/|ϵ|)h(h-1)(h-1)!.
Now, we show thatA~k, k≥1, is a bounded sequence. Again, by the Stirling formula, we get a universal constant C4,2>0 such that
(2.29)A~k≤C4,2exp(2σ|ϵ|rb(β))∑h=3+∞‍(σ|ϵ|rb(β))hexp(h(1-σ|ϵ|rb(β)))(hh-1)h-11(2π(h-1))1/2≤C4,2exp(2σ|ϵ|rb(β))∑h=3+∞‍(3(σ/|ϵ|)rb(β)2exp(1-σ|ϵ|rb(β)))h.
From the assumption (2.8) and the estimates that for all m1,m2>0 two real numbers, we have
(2.30)supx≥0xm1exp(-m2x)=(m1m2)m1e-m1,
we get a constant 0<δ<1 such that
(2.31)A~k≤C4,2e31-δ(3(σ/|ϵ|)rb(β)2)3exp(-σ|ϵ|rb(β))≤36C4,223(1-δ)
for all k≥1.
Finally, from the equality (2.12) and estimates (2.13), (2.19), (2.26), (2.27) and (2.31), we get a constant C4,μ,ρ,1>0 depending only on μ,ρ such that ||h(t)Δk(k)(t)||β,σ,ϵ,d≤ChC4,μ,ρ,1||Δk(k)(t)||β,σ,ϵ,d for all k≥1. It remains to consider the case k=0.
Whenk=0, let f(t)=Δ0(t)∈L1(ℝ+), with supp(Δ0)∈[0,1]. By definition, we can write
(2.32)∥h(t)Δ0(t)∥β,σ,ϵ,d=∥h(t)Δ0(t)∥β,σ,ϵ=∫01‍|h(t)||Δ0(t)|exp(-σrb(β)|ϵ|t)dt≤Chρμ∥Δ0(t)∥β,σ,ϵ=Chρμ∥Δ0(t)∥β,σ,ϵ,d.In the next proposition, we study norm estimates for the multiplication by polynomials.Proposition 2.12.
Letσ and β≥0 such that
(2.33)32σ|ϵ|rb(β)e1-(σ/|ϵ|)rb(β)<1,|ϵ|<σ
and let s1,k2≥1 be integers. Then, for all f∈𝒟β-k2,σ,ϵ′, one has rs1f(r)∈𝒟β,σ,ϵ′. Moreover, there exists a constant C5>0 (depending on s1,σ) such that
(2.34)∥rs1f(r)∥β,σ,ϵ,d≤C5|ϵ|s1(β+1)bs1∥f(r)∥β-k2,σ,ϵ,d
for all f∈𝒟β-k2,σ,ϵ′.Proof.
The proof is an adaptation of Proposition2.11. Without loss of generality, we can assume that f has the following form f(t)=Δk(k)(t) where Δk∈L1(ℝ+) with supp(Δk)∈[k,k+1], for k≥1. We also put h(t)=ts1. Let gk,j(t)=h(k-j)(t)Δk(t). Then, supp(gk,j(t))⊂[k,k+1]. From the Leibniz formula, we get the identity
(2.35)h(t)Δk(k)(t)=∑j=0k‍k!j!(k-j)!gk,j(j)(t).
On the other hand, one can rewrite gk,j(j)(t)=(𝒫[k-j]gk,j)(k), where supp(𝒫[k-j]gk,j)∈[k,+∞) and 𝒫[q] denotes the qth iteration of 𝒫.
Due to Lemma2.5, gk,j(j) can be written gk,j(j)=∑l=k+∞‍(Δ~l,j)(l), with Δ~l,j=Gl,j1[l,l+1], Gl,j=𝒫(Gl-1,j1[l,+∞)) and Gk,j=𝒫[k-j]gk,j. Therefore, we get the following identity as:
(2.36)h(t)Δk(k)(t)=(h(t)Δk(t))(k)+∑j=0k-1‍k!j!(k-j)!Δ~k,j(k)+∑j=0k-1‍k!j!(k-j)!∑l=k+1+∞‍Δ~l,j(l).
(1) We first give estimates for||(h(t)Δk(t))(k)||β,σ,ϵ,d. We write
(2.37)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d=(σrb(β)|ϵ|)k∫0+∞‍τs1|Δk(τ)|exp(-σ|ϵ|rb(β)τ)dτ=(σrb(β-k2)|ϵ|)k(rb(β)rb(β-k2))k×∫kk+1‍τs1exp(-σ|ϵ|(rb(β)-rb(β-k2))τ)×|Δk(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ≤A(ϵ,β)(σrb(β-k2)|ϵ|)k∫kk+1‍|Δk(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ,
where
(2.38)A(ϵ,β)=supk≥1((rb(β)rb(β-k2))k(k+1)s1exp(-σ|ϵ|(rb(β)-rb(β-k2))k)).
Now, we gives estimates for A(ϵ,β). We write
(2.39)(rb(β)rb(β-k2))k(k+1)s1exp(-σ|ϵ|(rb(β)-rb(β-k2))k)=(k+1)s1exp(-kσ|ϵ|(ψ(rb(β))-ψ(rb(β-k2))))≤2s1ks1exp(-kσ|ϵ|(ψ(rb(β))-ψ(rb(β-k2)))),
where ψ(x)=x-(|ϵ|/σ)log(x) for all k≥1. From the Taylor formula applied to ψ on [rb(β-k2),rb(β)], we get that
(2.40)ψ(rb(β))-ψ(rb(β-k2))≥(1-|ϵ|σ)(rb(β)-rb(β-k2))≥(1-|ϵ|σ)k2(β+1)b
for all β≥k2. Now, we recall that for all m1,m2>0 two real numbers, we have
(2.41)supx≥0xm1exp(-m2x)=(m1m2)m1e-m1.
From (2.39), (2.40) and (2.41), we deduce that
(2.42)A(ϵ,β)≤2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1
for all β≥k2. From (2.37) and (2.42), we deduce that
(2.43)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d≤2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d.
(2) We give estimates for||Δ~l,j||β,σ,ϵ for all 0≤j≤k-1 and all l≥k. From Lemma 2.6, we have the estimates
(2.44)∥Δ~k+n,j∥β,σ,ϵ≤(|ϵ|σrb(β))n∥𝒫[k-j]gk,j∥β,σ,ϵ,∥Δ~l,j∥β,σ,ϵ≤e(2-(l-k))(σ/|ϵ|)rb(β)(l-k)l-k-1(l-k-1)!∥𝒫[k-j]gk,j∥β,σ,ϵ,
for n=0,1,2 and all l≥k+3. Now, we give estimates for ||𝒫[k-j]gk,j||β,σ,ϵ. Using the Taylor formula with integral remainder, we have that
(2.45)|𝒫[k-j]gk,j(t)|≤1(k-j-1)!∫kt‍(t-s)k-j-1|h(k-j)(s)Δk(s)|ds,
and from the classical identity
(2.46)∫s+∞‍exp(-σ|ϵ|rb(β)t)(t-s)k-j-1dt=exp(-σ|ϵ|rb(β)s)(k-j)!((σ/|ϵ|)rb(β))k-j
we get from the Fubini theorem that
(2.47)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ=∫k+∞‍|𝒫[k-j]gk,j(t)|exp(-σ|ϵ|rb(β)t)dt≤∫k+∞‍(∫s∞‍(t-s)k-j-1(k-j-1)!exp(-σ|ϵ|rb(β)t)dt)|h(k-j)(s)Δk(s)|ds=(1(σ/|ϵ|)rb(β))k-j(k-j)∫k∞‍exp(-σ|ϵ|rb(β)s)|h(k-j)(s)Δk(s)|ds.
Again, we write
(2.48)∫k∞‍exp(-σ|ϵ|rb(β)s)|h(k-j)(s)Δk(s)|ds=∫k∞‍|h(k-j)(s)|exp(-σ|ϵ|(rb(β)-rb(β-k2))s)|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds.
From the expression of h, we have that
(2.49)|h(k-j)(s)|≤s1!ss1sk-j≤s1!ss1kk-j
for all s≥k, if 1≤k-j≤s1, and h(k-j)(s)=0, if k-j>s1. Using (2.49) in the right-hand side of the equality (2.48), we deduce from (2.47) that
(2.50)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ≤s1!(k-j)kk-j(|ϵ|σrb(β))k-j×∫kk+1‍ss1exp(-σ|ϵ|(rb(β)-rb(β-k2))s)×|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds
if 1≤k-j≤s1, and ||𝒫[k-j]gk,j(t)||β,σ,ϵ=0 if k-j>s1.
(3) We give estimates for∑j=0k-1‍k!||Δ~k+n,j(k+n)||β,σ,ϵ,d/(j!(k-j)!), for n=0,1,2. From the estimates (2.44) and (2.50), we get that
(2.51)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j)!s1!(k-j)kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-σ|ϵ|(rb(β)-rb(β-k2))s)×|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds.
From (2.37) and (2.42), we deduce from (2.51) that
(2.52)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤Ak2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d,
where
(2.53)Ak=∑j=k-s1,j≥0k-1‍k!s1!j!(k-j-1)!kk-j(|ϵ|σrb(β))k-j
for all k≥1, and n=0,1,2. Now, we show that Ak, k≥1, is a bounded sequence. We have
(2.54)Ak≤s1!k!kk(∑m=0s1-1‍kk-s1+m(k-s1+m)!(s1-m-1)!)
for all k≥s1. From the Stirling formula which asserts that k!~kke-k(2πk)1/2 as k→+∞, we get a universal constant C1>0 and a constant C2>0 (depending on s1, m) such that
(2.55)k!kk≤C1e-k(2πk)1/2,kk-s1+m(k-s1+m)!≤C1k!ek(k-s1+m)!(2πk)1/2ks1-m≤C2ek(2πk)1/2
for all k≥1. From (2.54), (2.55), we get a constant C3>0 (depending on s1) such that
(2.56)Ak≤C3
for all k≥1.
(4) We give estimates for∑j=0k-1‍(k!/j!(k-j)!)∑l=k+3+∞‍||Δ~l,j(l)||β,σ,ϵ,d. From the estimates (2.44) and (2.50), we get that
(2.57)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j-1)!s1!kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-σ|ϵ|(rb(β)-rb(β-k2))s)×|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds×∑l=k+3+∞‍(σ|ϵ|rb(β))l-kexp((2-(l-k))σ|ϵ|rb(β))(l-k)l-k-1(l-k-1)!.
Again from (2.37) and (2.42), we deduce from (2.57) that
(2.58)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤Bk2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d,
where Bk=AkA~k and
(2.59)A~k=∑l=k+3+∞‍(σ|ϵ|rb(β))l-kexp((2-(l-k))σ|ϵ|rb(β))(l-k)l-k-1(l-k-1)!=∑h=3+∞‍(σ|ϵ|rb(β))hexp((2-h)σ|ϵ|rb(β))hh-1(h-1)!
for all k≥1. Now, we remind from (2.31) that A~k is a bounded sequence.
Finally, from (2.31), (2.36), (2.43), (2.52), (2.56), and (2.58), we deduce a constant C5>0 (depending on s1, σ) such that
(2.60)∥h(t)Δk(k)(t)∥β,σ,ϵ,d≤C5|ϵ|s1(β+1)bs1∥Δk(k)∥β-k2,σ,ϵ,d,
which gives the result. It remains to consider the case k=0.
Whenk=0, let f(t)=Δ0(t)∈L1(ℝ+), with supp(Δ0)∈[0,1]. By definition, we can write
(2.61)∥h(t)Δ0(t)∥β,σ,ϵ,d=∥h(t)Δ0(t)∥β,σ,ϵ=∫01‍τs1exp(-σ|ϵ|(rb(β)-rb(β-k2))τ)|Δ0(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ.
Using (2.41), we deduce from (2.61) that
(2.62)∥h(t)Δ0(t)∥β,σ,ϵ,d≤(s1e-1σk2)s1|ϵ|s1(β+1)bs1∫01‍|Δ0(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ=(s1e-1σk2)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d.
Hence there exists a constant C5,1>0 (depending on s1, σ) such that
(2.63)∥h(t)f(t)∥β,σ,ϵ,d≤C5,1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d,
which yields the result.Proposition 2.13.
Letσ>σ~>0 be real numbers such that
(2.64)32σ|ϵ|rb(β)e1-(σ/|ϵ|)rb(β)<1,|ϵ|<σ~.
Let s1≥0 be a nonnegative integer. Then, for all f∈𝒟β,σ~,ϵ′, one has rs1f(r)∈𝒟β,σ,ϵ′. Moreover, there exists a constant C6>0 (depending on s1, σ, σ~) such that
(2.65)∥rs1f(r)∥β,σ,ϵ,d≤C6|ϵ|s1∥f(r)∥β,σ~,ϵ,d
for all f∈𝒟β,σ,ϵ′.Proof.
The line of reasoning will follow the proof of Proposition2.12. We start from the identity (2.36).
(
1
) We first give estimates for ||(h(t)Δk(t))(k)||β,σ,ϵ,d. We write
(2.66)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d=(σrb(β)|ϵ|)k∫0+∞‍τs1|Δk(τ)|exp(-σ|ϵ|rb(β)τ)dτ=(σ~rb(β)|ϵ|)k(σσ~)k∫kk+1‍τs1exp(-(σ-σ~)|ϵ|rb(β)τ)×|Δk(τ)|exp(-σ~|ϵ|rb(β)τ)dτ≤A~(ϵ,β)(σ~rb(β)|ϵ|)k∫kk+1‍|Δk(τ)|exp(-σ~|ϵ|rb(β)τ)dτ,
where
(2.67)A~(ϵ,β)=supk≥1((σσ~)k(k+1)s1exp(-(σ-σ~)|ϵ|rb(β)k)).
Now, we give estimates for A~(ϵ,β). We write
(2.68)(σσ~)k(k+1)s1exp(-(σ-σ~)|ϵ|rb(β)k)=(k+1)s1exp(-krb(β)|ϵ|(φ(σ)-φ(σ~)))≤2s1ks1exp(-krb(β)|ϵ|(φ(σ)-φ(σ~))),
where φ(x)=x-(|ϵ|/rb(β))log(x), for all k≥1. From the Taylor formula applied to φ on [σ~,σ], we get that
(2.69)φ(σ)-φ(σ~)≥(1-|ϵ|σ~)(σ-σ~).
From (2.68), (2.69), and (2.41), we deduce that
(2.70)A~(ϵ,β)≤2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1.
From (2.66) and (2.70), we get that
(2.71)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d≤2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d.
(
2
) We give estimates for ||Δ~l,j||β,σ,ϵ, for all 0≤j≤k-1, all l≥k. We start from the formula (2.44) and (2.47). We write
(2.72)∫k∞‍exp(-σ|ϵ|rb(β)s)|h(k-j)(s)Δk(s)|ds=∫k∞‍|h(k-j)(s)|exp(-(σ-σ~)|ϵ|rb(β)s)|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds.
We get that
(2.73)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ≤s1!(k-j)kk-j(|ϵ|σrb(β))k-j∫kk+1‍ss1exp(-(σ-σ~)|ϵ|rb(β)s)|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds
if 1≤k-j≤s1, and ||𝒫[k-j]gk,j(t)||β,σ,ϵ=0 if k-j>s1.
(
3
) We give estimates for ∑j=0k-1‍k!||Δ~k+n,j(k+n)||β,σ,ϵ,d/(j!(k-j)!), for n=0,1,2. From the estimates (2.44) and (2.73), we get that
(2.74)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j)!s1!(k-j)kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-(σ-σ~)|ϵ|rb(β)s)×|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds.
From (2.66) and (2.70), we deduce from (2.74) that
(2.75)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤Ak2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d,
where Ak is the bounded sequence given in the proof of Proposition 2.12.
We give estimates for∑j=0k-1‍(k!/j!(k-j)!)∑l=k+3+∞‍||Δ~l,j(l)||β,σ,ϵ,d. From the estimates (2.44) and (2.73), we get that
(2.76)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j-1)!s1!kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-(σ-σ~)|ϵ|rb(β)s)×|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds×∑l=k+3+∞‍(σ|ϵ|rb(β))l-kexp((2-(l-k))σ|ϵ|rb(β))(l-k)l-k-1(l-k-1)!.
From (2.66) and (2.70), we deduce from (2.76), that
(2.77)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤Bk2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d,
where Bk is the bounded sequence given in the proof of Proposition 2.12.
Finally, from (2.31), (2.36), (2.56), (2.71), (2.75), and (2.77), we deduce a constant C6>0 (depending on s1, σ, σ~) such that
(2.78)∥h(t)Δk(k)(t)∥β,σ,d≤C6|ϵ|s1∥Δk(k)∥β,σ~,ϵ,d,
which gives the result. It remains to consider the case k=0.
Whenk=0, let f(t)=Δ0(t)∈L1(ℝ+), with supp(Δ0)∈[0,1]. By definition, we can write
(2.79)∥h(t)Δ0(t)∥β,σ,ϵ,d=∥h(t)Δ0(t)∥β,σ,ϵ=∫01‍τs1exp(-(σ-σ~)|ϵ|rb(β)τ)|Δ0(τ)|exp(-σ~|ϵ|rb(β)τ)dτ.
Using (2.41), we deduce from (2.79) that
(2.80)∥h(t)Δ0(t)∥β,σ,ϵ,d≤(s1e-1σ-σ~)s1|ϵ|s1∫01‍|Δ0(τ)|exp(-σ~|ϵ|rb(β)τ)dτ=(s1e-1σ-σ~)s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d.
Hence, there exists a constant C6,1>0 (depending on s1, σ, σ~) such that
(2.81)∥h(t)f(t)∥β,σ,ϵ,d≤C6,1|ϵ|s1∥f(t)∥β,σ~,ϵ,d,
which yields the result.


### 2.2. Banach Spaces of Formal Power Series with Coefficients in Spaces of Distributions

Definition 2.14.
Letδ>0 be a real number. We denote by 𝒟′(σ,ϵ,δ) the vector space of formal series v(r,z)=∑β≥0‍vβ(r)zβ/β! such that vβ(r)∈𝒟β,σ,ϵ′ for all β≥0 and
(2.82)∥v(r,z)∥(σ,ϵ,d,δ)∶=∑β≥0‍∥vβ(r)∥β,σ,ϵ,dδββ!
is finite. One can check that the normed space (𝒟'(σ,ϵ,δ),||·||(σ,ϵ,d,δ)) is a Banach space.In the next proposition, we study some parameter depending linear operators acting on the space𝒟′(σ,ϵ,δ).Proposition 2.15.
Lets1,s2,k1,k2≥0 be positive integers. Assume that the condition
(2.83)k2≥bs1
holds. Then, if
(2.84)|ϵ|<σ,3(σ/|ϵ|)ζ(b)2e1-σ/|ϵ|<1,
the operator τs1∂τ-k1∂z-k2 is a bounded linear operator from the space (𝒟′(σ,ϵ,δ),||·||(σ,ϵ,d,δ)) into itself. Moreover, there exists a constant C7>0 (depending on b, s1, k2, σ) such that
(2.85)∥rs1∂r-k1∂z-k2v(r,z)∥(σ,ϵ,d,δ)≤|ϵ|s1+k1C7δk2∥v(r,z)∥(σ,ϵ,d,δ)
for all v∈𝒟'(σ,ϵ,δ).Proof.
Letv(r,z)∈𝒟'(σ,ϵ,δ). By definition, we have
(2.86)∥rs1∂r-k1∂z-k2v(r,z)∥(σ,ϵ,d,δ)=∑β≥k2‍∥rs1∂r-k1vβ-k2(r)∥β,σ,ϵ,dδββ!.
From Corollary 2.10 and Proposition 2.12, we get a constant C3,5>0 (depending on s1,σ) such that
(2.87)∥rs1∂r-k1∂z-k2v(r,z)∥(σ,ϵ,d,δ)≤C3,5∑β≥k2‍|ϵ|s1+k1(β+1)bs1(β-k2)!β!×∥vβ-k2(r)∥β-k2,σ,ϵ,dδk2δβ-k2(β-k2)!.
From the assumptions (2.83), we get a constant Cb,s1,k2>0 (depending on b, s1, k2) such that
(2.88)(β+1)bs1(β-k2)!β!≤Cb,s1,k2
for all β≥k2. Finally, from the estimates (2.87) and (2.88), we get the inequality (2.85).In the next proposition, we study linear operators of multiplication by bounded holomorphic andC∞ functions.Proposition 2.16.
For allβ≥0, let hβ(τ) be a C∞ function with respect to r on ℝ+ such that there exist A,B,ρ,μ>0 with
(2.89)|hβ(q)(r)|≤AB-ββ!q!(ρ(r+μ))q+1
for all r∈ℝ+. One consider the series
(2.90)h(r,z)=∑β≥0‍hβ(r)zββ!,
which is convergent for all |z|<B, all r∈ℝ+. Let 0<δ<B. Then, if
(2.91)|ϵ|<σ,|ϵ|<ρσ,3(σ/|ϵ|)ζ(b)2e1-σ/|ϵ|<1,
the linear operator of multiplication by h(r,z) is continuous from (𝒟'(σ,ϵ,δ),||·||(σ,ϵ,δ)) into itself. Moreover, there exists a constant C8 (depending on μ, ρ, B) such that
(2.92)∥h(r,z)v(r,z)∥(σ,ϵ,d,δ)≤C8A∥v(r,z)∥(σ,ϵ,d,δ)
for all v(r,z)∈𝒟'(σ,ϵ,δ) satisfying (2.91).Proof.
Letv(r,z)=∑β≥0‍vβ(r)zβ/β!∈𝒟'(σ,ϵ,δ). By definition, we have that
(2.93)∥h(τ,z)v(r,z)∥(σ,ϵ,d,δ)≤∑β≥0‍(∑β1+β2=β‍∥hβ1(r)vβ2(r)∥β,σ,ϵ,dβ!β1!β2!)δββ!.
From Proposition 2.11 and Remark 2.4, we deduce that there exists C4>0 (depending on μ,ρ) such that
(2.94)∥hβ1(r)vβ2(r)∥β,σ,ϵ,d≤C4AB-β1β1!∥vβ2(r)∥β,σ,ϵ,d≤C4AB-β1β1!∥vβ2(r)∥β2,σ,ϵ,d
for all β1,β2≥0 such that β1+β2=β. From (2.93) and (2.94), we deduce that
(2.95)∥h(r,z)v(r,z)∥(σ,ϵ,d,δ)≤C4A(∑β≥0‍(δB)β)∥v(r,z)∥(σ,ϵ,d,δ)
which yields (2.92).


### 2.3. Cauchy Problems in Analytic Functions Spaces with Dependence on Initial Data

In this section, we recall the well-know-Cauchy Kowalevski theorem in some spaces of analytic functions for which the dependence on the coefficients and initial data can be obtained.The following Banach spaces were used in [26].Definition 2.17.
LetT,X be real numbers such that T,X>0. We define a vector space G(T,X) of holomorphic functions on a neighborhood of the origin in ℂ2. A formal series U(t,x)∈ℂ[[t,x]],
(2.96)U(t,x)=∑l,β≥0‍ul,βtll!xββ!
belongs to G(T,X) if the series
(2.97)∑l,β≥0‍|ul,β|(l+β)!TlXβ
converges. We also define a norm on G(T,X) as
(2.98)∥U(t,x)∥(T,X)=∑l,β≥0‍|ul,β|(l+β)!TlXβ.
One can easily show that (G(T,X),||·||(T,X)) is a Banach space.Remark 2.18.
LetU(t,x) be in G(T0,X0) for given T0,X0>0. Then, U(t,x) also belongs to the spaces G(T,X) for all T≤T0 and X≤X0. Moreover, the maps T→||U(t,x)||(T,X) and X→||U(t,x)||(T,X) are increasing functions from [0,T0] (resp., [0,X0]) into ℝ+.We depart from some preliminary lemma from [26]. In the following, for u(t,x)∈ℂ[[t,x]], we denote by ∂x-1u(t,x) the formal series ∫0x‍u(t,τ)dτ.Lemma 2.19.
Leth0,h1∈ℕ such that h0≤h1. The operator ∂th0∂x-h1 is a bounded linear operator from (G(T,X),||·||(T,X)) into itself. Moreover, there exists a universal constant C10>0 such that the estimates
(2.99)∥∂th0∂x-h1U(t,x)∥(T,X)≤C10T-h0Xh1∥U(t,x)∥(T,X)
hold for all U(t,x)∈G(T,X).Lemma 2.20.
LetA(t,x)=∑l,β≥0‍al,βtlxβ/l!β! be an analytic function on an open polydisc containing D(0,T)×D(0,X) and let U(t,x) be in G(T,X). Then, the product A(t,x)U(t,x) belongs to G(T,X). Moreover,
(2.100)∥A(t,x)U(t,x)∥(T,X)≤|A|(T,X)∥U(t,x)∥(T,X)
where |A|(T,X)=∑l,β≥0‍|al,β|TlXβ/l!β!.Proof.
Let(2.101)U(t,x)=∑l,β≥0‍ul,βtll!xββ!.
We have
(2.102)A(t,x)U(t,x)=∑l,β≥0‍vl,βtll!xββ!,
where
(2.103)vl,β=∑l1+l2=l‍∑β1+β2=β‍(al1,β1l1!β1!ul2,β2l2!β2!β!l!)
for all l,β≥0. By definition, we have
(2.104)|A|(T,X)∥U2(t,x)∥(T,X)=∑l,β≥0‍(∑l1+l2=l‍∑β1+β2=β‍|al1,β1||ul2,β2|l1!β1!(l2+β2)!)TlXβ,∥A(t,x)U(t,x)∥(T,X)=∑l,β≥0‍|∑l1+l2=l‍∑β1+β2=β‍al1,β1ul2,β2l!β!l1!β1!l2!β2!|TlXβ(l+β)!.
On the other side, the next inequalities are well known:
(2.105)l!β!l1!β1!l2!β2!≤(l+β)!(l1+β1)!(l2+β2)!≤(l+β)!l1!β1!(l2+β2)!
for all l1,l2≥0 such that l1+l2=l and β1,β2≥0 such that β1+β2=β.
Finally, from (2.105), we deduce that ||A(t,x)U(t,x)||(T,X) converges and that the estimates (2.100) hold.Lemma 2.21.
Leth1,h2∈ℕ and let U(t,x) be in G(T0,X0) for given T0,X0>0. Then, there exist T,X>0 small enough (depending on T0,X0) such that the formal series ∂th1∂xh2U(t,x) belongs to G(T,X). Moreover, there exists a constant C11>0 (depending on h1,h2) such that
(2.106)∥(∂th1∂xh2U)(t,x)∥(T,X)≤C11T-h1X-h2∥U(t,x)∥(T0,X0)
for all U(t,x)∈G(T0,X0).Let𝒞1 be a finite subset of ℕ2. For all (l0,l1)∈𝒞1, let cl0,l1(t,x)=∑l,β≥0‍cl0,l1,l,βtlxβ/l!β! be analytic functions on some polydisc containing the closed polydisc D-(0,T0)×D-(0,X0) for some T0,X0>0. As in Lemma 2.20, we define
(2.107)|cl0,l1|(t,x)=∑l,β≥0‍|cl0,l1,l,β|tlxβl!β!,
which converges on D-(0,T0)×D-(0,X0). We also consider d(t,x)∈G(Td,Xd), for some Td,Xd>0. The following proposition holds.Proposition 2.22.
LetS≥1 be an integer. One make the following assumptions. For all (l0,l1)∈𝒞1, one has
(2.108)S>l1,S≥l0+l1.
One consider the following Cauchy problem:
(2.109)∂xSU(t,x)=∑(l0,l1)∈𝒞1‍cl0,l1(t,x)∂tl0∂xl1U(t,x)+d(t,x)
for the given initial conditions
(2.110)(∂xjU)(t,0)=Uj(t),0≤j≤S-1,
which are analytic functions on some disc containing the closed discD-(0,T0). If Uj(t)=∑l≥0‍Uj,ltl/l!, we define |Uj|(t)=∑l≥0‍|Uj,l|tl/l!, which converges for all t∈D-(0,T0).
Then, there existT1>0 with 0<T1<min(T0,Td) (depending on T0,Td,𝒞1) and X1>0 with 0<X1<min(X0,Xd) (depending on S, T0, 𝒞1, max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)) such that the problem (2.109), (2.110) has a unique formal solution U(t,x)∈G(T1,X1). Moreover, there exist constants C12,1,C12,2,C12,3>0 (depending on S,T0,X0,𝒞1) such that
(2.111)∥U(t,x)∥(T1,X1)≤max0≤j≤S-1|Uj|(T0)(C12,1max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)+C12,2)+C12,3∥d(t,x)∥(Td,Xd).Proof.
We denote by𝒫 the linear operator from ℂ[[t,x]] into itself defined by
(2.112)𝒫(H(t,x))∶=∂xSH(t,x)-∑(l0,l1)∈𝒞1‍cl0,l1(t,x)∂tl0∂xl1H(t,x),
and 𝒜 denotes the linear map from ℂ[[t,x]] into itself:
(2.113)𝒜(H(t,x))∶=∑(l0,l1)∈𝒞1‍cl0,l1(t,x)∂tl0∂xl1-SH(t,x)
for all H(t,x)∈ℂ[[t,x]]. By construction, we have that 𝒫∘∂x-S=id-𝒜, where id represents the identity map H↦H from ℂ[[t,x]] into itself.
Now, we show that for any givenT1>0 such that 0<T1≤T0, there exists X𝒜,T1>0 with 0<X𝒜,T1≤X0 (depending on S, T1, 𝒞1, max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)) such that id-𝒜 is an invertible map from G(T1,X) into itself for all 0<X≤X𝒜,T1. Moreover, the following inequality
(2.114)∥(id-𝒜)-1C(t,x)∥(T1,X)≤2∥C(t,x)∥(T1,X)
holds for all C(t,x)∈G(T1,X), for any 0<X≤X𝒜,T1. Indeed, from the assumption (2.108) and Lemmas 2.19 and 2.20, we get a universal constant C10,1>0 such that
(2.115)∥𝒜(C(t,x))∥(T1,X)≤C10,1(∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X)T1-l0XS-l1)∥C(t,x)∥(T1,X)≤C10,1max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)(∑(l0,l1)∈𝒞1‍T1-l0X𝒜,T1S-l1)∥C(t,x)∥(T1,X)∶=NT1,X𝒜,T1∥C(t,x)∥(T1,X)
for all C(t,x)∈G(T1,X). Since S>l1, for all (l0,l1)∈𝒞1, for the given T1>0 one can choose X𝒜,T1 small enough such that NT1,X𝒜,T1≤1/2. Therefore, the inequality (2.114) holds.
Letw(t,x)=∑j=0S-1‍Uj(t)xj/j!. From the hypothesis (2.110), we deduce that 𝒫(w(t,x)) and w(t,x) belong to G(T1,X0), for some 0<T1<T0 (depending on 𝒞1, T0). Indeed, from Lemmas 2.20 and 2.21 we get constants C11,1>0, 0<T1<T0 (depending on 𝒞1, T0) such that
(2.116)∥𝒫(w(t,x))∥(T1,X0)≤∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X0)(∑j=0S-1-l1‍∥∂tl0Uj+l1(t)∥(T1,X0)X0jj!)≤C11,1∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X0)T1-l0(∑j=0S-1-l1‍∥Uj+l1(t)∥(T0,X0)X0jj!)≤C11,1∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X0)T1-l0(∑j=0S-1-l1‍|Uj+l1|(T0)X0jj!)≤C11,1max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)max0≤j≤S-1|Uj|(T0)×∑(l0,l1)∈𝒞1‍T1-l0(∑j=0S-1-l1‍X0jj!),(2.117)∥(w(t,x))∥(T1,X0)≤∑j=0S-1‍∥Uj(t)∥(T1,X0)X0jj!≤∑j=0S-1‍|Uj|(T1)X0jj!≤max0≤j≤S-1|Uj|(T0)∑j=0S-1‍X0jj!.
Now, for this constructed T1>0 satisfying (2.116) that we choose in such a way that T1<Td also holds, we select X1>0 such that 0<X1<min(X𝒜,T1,Xd). From the estimates (2.116) and Remark 2.4, we deduce that 𝒫(w(t,x)), w(t,x), and d(t,x) belong to G(T1,X1). From (2.114), we deduce the existence of a unique H(t,x)∈G(T1,X1) such that
(2.118)(𝒫∘∂x-S)H(t,x)=-𝒫(w(t,x))+d(t,x).
Now, we put U(t,x)=∂x-SH(t,x)+w(t,x). By Lemma 2.19, we deduce that U(t,x)∈G(T1,X1) and solves the problem (2.109), (2.110). Moreover, from (2.114) and (2.116), we get constants C12,1,C12,2,C12,3>0 (depending on S, T0, X0, 𝒞1) such that (2.111) holds, which yields the result.


## 2.1. Weighted Banach Spaces of Distributions

We define𝒟(ℝ+) to be the space of complex valued C∞-functions with compact support in ℝ+, where ℝ+ is the set of the positive real numbers x>0. We also denote by 𝒟'(ℝ+) the space of distributions on ℝ+. For f∈𝒟'(ℝ+), we write f(k) the k-derivative of f in the sense of distribution, for k≥0, with the convention f(0)=f.Definition 2.1.
A distributionf∈𝒟'(ℝ+) is called staircase if f can be written in the form
(2.1)f=∑k=0∞‍(Δk(f))(k),
for unique integrable functions Δk(f)∈L1(ℝ+) such that the support supp(Δk(f)) of Δk(f) is in [k,k+1] for all k≥0.Remark 2.2.
Given a compact setK∈ℝ+, a general distribution Λ∈𝒟′(ℝ+) can always be written as a k-derivative of a continuous function on ℝ+ restricted to the test functions with support in K, where k depends on K, see [24].Definition 2.3.
Letσ>0 be a real number, b>1 an integer and let rb(β)=∑n=0β‍1/(n+1)b for all integers β≥0. Let ℰ be an open sector centered at 0 and let ϵ∈ℰ. We denote by Lβ,σ,ϵ the vector space of all locally integrable functions f∈Lloc1(ℝ+) such that
(2.2)∥f(r)∥β,σ,ϵ∶=∫0∞‍|f(τ)|exp(-σ|ϵ|rb(β)τ)dτ
is finite. We denote by 𝒟β,σ,ϵ′ the vector space of staircase distributions f=∑k=0∞‍(Δk(f))(k) such that
(2.3)∥f∥β,σ,ϵ,d=∑k=0+∞‍(σ|ϵ|rb(β))k∥Δk(f)∥β,σ,ϵ
is finite.Remark 2.4.
Letϵ,σ,β such that |ϵ|<σrb(β). If f∈𝒟β,σ,ϵ′, then f∈𝒟β′,σ,ϵ′ for all β′≥β and we have that h↦||f||h,σ,ϵ,d is a decreasing sequence on [β,+∞). Likewise, if f∈𝒟β,σ~,ϵ′, then f∈𝒟β,σ,ϵ′ for all σ≥σ~ and we have that σ↦||f||β,σ,ϵ,d is a decreasing sequence on [σ~,+∞).Letℋ be the Heaviside one-step function defined by ℋ(r)=1, if r≥0 and ℋ(r)=0, if r<0. Let 𝒫 the operator defined on distributions T∈𝒟'(ℝ+) by 𝒫T=ℋ*T. For a subset A⊂ℝ, we denote by 1A the function which is equal to 1 on A and 0 elsewhere.The proofs of the following Lemmas2.5 and 2.6, Propositions 2.7, 2.8, 2.9, and Corollary 2.10 are given in the appendix of [25], see also [8].Lemma 2.5.
Letk≥0 and f=F(k)∈𝒟′(ℝ+), where F∈L1(ℝ+) and supp(F)⊂[k,+∞). Then f is a staircase distribution and the decomposition of f has the following terms Δ0=Δ1=⋯=Δk-1=0, Δk=F1[k,k+1] and for n≥1, Δk+n=Gn1[k+n,k+n+1] where Gn=𝒫(Gn-11[k+n,+∞)) and G0=F.Lemma 2.6.
Letf be as in Lemma 2.5 and ϵ,σ,β such that |ϵ|<σrb(β). Then, one has
(2.4)∥Δk+n∥β,σ,ϵ≤(σ|ϵ|rb(β))-n∥F∥β,σ,ϵ,
if n=0,1,2 and for n≥3,
(2.5)∥Δk+n∥β,σ,ϵ≤e(2-n)(σ/|ϵ|)rb(β)nn-1(n-1)!∥F∥β,σ,ϵ.Proposition 2.7.
Letf∈Lβ,σ/2,ϵ and ϵ,σ,β such that |ϵ|<σrb(β)/2. Then f belongs to 𝒟′β,σ,ϵ and the decomposition (2.1) of f has the following terms Δn=Gn1[n,n+1] with Gn=𝒫(Gn-11[n,+∞)) and G0=f, for n≥0. Moreover, there exists a universal constant C1>0 such that ||f||β,σ,ϵ,d≤C1||f||β,σ/2,ϵ.Proposition 2.8.
The set𝒟(ℝ+) of C∞-functions with compact support in ℝ+ is dense in 𝒟β,σ,ϵ′ for all β≥0, σ>0 and ϵ∈ℰ.Proposition 2.9.
Letϵ,σ,β such that |ϵ|<σrb(β). For all f,f~∈𝒟β,σ,ϵ′, we have f*f~∈𝒟β,σ,ϵ′. Moreover, there exists a universal constant C2>0 such that
(2.6)∥f*f~∥β,σ,ϵ,d≤C2∥f∥β,σ,ϵ,d∥f~∥β,σ,ϵ,d
for all f,f~∈𝒟β,σ,ϵ′.In this paper, for all integersk≥1, we will denote ∂r-kf(r) the convolution ℋ*k*f for all f∈𝒟β,σ,ϵ′ where ℋ*k stands for the convolution product of ℋ with itself k-1 times for k≥2 and with the convention that ℋ*1=ℋ. From Propositions 2.7 and 2.9, we deduce the following.Corollary 2.10.
Letϵ,σ,β be such that |ϵ|<σrb(β) and let k≥1 be an integer. For all f∈𝒟β,σ,ϵ′, one has ∂r-kf(r)∈𝒟β,σ,ϵ′. Moreover, there exists a universal constant C3>0 such that
(2.7)∥∂r-kf(r)∥β,σ,ϵ,d≤C3(|ϵ|σrb(β))k∥f(r)∥β,σ,ϵ,d
for all f∈𝒟β,σ,ϵ′.In the next proposition, we study norm estimates for the multiplication by bounded analytic functions.Proposition 2.11.
Letσ and β≥0 such that
(2.8)32σ|ϵ|rb(β)e1-(σ/|ϵ|)rb(β)<1,|ϵ|<σrb(β),
and let h be a C∞-function on ℝ+ such that there exist constants Ch>0,μ>0 and ρ>|ϵ|/(σrb(β)) such that
(2.9)|h(q)(r)|≤Chq!(ρ(r+μ))(q+1)
for all r∈ℝ+. Then, for all f∈𝒟β,σ,ϵ′, we have h(r)f(r)∈𝒟β,σ,ϵ′. Moreover, there exists a constant C4>0 (depending on μ,ρ) such that
(2.10)∥h(r)f(r)∥β,σ,ϵ,d≤C4Ch∥f(r)∥β,σ,ϵ,d
for all f∈𝒟β,σ,ϵ′.Proof.
The proof can be found in [14] and is inspired from [25, Lemma 2.9.1], but for the sake of completeness, we sketch it below. Without loss of generality, we can assume that f has the following form f(t)=Δk(k)(t), where Δk∈L1(ℝ+) with supp(Δk)∈[k,k+1], for k≥1. Put gk,j(t)=h(k-j)(t)Δk(t). Then, supp(gk,j(t))⊂[k,k+1].
From the Leibniz formula, we get the identity(2.11)h(t)Δk(k)(t)=∑j=0k‍k!j!(k-j)!gk,j(j)(t).
On the other hand, one can rewrite gk,j(j)(t)=(𝒫[k-j]gk,j)(k), where supp(𝒫[k-j]gk,j)∈[k,+∞) and 𝒫[q] denotes the qth iteration of 𝒫.
Due to Lemma2.5, gk,j(j) can be written gk,j(j)=∑l=k+∞‍(Δ~l,j)(l), with Δ~l,j=Gl,j1[l,l+1],Gl,j=𝒫(Gl-1,j1[l,+∞)) and Gk,j=𝒫[k-j]gk,j.
Therefore, we get the following identity(2.12)h(t)Δk(k)(t)=(h(t)Δk(t))(k)+∑j=0k-1‍k!j!(k-j)!Δ~k,j(k)+∑j=0k-1‍k!j!(k-j)!∑l=k+1+∞‍Δ~l,j(l).
First of all, we have(2.13)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d=(σrb(β)|ϵ|)k∫0+∞‍|h(t)Δk(t)|e-σrb(β)t/|ϵ|dt≤Chρμ(σrb(β)|ϵ|)k∥Δk(t)∥β,σ,ϵ,
where Ch>0 is given in (2.9). From Lemma 2.6, we have the estimates
(2.14)∥Δ~k+n,j∥β,σ,ϵ≤(|ϵ|σrb(β))n∥𝒫[k-j]gk,j∥β,σ,ϵ,∥Δ~l,j∥β,σ,ϵ≤e(2-(l-k))(σ/|ϵ|)rb(β)(l-k)l-k-1(l-k-1)!∥𝒫[k-j]gk,j∥β,σ,ϵ,
for n=0,1,2 and all l≥k+3. Now, we give estimates for ||𝒫[k-j]gk,j||β,σ,ϵ.
Using the Taylor formula with integral remainder and the hypothesis (2.9), we get
(2.15)|𝒫[k-j]gk,j(t)|≤Ch(k-j)!(k-j-1)!∫kt‍(t-s)k-j-1(ρ(s+μ))1+(k-j)|Δk(s)|ds.
Hence, from the Fubini theorem and the identity
(2.16)∫s+∞‍e-(σrb(β)/|ϵ|)t(t-s)k-j-1dt=e-(σrb(β)/|ϵ|)s(k-j)!(|ϵ|σrb(β))(k-j),
we deduce
(2.17)∫k+∞‍e-(σrb(β)/|ϵ|)t|𝒫[k-j]gk,j(t)|dt≤Ch(k-j)(k-j)!(|ϵ|σrb(β))(k-j)∫k+∞‍e-(σrb(β)/|ϵ|)s|Δk(s)|(ρ(s+μ))1+(k-j)ds
and hence
(2.18)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ≤Ch(k-j)(k-j)!(ρ(k+μ))1+k-j(|ϵ|σrb(β))(k-j)∥Δk(s)∥β,σ,ϵ.
From (2.14) and (2.18), we obtain
(2.19)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)(t)∥β,σ,ϵ,d≤ChAk(σrb(β)|ϵ|)k∥Δk(s)∥β,σ,ϵ,
for n=0,1,2, all k≥1, where
(2.20)Ak=∑j=0k-1‍k!(k-j)j!(ρ(k+μ))1+k-j(|ϵ|σrb(β))(k-j).
Now, we need to estimate Ak. Due to the Stirling formula, k!~kke-k(2πk)1/2 as k tends to infinity, there exists a universal constant C4,1>0 such that
(2.21)Ak≤C4,1kk(k+μ)k1ρ(k+μ)(2πk)1/2e-k∑j=0k-1‍(k-j)((k+μ)(σrb(β)/|ϵ|)ρ)jj!((σrb(β)/|ϵ|)ρ)k
for all k≥1. Using the hypothesis σrb(β)ρ/|ϵ|≥1, we have
(2.22)∑j=0k-1‍(k-j)(k+μ)jj!((σrb(β)/|ϵ|)ρ)j((σrb(β)/|ϵ|)ρ)k≤∑j=0k-1‍(k-j)(k+μ)jj!=k(k+μ)k-1(k-1)!-μ∑j=0k-2‍(k+μ)jj!.
Using again the Stirling formula, we get a constant C4,μ>0 (depending on μ) such that
(2.23)k(k+μ)k-1(k-1)!≤C4,μk1/2ek
for all k≥1. Moreover,
(2.24)μ∑j=0k-2‍(k+μ)jj!≤μ∑j=0+∞‍(k+μ)jj!=μek+μ.
Hence,
(2.25)∑j=0k-1‍(k-j)(k+μ)jj!((σrb(β)/|ϵ|)ρ)j((σrb(β)/|ϵ|)ρ)k≤(C4,μk1/2+μeμ)ek
for all k≥1. Finally, we obtain a constant C4,μ,ρ>0 depending only on ρ,μ such that
(2.26)Ak≤C4,μ,ρ
for all k≥1. From (2.14) and (2.18), we have
(2.27)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤ChAkA~k(σrb(β)|ϵ|)k∥Δk(s)∥β,σ,ϵ,
where
(2.28)A~k=∑l=k+3+∞‍(σrb(β)|ϵ|)l-ke(2-(l-k))(σrb(β)/|ϵ|)(l-k)(l-k-1)(l-k-1)!=∑h=3∞‍(σrb(β)|ϵ|)he(2-h)(σrb(β)/|ϵ|)h(h-1)(h-1)!.
Now, we show thatA~k, k≥1, is a bounded sequence. Again, by the Stirling formula, we get a universal constant C4,2>0 such that
(2.29)A~k≤C4,2exp(2σ|ϵ|rb(β))∑h=3+∞‍(σ|ϵ|rb(β))hexp(h(1-σ|ϵ|rb(β)))(hh-1)h-11(2π(h-1))1/2≤C4,2exp(2σ|ϵ|rb(β))∑h=3+∞‍(3(σ/|ϵ|)rb(β)2exp(1-σ|ϵ|rb(β)))h.
From the assumption (2.8) and the estimates that for all m1,m2>0 two real numbers, we have
(2.30)supx≥0xm1exp(-m2x)=(m1m2)m1e-m1,
we get a constant 0<δ<1 such that
(2.31)A~k≤C4,2e31-δ(3(σ/|ϵ|)rb(β)2)3exp(-σ|ϵ|rb(β))≤36C4,223(1-δ)
for all k≥1.
Finally, from the equality (2.12) and estimates (2.13), (2.19), (2.26), (2.27) and (2.31), we get a constant C4,μ,ρ,1>0 depending only on μ,ρ such that ||h(t)Δk(k)(t)||β,σ,ϵ,d≤ChC4,μ,ρ,1||Δk(k)(t)||β,σ,ϵ,d for all k≥1. It remains to consider the case k=0.
Whenk=0, let f(t)=Δ0(t)∈L1(ℝ+), with supp(Δ0)∈[0,1]. By definition, we can write
(2.32)∥h(t)Δ0(t)∥β,σ,ϵ,d=∥h(t)Δ0(t)∥β,σ,ϵ=∫01‍|h(t)||Δ0(t)|exp(-σrb(β)|ϵ|t)dt≤Chρμ∥Δ0(t)∥β,σ,ϵ=Chρμ∥Δ0(t)∥β,σ,ϵ,d.In the next proposition, we study norm estimates for the multiplication by polynomials.Proposition 2.12.
Letσ and β≥0 such that
(2.33)32σ|ϵ|rb(β)e1-(σ/|ϵ|)rb(β)<1,|ϵ|<σ
and let s1,k2≥1 be integers. Then, for all f∈𝒟β-k2,σ,ϵ′, one has rs1f(r)∈𝒟β,σ,ϵ′. Moreover, there exists a constant C5>0 (depending on s1,σ) such that
(2.34)∥rs1f(r)∥β,σ,ϵ,d≤C5|ϵ|s1(β+1)bs1∥f(r)∥β-k2,σ,ϵ,d
for all f∈𝒟β-k2,σ,ϵ′.Proof.
The proof is an adaptation of Proposition2.11. Without loss of generality, we can assume that f has the following form f(t)=Δk(k)(t) where Δk∈L1(ℝ+) with supp(Δk)∈[k,k+1], for k≥1. We also put h(t)=ts1. Let gk,j(t)=h(k-j)(t)Δk(t). Then, supp(gk,j(t))⊂[k,k+1]. From the Leibniz formula, we get the identity
(2.35)h(t)Δk(k)(t)=∑j=0k‍k!j!(k-j)!gk,j(j)(t).
On the other hand, one can rewrite gk,j(j)(t)=(𝒫[k-j]gk,j)(k), where supp(𝒫[k-j]gk,j)∈[k,+∞) and 𝒫[q] denotes the qth iteration of 𝒫.
Due to Lemma2.5, gk,j(j) can be written gk,j(j)=∑l=k+∞‍(Δ~l,j)(l), with Δ~l,j=Gl,j1[l,l+1], Gl,j=𝒫(Gl-1,j1[l,+∞)) and Gk,j=𝒫[k-j]gk,j. Therefore, we get the following identity as:
(2.36)h(t)Δk(k)(t)=(h(t)Δk(t))(k)+∑j=0k-1‍k!j!(k-j)!Δ~k,j(k)+∑j=0k-1‍k!j!(k-j)!∑l=k+1+∞‍Δ~l,j(l).
(1) We first give estimates for||(h(t)Δk(t))(k)||β,σ,ϵ,d. We write
(2.37)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d=(σrb(β)|ϵ|)k∫0+∞‍τs1|Δk(τ)|exp(-σ|ϵ|rb(β)τ)dτ=(σrb(β-k2)|ϵ|)k(rb(β)rb(β-k2))k×∫kk+1‍τs1exp(-σ|ϵ|(rb(β)-rb(β-k2))τ)×|Δk(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ≤A(ϵ,β)(σrb(β-k2)|ϵ|)k∫kk+1‍|Δk(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ,
where
(2.38)A(ϵ,β)=supk≥1((rb(β)rb(β-k2))k(k+1)s1exp(-σ|ϵ|(rb(β)-rb(β-k2))k)).
Now, we gives estimates for A(ϵ,β). We write
(2.39)(rb(β)rb(β-k2))k(k+1)s1exp(-σ|ϵ|(rb(β)-rb(β-k2))k)=(k+1)s1exp(-kσ|ϵ|(ψ(rb(β))-ψ(rb(β-k2))))≤2s1ks1exp(-kσ|ϵ|(ψ(rb(β))-ψ(rb(β-k2)))),
where ψ(x)=x-(|ϵ|/σ)log(x) for all k≥1. From the Taylor formula applied to ψ on [rb(β-k2),rb(β)], we get that
(2.40)ψ(rb(β))-ψ(rb(β-k2))≥(1-|ϵ|σ)(rb(β)-rb(β-k2))≥(1-|ϵ|σ)k2(β+1)b
for all β≥k2. Now, we recall that for all m1,m2>0 two real numbers, we have
(2.41)supx≥0xm1exp(-m2x)=(m1m2)m1e-m1.
From (2.39), (2.40) and (2.41), we deduce that
(2.42)A(ϵ,β)≤2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1
for all β≥k2. From (2.37) and (2.42), we deduce that
(2.43)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d≤2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d.
(2) We give estimates for||Δ~l,j||β,σ,ϵ for all 0≤j≤k-1 and all l≥k. From Lemma 2.6, we have the estimates
(2.44)∥Δ~k+n,j∥β,σ,ϵ≤(|ϵ|σrb(β))n∥𝒫[k-j]gk,j∥β,σ,ϵ,∥Δ~l,j∥β,σ,ϵ≤e(2-(l-k))(σ/|ϵ|)rb(β)(l-k)l-k-1(l-k-1)!∥𝒫[k-j]gk,j∥β,σ,ϵ,
for n=0,1,2 and all l≥k+3. Now, we give estimates for ||𝒫[k-j]gk,j||β,σ,ϵ. Using the Taylor formula with integral remainder, we have that
(2.45)|𝒫[k-j]gk,j(t)|≤1(k-j-1)!∫kt‍(t-s)k-j-1|h(k-j)(s)Δk(s)|ds,
and from the classical identity
(2.46)∫s+∞‍exp(-σ|ϵ|rb(β)t)(t-s)k-j-1dt=exp(-σ|ϵ|rb(β)s)(k-j)!((σ/|ϵ|)rb(β))k-j
we get from the Fubini theorem that
(2.47)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ=∫k+∞‍|𝒫[k-j]gk,j(t)|exp(-σ|ϵ|rb(β)t)dt≤∫k+∞‍(∫s∞‍(t-s)k-j-1(k-j-1)!exp(-σ|ϵ|rb(β)t)dt)|h(k-j)(s)Δk(s)|ds=(1(σ/|ϵ|)rb(β))k-j(k-j)∫k∞‍exp(-σ|ϵ|rb(β)s)|h(k-j)(s)Δk(s)|ds.
Again, we write
(2.48)∫k∞‍exp(-σ|ϵ|rb(β)s)|h(k-j)(s)Δk(s)|ds=∫k∞‍|h(k-j)(s)|exp(-σ|ϵ|(rb(β)-rb(β-k2))s)|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds.
From the expression of h, we have that
(2.49)|h(k-j)(s)|≤s1!ss1sk-j≤s1!ss1kk-j
for all s≥k, if 1≤k-j≤s1, and h(k-j)(s)=0, if k-j>s1. Using (2.49) in the right-hand side of the equality (2.48), we deduce from (2.47) that
(2.50)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ≤s1!(k-j)kk-j(|ϵ|σrb(β))k-j×∫kk+1‍ss1exp(-σ|ϵ|(rb(β)-rb(β-k2))s)×|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds
if 1≤k-j≤s1, and ||𝒫[k-j]gk,j(t)||β,σ,ϵ=0 if k-j>s1.
(3) We give estimates for∑j=0k-1‍k!||Δ~k+n,j(k+n)||β,σ,ϵ,d/(j!(k-j)!), for n=0,1,2. From the estimates (2.44) and (2.50), we get that
(2.51)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j)!s1!(k-j)kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-σ|ϵ|(rb(β)-rb(β-k2))s)×|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds.
From (2.37) and (2.42), we deduce from (2.51) that
(2.52)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤Ak2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d,
where
(2.53)Ak=∑j=k-s1,j≥0k-1‍k!s1!j!(k-j-1)!kk-j(|ϵ|σrb(β))k-j
for all k≥1, and n=0,1,2. Now, we show that Ak, k≥1, is a bounded sequence. We have
(2.54)Ak≤s1!k!kk(∑m=0s1-1‍kk-s1+m(k-s1+m)!(s1-m-1)!)
for all k≥s1. From the Stirling formula which asserts that k!~kke-k(2πk)1/2 as k→+∞, we get a universal constant C1>0 and a constant C2>0 (depending on s1, m) such that
(2.55)k!kk≤C1e-k(2πk)1/2,kk-s1+m(k-s1+m)!≤C1k!ek(k-s1+m)!(2πk)1/2ks1-m≤C2ek(2πk)1/2
for all k≥1. From (2.54), (2.55), we get a constant C3>0 (depending on s1) such that
(2.56)Ak≤C3
for all k≥1.
(4) We give estimates for∑j=0k-1‍(k!/j!(k-j)!)∑l=k+3+∞‍||Δ~l,j(l)||β,σ,ϵ,d. From the estimates (2.44) and (2.50), we get that
(2.57)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j-1)!s1!kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-σ|ϵ|(rb(β)-rb(β-k2))s)×|Δk(s)|exp(-σ|ϵ|rb(β-k2)s)ds×∑l=k+3+∞‍(σ|ϵ|rb(β))l-kexp((2-(l-k))σ|ϵ|rb(β))(l-k)l-k-1(l-k-1)!.
Again from (2.37) and (2.42), we deduce from (2.57) that
(2.58)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤Bk2s1(s1e-1(1-|ϵ|/σ)k2σ)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d,
where Bk=AkA~k and
(2.59)A~k=∑l=k+3+∞‍(σ|ϵ|rb(β))l-kexp((2-(l-k))σ|ϵ|rb(β))(l-k)l-k-1(l-k-1)!=∑h=3+∞‍(σ|ϵ|rb(β))hexp((2-h)σ|ϵ|rb(β))hh-1(h-1)!
for all k≥1. Now, we remind from (2.31) that A~k is a bounded sequence.
Finally, from (2.31), (2.36), (2.43), (2.52), (2.56), and (2.58), we deduce a constant C5>0 (depending on s1, σ) such that
(2.60)∥h(t)Δk(k)(t)∥β,σ,ϵ,d≤C5|ϵ|s1(β+1)bs1∥Δk(k)∥β-k2,σ,ϵ,d,
which gives the result. It remains to consider the case k=0.
Whenk=0, let f(t)=Δ0(t)∈L1(ℝ+), with supp(Δ0)∈[0,1]. By definition, we can write
(2.61)∥h(t)Δ0(t)∥β,σ,ϵ,d=∥h(t)Δ0(t)∥β,σ,ϵ=∫01‍τs1exp(-σ|ϵ|(rb(β)-rb(β-k2))τ)|Δ0(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ.
Using (2.41), we deduce from (2.61) that
(2.62)∥h(t)Δ0(t)∥β,σ,ϵ,d≤(s1e-1σk2)s1|ϵ|s1(β+1)bs1∫01‍|Δ0(τ)|exp(-σ|ϵ|rb(β-k2)τ)dτ=(s1e-1σk2)s1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d.
Hence there exists a constant C5,1>0 (depending on s1, σ) such that
(2.63)∥h(t)f(t)∥β,σ,ϵ,d≤C5,1|ϵ|s1(β+1)bs1∥f(t)∥β-k2,σ,ϵ,d,
which yields the result.Proposition 2.13.
Letσ>σ~>0 be real numbers such that
(2.64)32σ|ϵ|rb(β)e1-(σ/|ϵ|)rb(β)<1,|ϵ|<σ~.
Let s1≥0 be a nonnegative integer. Then, for all f∈𝒟β,σ~,ϵ′, one has rs1f(r)∈𝒟β,σ,ϵ′. Moreover, there exists a constant C6>0 (depending on s1, σ, σ~) such that
(2.65)∥rs1f(r)∥β,σ,ϵ,d≤C6|ϵ|s1∥f(r)∥β,σ~,ϵ,d
for all f∈𝒟β,σ,ϵ′.Proof.
The line of reasoning will follow the proof of Proposition2.12. We start from the identity (2.36).
(
1
) We first give estimates for ||(h(t)Δk(t))(k)||β,σ,ϵ,d. We write
(2.66)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d=(σrb(β)|ϵ|)k∫0+∞‍τs1|Δk(τ)|exp(-σ|ϵ|rb(β)τ)dτ=(σ~rb(β)|ϵ|)k(σσ~)k∫kk+1‍τs1exp(-(σ-σ~)|ϵ|rb(β)τ)×|Δk(τ)|exp(-σ~|ϵ|rb(β)τ)dτ≤A~(ϵ,β)(σ~rb(β)|ϵ|)k∫kk+1‍|Δk(τ)|exp(-σ~|ϵ|rb(β)τ)dτ,
where
(2.67)A~(ϵ,β)=supk≥1((σσ~)k(k+1)s1exp(-(σ-σ~)|ϵ|rb(β)k)).
Now, we give estimates for A~(ϵ,β). We write
(2.68)(σσ~)k(k+1)s1exp(-(σ-σ~)|ϵ|rb(β)k)=(k+1)s1exp(-krb(β)|ϵ|(φ(σ)-φ(σ~)))≤2s1ks1exp(-krb(β)|ϵ|(φ(σ)-φ(σ~))),
where φ(x)=x-(|ϵ|/rb(β))log(x), for all k≥1. From the Taylor formula applied to φ on [σ~,σ], we get that
(2.69)φ(σ)-φ(σ~)≥(1-|ϵ|σ~)(σ-σ~).
From (2.68), (2.69), and (2.41), we deduce that
(2.70)A~(ϵ,β)≤2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1.
From (2.66) and (2.70), we get that
(2.71)∥(h(t)Δk(t))(k)∥β,σ,ϵ,d≤2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d.
(
2
) We give estimates for ||Δ~l,j||β,σ,ϵ, for all 0≤j≤k-1, all l≥k. We start from the formula (2.44) and (2.47). We write
(2.72)∫k∞‍exp(-σ|ϵ|rb(β)s)|h(k-j)(s)Δk(s)|ds=∫k∞‍|h(k-j)(s)|exp(-(σ-σ~)|ϵ|rb(β)s)|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds.
We get that
(2.73)∥𝒫[k-j]gk,j(t)∥β,σ,ϵ≤s1!(k-j)kk-j(|ϵ|σrb(β))k-j∫kk+1‍ss1exp(-(σ-σ~)|ϵ|rb(β)s)|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds
if 1≤k-j≤s1, and ||𝒫[k-j]gk,j(t)||β,σ,ϵ=0 if k-j>s1.
(
3
) We give estimates for ∑j=0k-1‍k!||Δ~k+n,j(k+n)||β,σ,ϵ,d/(j!(k-j)!), for n=0,1,2. From the estimates (2.44) and (2.73), we get that
(2.74)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j)!s1!(k-j)kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-(σ-σ~)|ϵ|rb(β)s)×|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds.
From (2.66) and (2.70), we deduce from (2.74) that
(2.75)∑j=0k-1‍k!j!(k-j)!∥Δ~k+n,j(k+n)∥β,σ,ϵ,d≤Ak2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d,
where Ak is the bounded sequence given in the proof of Proposition 2.12.
We give estimates for∑j=0k-1‍(k!/j!(k-j)!)∑l=k+3+∞‍||Δ~l,j(l)||β,σ,ϵ,d. From the estimates (2.44) and (2.73), we get that
(2.76)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤∑j≥0,j≥k-s1k-1‍k!j!(k-j-1)!s1!kk-j(|ϵ|σrb(β))k-j×(σrb(β)|ϵ|)k∫kk+1‍ss1exp(-(σ-σ~)|ϵ|rb(β)s)×|Δk(s)|exp(-σ~|ϵ|rb(β)s)ds×∑l=k+3+∞‍(σ|ϵ|rb(β))l-kexp((2-(l-k))σ|ϵ|rb(β))(l-k)l-k-1(l-k-1)!.
From (2.66) and (2.70), we deduce from (2.76), that
(2.77)∑j=0k-1‍k!j!(k-j)!∑l=k+3+∞‍∥Δ~l,j(l)∥β,σ,ϵ,d≤Bk2s1(s1e-1(1-|ϵ|/σ~)(σ-σ~))s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d,
where Bk is the bounded sequence given in the proof of Proposition 2.12.
Finally, from (2.31), (2.36), (2.56), (2.71), (2.75), and (2.77), we deduce a constant C6>0 (depending on s1, σ, σ~) such that
(2.78)∥h(t)Δk(k)(t)∥β,σ,d≤C6|ϵ|s1∥Δk(k)∥β,σ~,ϵ,d,
which gives the result. It remains to consider the case k=0.
Whenk=0, let f(t)=Δ0(t)∈L1(ℝ+), with supp(Δ0)∈[0,1]. By definition, we can write
(2.79)∥h(t)Δ0(t)∥β,σ,ϵ,d=∥h(t)Δ0(t)∥β,σ,ϵ=∫01‍τs1exp(-(σ-σ~)|ϵ|rb(β)τ)|Δ0(τ)|exp(-σ~|ϵ|rb(β)τ)dτ.
Using (2.41), we deduce from (2.79) that
(2.80)∥h(t)Δ0(t)∥β,σ,ϵ,d≤(s1e-1σ-σ~)s1|ϵ|s1∫01‍|Δ0(τ)|exp(-σ~|ϵ|rb(β)τ)dτ=(s1e-1σ-σ~)s1|ϵ|s1∥f(t)∥β,σ~,ϵ,d.
Hence, there exists a constant C6,1>0 (depending on s1, σ, σ~) such that
(2.81)∥h(t)f(t)∥β,σ,ϵ,d≤C6,1|ϵ|s1∥f(t)∥β,σ~,ϵ,d,
which yields the result.


## 2.2. Banach Spaces of Formal Power Series with Coefficients in Spaces of Distributions

Definition 2.14.
Letδ>0 be a real number. We denote by 𝒟′(σ,ϵ,δ) the vector space of formal series v(r,z)=∑β≥0‍vβ(r)zβ/β! such that vβ(r)∈𝒟β,σ,ϵ′ for all β≥0 and
(2.82)∥v(r,z)∥(σ,ϵ,d,δ)∶=∑β≥0‍∥vβ(r)∥β,σ,ϵ,dδββ!
is finite. One can check that the normed space (𝒟'(σ,ϵ,δ),||·||(σ,ϵ,d,δ)) is a Banach space.In the next proposition, we study some parameter depending linear operators acting on the space𝒟′(σ,ϵ,δ).Proposition 2.15.
Lets1,s2,k1,k2≥0 be positive integers. Assume that the condition
(2.83)k2≥bs1
holds. Then, if
(2.84)|ϵ|<σ,3(σ/|ϵ|)ζ(b)2e1-σ/|ϵ|<1,
the operator τs1∂τ-k1∂z-k2 is a bounded linear operator from the space (𝒟′(σ,ϵ,δ),||·||(σ,ϵ,d,δ)) into itself. Moreover, there exists a constant C7>0 (depending on b, s1, k2, σ) such that
(2.85)∥rs1∂r-k1∂z-k2v(r,z)∥(σ,ϵ,d,δ)≤|ϵ|s1+k1C7δk2∥v(r,z)∥(σ,ϵ,d,δ)
for all v∈𝒟'(σ,ϵ,δ).Proof.
Letv(r,z)∈𝒟'(σ,ϵ,δ). By definition, we have
(2.86)∥rs1∂r-k1∂z-k2v(r,z)∥(σ,ϵ,d,δ)=∑β≥k2‍∥rs1∂r-k1vβ-k2(r)∥β,σ,ϵ,dδββ!.
From Corollary 2.10 and Proposition 2.12, we get a constant C3,5>0 (depending on s1,σ) such that
(2.87)∥rs1∂r-k1∂z-k2v(r,z)∥(σ,ϵ,d,δ)≤C3,5∑β≥k2‍|ϵ|s1+k1(β+1)bs1(β-k2)!β!×∥vβ-k2(r)∥β-k2,σ,ϵ,dδk2δβ-k2(β-k2)!.
From the assumptions (2.83), we get a constant Cb,s1,k2>0 (depending on b, s1, k2) such that
(2.88)(β+1)bs1(β-k2)!β!≤Cb,s1,k2
for all β≥k2. Finally, from the estimates (2.87) and (2.88), we get the inequality (2.85).In the next proposition, we study linear operators of multiplication by bounded holomorphic andC∞ functions.Proposition 2.16.
For allβ≥0, let hβ(τ) be a C∞ function with respect to r on ℝ+ such that there exist A,B,ρ,μ>0 with
(2.89)|hβ(q)(r)|≤AB-ββ!q!(ρ(r+μ))q+1
for all r∈ℝ+. One consider the series
(2.90)h(r,z)=∑β≥0‍hβ(r)zββ!,
which is convergent for all |z|<B, all r∈ℝ+. Let 0<δ<B. Then, if
(2.91)|ϵ|<σ,|ϵ|<ρσ,3(σ/|ϵ|)ζ(b)2e1-σ/|ϵ|<1,
the linear operator of multiplication by h(r,z) is continuous from (𝒟'(σ,ϵ,δ),||·||(σ,ϵ,δ)) into itself. Moreover, there exists a constant C8 (depending on μ, ρ, B) such that
(2.92)∥h(r,z)v(r,z)∥(σ,ϵ,d,δ)≤C8A∥v(r,z)∥(σ,ϵ,d,δ)
for all v(r,z)∈𝒟'(σ,ϵ,δ) satisfying (2.91).Proof.
Letv(r,z)=∑β≥0‍vβ(r)zβ/β!∈𝒟'(σ,ϵ,δ). By definition, we have that
(2.93)∥h(τ,z)v(r,z)∥(σ,ϵ,d,δ)≤∑β≥0‍(∑β1+β2=β‍∥hβ1(r)vβ2(r)∥β,σ,ϵ,dβ!β1!β2!)δββ!.
From Proposition 2.11 and Remark 2.4, we deduce that there exists C4>0 (depending on μ,ρ) such that
(2.94)∥hβ1(r)vβ2(r)∥β,σ,ϵ,d≤C4AB-β1β1!∥vβ2(r)∥β,σ,ϵ,d≤C4AB-β1β1!∥vβ2(r)∥β2,σ,ϵ,d
for all β1,β2≥0 such that β1+β2=β. From (2.93) and (2.94), we deduce that
(2.95)∥h(r,z)v(r,z)∥(σ,ϵ,d,δ)≤C4A(∑β≥0‍(δB)β)∥v(r,z)∥(σ,ϵ,d,δ)
which yields (2.92).


## 2.3. Cauchy Problems in Analytic Functions Spaces with Dependence on Initial Data

In this section, we recall the well-know-Cauchy Kowalevski theorem in some spaces of analytic functions for which the dependence on the coefficients and initial data can be obtained.The following Banach spaces were used in [26].Definition 2.17.
LetT,X be real numbers such that T,X>0. We define a vector space G(T,X) of holomorphic functions on a neighborhood of the origin in ℂ2. A formal series U(t,x)∈ℂ[[t,x]],
(2.96)U(t,x)=∑l,β≥0‍ul,βtll!xββ!
belongs to G(T,X) if the series
(2.97)∑l,β≥0‍|ul,β|(l+β)!TlXβ
converges. We also define a norm on G(T,X) as
(2.98)∥U(t,x)∥(T,X)=∑l,β≥0‍|ul,β|(l+β)!TlXβ.
One can easily show that (G(T,X),||·||(T,X)) is a Banach space.Remark 2.18.
LetU(t,x) be in G(T0,X0) for given T0,X0>0. Then, U(t,x) also belongs to the spaces G(T,X) for all T≤T0 and X≤X0. Moreover, the maps T→||U(t,x)||(T,X) and X→||U(t,x)||(T,X) are increasing functions from [0,T0] (resp., [0,X0]) into ℝ+.We depart from some preliminary lemma from [26]. In the following, for u(t,x)∈ℂ[[t,x]], we denote by ∂x-1u(t,x) the formal series ∫0x‍u(t,τ)dτ.Lemma 2.19.
Leth0,h1∈ℕ such that h0≤h1. The operator ∂th0∂x-h1 is a bounded linear operator from (G(T,X),||·||(T,X)) into itself. Moreover, there exists a universal constant C10>0 such that the estimates
(2.99)∥∂th0∂x-h1U(t,x)∥(T,X)≤C10T-h0Xh1∥U(t,x)∥(T,X)
hold for all U(t,x)∈G(T,X).Lemma 2.20.
LetA(t,x)=∑l,β≥0‍al,βtlxβ/l!β! be an analytic function on an open polydisc containing D(0,T)×D(0,X) and let U(t,x) be in G(T,X). Then, the product A(t,x)U(t,x) belongs to G(T,X). Moreover,
(2.100)∥A(t,x)U(t,x)∥(T,X)≤|A|(T,X)∥U(t,x)∥(T,X)
where |A|(T,X)=∑l,β≥0‍|al,β|TlXβ/l!β!.Proof.
Let(2.101)U(t,x)=∑l,β≥0‍ul,βtll!xββ!.
We have
(2.102)A(t,x)U(t,x)=∑l,β≥0‍vl,βtll!xββ!,
where
(2.103)vl,β=∑l1+l2=l‍∑β1+β2=β‍(al1,β1l1!β1!ul2,β2l2!β2!β!l!)
for all l,β≥0. By definition, we have
(2.104)|A|(T,X)∥U2(t,x)∥(T,X)=∑l,β≥0‍(∑l1+l2=l‍∑β1+β2=β‍|al1,β1||ul2,β2|l1!β1!(l2+β2)!)TlXβ,∥A(t,x)U(t,x)∥(T,X)=∑l,β≥0‍|∑l1+l2=l‍∑β1+β2=β‍al1,β1ul2,β2l!β!l1!β1!l2!β2!|TlXβ(l+β)!.
On the other side, the next inequalities are well known:
(2.105)l!β!l1!β1!l2!β2!≤(l+β)!(l1+β1)!(l2+β2)!≤(l+β)!l1!β1!(l2+β2)!
for all l1,l2≥0 such that l1+l2=l and β1,β2≥0 such that β1+β2=β.
Finally, from (2.105), we deduce that ||A(t,x)U(t,x)||(T,X) converges and that the estimates (2.100) hold.Lemma 2.21.
Leth1,h2∈ℕ and let U(t,x) be in G(T0,X0) for given T0,X0>0. Then, there exist T,X>0 small enough (depending on T0,X0) such that the formal series ∂th1∂xh2U(t,x) belongs to G(T,X). Moreover, there exists a constant C11>0 (depending on h1,h2) such that
(2.106)∥(∂th1∂xh2U)(t,x)∥(T,X)≤C11T-h1X-h2∥U(t,x)∥(T0,X0)
for all U(t,x)∈G(T0,X0).Let𝒞1 be a finite subset of ℕ2. For all (l0,l1)∈𝒞1, let cl0,l1(t,x)=∑l,β≥0‍cl0,l1,l,βtlxβ/l!β! be analytic functions on some polydisc containing the closed polydisc D-(0,T0)×D-(0,X0) for some T0,X0>0. As in Lemma 2.20, we define
(2.107)|cl0,l1|(t,x)=∑l,β≥0‍|cl0,l1,l,β|tlxβl!β!,
which converges on D-(0,T0)×D-(0,X0). We also consider d(t,x)∈G(Td,Xd), for some Td,Xd>0. The following proposition holds.Proposition 2.22.
LetS≥1 be an integer. One make the following assumptions. For all (l0,l1)∈𝒞1, one has
(2.108)S>l1,S≥l0+l1.
One consider the following Cauchy problem:
(2.109)∂xSU(t,x)=∑(l0,l1)∈𝒞1‍cl0,l1(t,x)∂tl0∂xl1U(t,x)+d(t,x)
for the given initial conditions
(2.110)(∂xjU)(t,0)=Uj(t),0≤j≤S-1,
which are analytic functions on some disc containing the closed discD-(0,T0). If Uj(t)=∑l≥0‍Uj,ltl/l!, we define |Uj|(t)=∑l≥0‍|Uj,l|tl/l!, which converges for all t∈D-(0,T0).
Then, there existT1>0 with 0<T1<min(T0,Td) (depending on T0,Td,𝒞1) and X1>0 with 0<X1<min(X0,Xd) (depending on S, T0, 𝒞1, max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)) such that the problem (2.109), (2.110) has a unique formal solution U(t,x)∈G(T1,X1). Moreover, there exist constants C12,1,C12,2,C12,3>0 (depending on S,T0,X0,𝒞1) such that
(2.111)∥U(t,x)∥(T1,X1)≤max0≤j≤S-1|Uj|(T0)(C12,1max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)+C12,2)+C12,3∥d(t,x)∥(Td,Xd).Proof.
We denote by𝒫 the linear operator from ℂ[[t,x]] into itself defined by
(2.112)𝒫(H(t,x))∶=∂xSH(t,x)-∑(l0,l1)∈𝒞1‍cl0,l1(t,x)∂tl0∂xl1H(t,x),
and 𝒜 denotes the linear map from ℂ[[t,x]] into itself:
(2.113)𝒜(H(t,x))∶=∑(l0,l1)∈𝒞1‍cl0,l1(t,x)∂tl0∂xl1-SH(t,x)
for all H(t,x)∈ℂ[[t,x]]. By construction, we have that 𝒫∘∂x-S=id-𝒜, where id represents the identity map H↦H from ℂ[[t,x]] into itself.
Now, we show that for any givenT1>0 such that 0<T1≤T0, there exists X𝒜,T1>0 with 0<X𝒜,T1≤X0 (depending on S, T1, 𝒞1, max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)) such that id-𝒜 is an invertible map from G(T1,X) into itself for all 0<X≤X𝒜,T1. Moreover, the following inequality
(2.114)∥(id-𝒜)-1C(t,x)∥(T1,X)≤2∥C(t,x)∥(T1,X)
holds for all C(t,x)∈G(T1,X), for any 0<X≤X𝒜,T1. Indeed, from the assumption (2.108) and Lemmas 2.19 and 2.20, we get a universal constant C10,1>0 such that
(2.115)∥𝒜(C(t,x))∥(T1,X)≤C10,1(∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X)T1-l0XS-l1)∥C(t,x)∥(T1,X)≤C10,1max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)(∑(l0,l1)∈𝒞1‍T1-l0X𝒜,T1S-l1)∥C(t,x)∥(T1,X)∶=NT1,X𝒜,T1∥C(t,x)∥(T1,X)
for all C(t,x)∈G(T1,X). Since S>l1, for all (l0,l1)∈𝒞1, for the given T1>0 one can choose X𝒜,T1 small enough such that NT1,X𝒜,T1≤1/2. Therefore, the inequality (2.114) holds.
Letw(t,x)=∑j=0S-1‍Uj(t)xj/j!. From the hypothesis (2.110), we deduce that 𝒫(w(t,x)) and w(t,x) belong to G(T1,X0), for some 0<T1<T0 (depending on 𝒞1, T0). Indeed, from Lemmas 2.20 and 2.21 we get constants C11,1>0, 0<T1<T0 (depending on 𝒞1, T0) such that
(2.116)∥𝒫(w(t,x))∥(T1,X0)≤∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X0)(∑j=0S-1-l1‍∥∂tl0Uj+l1(t)∥(T1,X0)X0jj!)≤C11,1∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X0)T1-l0(∑j=0S-1-l1‍∥Uj+l1(t)∥(T0,X0)X0jj!)≤C11,1∑(l0,l1)∈𝒞1‍|cl0,l1|(T1,X0)T1-l0(∑j=0S-1-l1‍|Uj+l1|(T0)X0jj!)≤C11,1max(l0,l1)∈𝒞1|cl0,l1|(T0,X0)max0≤j≤S-1|Uj|(T0)×∑(l0,l1)∈𝒞1‍T1-l0(∑j=0S-1-l1‍X0jj!),(2.117)∥(w(t,x))∥(T1,X0)≤∑j=0S-1‍∥Uj(t)∥(T1,X0)X0jj!≤∑j=0S-1‍|Uj|(T1)X0jj!≤max0≤j≤S-1|Uj|(T0)∑j=0S-1‍X0jj!.
Now, for this constructed T1>0 satisfying (2.116) that we choose in such a way that T1<Td also holds, we select X1>0 such that 0<X1<min(X𝒜,T1,Xd). From the estimates (2.116) and Remark 2.4, we deduce that 𝒫(w(t,x)), w(t,x), and d(t,x) belong to G(T1,X1). From (2.114), we deduce the existence of a unique H(t,x)∈G(T1,X1) such that
(2.118)(𝒫∘∂x-S)H(t,x)=-𝒫(w(t,x))+d(t,x).
Now, we put U(t,x)=∂x-SH(t,x)+w(t,x). By Lemma 2.19, we deduce that U(t,x)∈G(T1,X1) and solves the problem (2.109), (2.110). Moreover, from (2.114) and (2.116), we get constants C12,1,C12,2,C12,3>0 (depending on S, T0, X0, 𝒞1) such that (2.111) holds, which yields the result.


## 3. Laplace Transform on the Spaces𝒟'(σ,ϵ,δ)

We first introduce the definition of Laplace transform of a staircase distribution.Proposition 3.1.
(1) Letβ≥0 be an integer, σ>0 a real number, and ϵ∈ℰ. Let
(3.1)f(r)=∑k=0+∞‍(Δk(r))(k)∈𝒟β,σ,ϵ′
and choose θ∈[-π,π). Then, there exist ρθ>0, ρ>0 such that the function
(3.2)ℒθ(f)(t)=∑k=0+∞‍(eiθt)k+1∫0∞‍Δk(f)(r)exp(-reiθt)dr
is holomorphic on the sector Sθ,ρθ,|ϵ|ρ={t∈ℂ*/|θ-arg(t)|<ρθ,|t|<|ϵ|ρ} for all ϵ∈ℰ. Moreover, for all compacts K⊂Sθ,ρθ,|ϵ|ρ, there exists CK>0 (depending on K and σ) such that
(3.3)|ℒθ(f)(t)|≤CK∥f∥β,σ,ϵ,d
for all t∈K.
(2) Letδ>0 and let f(r,z)=∑β≥0‍fβ(r)zβ/β!∈𝒟'(σ,ϵ,δ). We define the Laplace transform of f(r,z) in direction θ∈[-π,π) to be the function
(3.4)ℒθ(f(r,z))(t)=∑β≥0‍ℒθ(fβ)(t)zββ!,
which defines a holomorphic function on Sθ,ρθ,|ϵ|ρ×D(0,δ), for some ρθ>0, ρ>0, for all ϵ∈ℰ. Moreover, for all compacts K⊂Sθ,ρθ,|ϵ|ρ, there exists CK>0 (depending on K and σ) such that
(3.5)|ℒθ(f(r,z))(t)|≤CK∥f(r,z)∥(σ,ϵ,d,δ)
for all (t,z)∈K×D(0,δ).Proof.
We prove part (1). The second part (2) is a direct application of (1). We have(3.6)|ℒθ(f)(t)|≤∑k=0+∞‍1|t|k+1∫0+∞‍|Δk(f)(r)|exp(-σrb(β)|ϵ|r)×exp(-r(cos(θ-arg(t))|t|-σ|ϵ|rb(β)))dr.
We choose δ1>0 and ρθ>0 such that cos(θ-arg(t))>δ1 for all t∈Sθ,ρθ,|ϵ|ρ. Moreover, we choose 0<δ2<δ1 and ρ>0 such that
(3.7)|t|<|ϵ|δ1-δ2σrb(β),|ϵ|e-δ2/|t||t|σrb(β)<1
for all t∈Sθ,ρθ,|ϵ|ρ. Let k≥0 an integer, for r∈[k,k+1], we get that
(3.8)exp(-r(cos(θ-arg(t))|t|-σ|ϵ|rb(β)))≤exp(-kδ2|t|).
We deduce that for k=0,
(3.9)1|t|∫0+∞‍|Δ0(f)(r)|exp(-σrb(β)|ϵ|r)×exp(-r(cos(θ-arg(t))|t|-σ|ϵ|rb(β)))dr≤1|t|∥Δ0(f)(r)∥β,σ,ϵ
and for k≥1,
(3.10)1|t|k+1∫0+∞‍|Δk(f)(r)|exp(-σrb(β)|ϵ|r)×exp(-r(cos(θ-arg(t))|t|-σ|ϵ|rb(β)))dr≤1|t|(|ϵ|e-δ2/|t||t|σrb(β))k(σ|ϵ|rb(β))k∥Δk(f)(r)∥β,σ,ϵ≤|ϵ|e-δ2/|t||t|2σrb(β)(σ|ϵ|rb(β))k∥Δk(f)(r)∥β,σ,ϵ.
From the estimates (3.9) and (3.10) we get the inequality (3.3).In the next proposition, we show that iff is a function, then the Laplace transform of f introduced in Proposition 3.1 coincides with the classical one.Proposition 3.2.
Letf(r)∈Lβ,σ/2,ϵ. Then, from Proposition 2.7, one knows that f∈𝒟β,σ,ϵ′. The Laplace transform ℒθ(f)(t) coincides with the classical Laplace transform of f in the direction θ defined by
(3.11)Tθ(f)(t)=eiθt∫0+∞‍f(r)exp(-reiθt)dr
for all t∈Sθ,ρθ,|ϵ|ρ.Proof.
From Proposition2.7, the staircase decomposition of f=∑k≥0‍(Δk(f))(k) has the following form Δk(r)=Gk(r)1[k,k+1], with Gk=𝒫(Gk-11[k,+∞)) and G0(r)=f(r) for all k≥0. We have to compute the integrals
(3.12)Ak=(eiθ)k+1tk+1∫kk+1‍Δk(r)exp(-reiθt)dr
for all k≥0. For k=0, we have that
(3.13)A0=eiθt∫01‍f(r)exp(-reiθt)dr.
For k=1, by one integration by parts, we get that
(3.14)A1=-eiθt[G1(r)exp(-reiθt)]12+eiθt∫12‍f(r)exp(-reiθt)dr,
and using successive integrations by parts, we get that
(3.15)Ak=∑m=1k‍-(eiθt)m[Gm(r)exp(-reiθt)]kk+1+eiθt∫kk+1‍f(r)exp(-reiθt)dr
for all k≥1. On the other hand, from the hypothesis that f(r)∈Lβ,σ/2,ϵ and from the fact that Gm(r)=0 for all r≤m, we have that the next telescopic sum
(3.16)∑k=1+∞‍-(eiθt)m[Gm(r)exp(-reiθt)]kk+1
is convergent and equal to zero for all m≥1. Finally, we deduce that ∑k≥0‍Ak=Tθ(f)(t).In the next proposition, we describe the action of multiplication by a polynomial and derivation on the Laplace transform.Proposition 3.3.
Letf(r)∈𝒟β,σ,ϵ′. Then, the following relations
(3.17)ℒθ(eiθ∂r-1f)(t)=tℒθ(f)(t),ℒθ(eiθrf(r))(t)=(t2∂t+t)ℒθ(f)(t)
hold for all t∈Sθ,ρθ,|ϵ|ρ. Let s,k0≥0 be two integers such that s≥2k0. Then, there exist a finite subset 𝒪s,k0⊂ℕ2 such that for all (q,p)∈𝒪s,k0, q+p=s-k0 and integers αq,ps,k0∈ℤ, for (q,p)∈𝒪s,k0 (depending on s, k0) such that
(3.18)ts∂tk0ℒθ(f)(t)=ℒθ(ei(s-k0)θ∑(q,p)∈𝒪s,k0‍αq,ps,k0rq∂r-pf(r))(t)
for all t∈Sθ,ρθ,|ϵ|ρ.Proof.
First of all, we have to check that the relations (3.17) and (3.18) hold when f∈𝒟(ℝ+). Since 𝒟(ℝ+) is dense in (𝒟β,σ,ϵ′,||·||β,σ,ϵ,d), from the inequality (3.3) and with the help of Corollary 2.10 and Proposition 2.12, we will get that (3.17) and (3.18) hold for all f∈𝒟β,σ,ϵ′. Now, let f∈𝒟(ℝ+). The first relation of (3.17) is obtained by integrating once by parts and the second formula of (3.17) is a consequence of the equality
(3.19)∂t(eiθt∫0+∞‍f(r)exp(-reiθt)dr)=-eiθt2∫0+∞‍f(r)exp(-reiθt)dr+e2iθt3∫0+∞‍rf(r)exp(-reiθt)dr
for all t∈Sθ,ρθ,|ϵ|ρ. To get the formula (3.18), we first show the following relation:
(3.20)∂t(ℒθ(f(r))(t)=ℒθ(e-iθ(r∂r2+∂r)f(r))(t)
for all t∈Sθ,ρθ,|ϵ|ρ. Indeed, using one integration by parts, we get that
(3.21)ℒθ(e-iθ(r∂r2+∂r)f(r))(t)=eiθt2∫0+∞‍∂rf(r)rexp(-reiθt)dr.
By a second integration by parts on the right-hand side of (3.21) and by comparison with (3.19), we get (3.20). Now, let s,k0∈ℕ be such that s≥2k0. Applying the first relation of (3.17) and (3.20), we get that
(3.22)ts∂tk0ℒθ(f)(t)=ℒθ(ei(s-k0)θ∂r-s(r∂r2+∂r)(k0)f(r))(t).
Now, we recall a variant of Lemmas 5 and 6 in [2].
Lemma  3.4.For allk0≥1, there exist constantsak,k0∈ℕ,k0≤k≤2k0such that(3.23)(r∂r2+∂r)k0u(r)=∑k=k02k0‍ak,k0rk-k0∂rku(r)for allC∞functionsu:ℝ+→ℂ.
Lemma  3.5.Leta,b,c≥0be positive integers such thata≥banda≥c. We putδ=a+b-c. Then, for allC∞functionu:ℝ+→ℂ, the function∂r-a(rb∂rcu(r))can be written in the form(3.24)∂r-a(rb∂rcu(r))=∑(b′,c′)∈𝒪δ‍αb′,c′rb′∂rc′u(r),where𝒪δis a finite subset ofℤ2such that for all(b′,c′)∈𝒪δ,b′-c′=δ,b′≥0,c′≤0, andαb′,c′∈ℤ.
Finally, we observe that the relation (3.18) follows from (3.22) and Lemmas 3.4 and 3.5.The next proposition can be found in [25, Appendix  A], see also [8].Proposition 3.6.
Letα≥1 and f(r)∈𝒟β,σ,ϵ′ with |ϵ|<σrb(β). Then, for every l≥0, the expression (f(t-αl)1[αl,+∞))(l) belongs to 𝒟β,σ,ϵ′. Moreover, there exist a universal constant A>0 and B(σ,b,ϵ)>0 (depending on σ, b, ϵ) such that
(3.25)∥(f(t-αl)1[αl,+∞))(l)∥β,σ,ϵ,d≤A(B(σ,b,ϵ))l∥f(r)∥β,σ,ϵ,d,
with B(σ,b,ϵ)→0 when ϵ→0.In the forthcoming proposition, we explain the action of multiplication by an exponential function on the Laplace transform.Proposition 3.7.
Letα≥1 and f(r)∈𝒟β,σ,ϵ′ with |ϵ|<σrb(β). From the latter proposition, one knows that Fl(r)=(f(r-αl)1[αl,+∞))(l) belongs to 𝒟β,σ,ϵ′. The following formula
(3.26)ℒθ(Fl)(t)=(eiθt)lexp(-αleiθt)ℒθ(f)(t)
holds for all t∈Sθ,ρθ,|ϵ|ρ.Proof.
Since𝒟(ℝ+) is dense in 𝒟β,σ,ϵ′, it is sufficient to prove that
(3.27)ℒθ(Fl)(t)=(eiθt)lexp(-αleiθt)Tθ(f)(t)
for all f∈𝒟(ℝ+), all t∈Sθ,ρθ,|ϵ|ρ. Then, we get the inequality (3.26) by using (3.3) and Proposition 3.6. Now, let f∈𝒟(ℝ+). We write
(3.28)(f(τ-αl)1[αl,+∞))(l)=∂τ-r(f(τ-αl)1[αl,+∞))(l+r),
where r≥0 is an integer chosen such that αl∈[l+r,l+r+1]. From our assumption, we have that τ↦f(τ-αl)1[αl,+∞) belongs to L1(ℝ+) and that supp(f(τ-αl)1[αl,+∞))⊂[l+r,+∞). By Lemma 2.5, we deduce that (f(τ-αl)1[αl,+∞))(l+r) is a staircase distribution ∑h≥0‍Δ~h,l(h)(τ) where the functions Δ~h,l(τ) are constructed as follows:
(3.29)Δ~j,l(τ)=0,for0≤j≤l+r-1,Δ~l+r,l(τ)=f(τ-αl)1[αl,+∞)1[l+r,l+r+1],
and for all n≥1, we have Δ~l+r+n,l(τ)=Gn(τ)1[l+r+n,l+r+n+1] where
(3.30)Gn(τ)=∂τ-1(Gn-1(τ)1[l+r+n,+∞)),G0=f(τ-αl)1[αl,+∞).
By definition, we have
(3.31)ℒθ((f(τ-αl)1[αl,+∞))(l+r))(t)=∑h=0∞‍(eiθt)h+1∫0∞‍Δ~h,l(τ)exp(-τeiθt)dτ.
Now, we will compute the integrals Ah,l=(eiθ/t)h+1∫0+∞‍Δ~h,l(τ)exp(-τeiθ/t)dτ for all h≥0. By construction, we have that Ah,l=0 for all 0≤h≤l+r-1. For h=l+r, we get
(3.32)Al+r,l=(eiθt)l+r+1∫αll+r+1‍f(τ-αl)exp(-τeiθt)dτ=(eiθt)l+r+1exp(-αleiθt)∫0(1-α)l+r+1‍f(s)exp(-seiθt)ds.
For h=l+r+1, by one integration by parts, we get that
(3.33)Al+r+1,l=[-(eiθt)l+r+1exp(-τeiθt)G1(τ)]l+r+1l+r+2+(eiθt)l+r+1∫l+r+1l+r+2‍f(τ-αl)exp(-τeiθt)dτ=[-(eiθt)l+r+1exp(-τeiθt)G1(τ)]l+r+1l+r+2+(eiθt)l+r+1exp(-αleiθt)∫(1-α)l+r+1(1-α)l+r+2‍f(s)exp(-seiθt)ds.
For h=l+r+n, with n≥1, by successive integrations by parts, we get that
(3.34)Al+r+n,l=∑q=1n‍[-(eiθt)l+r+qexp(-τeiθt)Gq(τ)]l+r+nl+r+n+1+(eiθt)l+r+1exp(-αleiθt)∫(1-α)l+r+n(1-α)l+r+n+1‍f(s)exp(-seiθt)ds.
Since Gq(l+r+q)=0, for all q≥1, we deduce that the telescopic sum
(3.35)∑n=q∞‍[(eiθt)l+r+qexp(-τeiθt)Gq(τ)]l+r+nl+r+n+1
is equal to 0. From the formula (3.32), (3.33), (3.34), and (3.35), we get that
(3.36)ℒθ((f(τ-αl)1[αl,+∞))(l+r))(t)=∑h=0+∞‍Ah,l=(eiθt)l+rexp(-αleiθt)eiθt∫0+∞‍f(s)exp(-seiθt)ds.
From Proposition 3.3, we have that
(3.37)ℒθ(Fl)(t)=tr(eiθ)-rℒθ((f(τ-αl)1[αl,+∞))(l+r))(t).
Finally, from (3.36) and (3.37), we get the equality (3.27).


## 4. Formal and Analytic Transseries Solutions for a Singularly Perturbed Cauchy Problem

### 4.1. Laplace Transform and Asymptotic Expansions

We recall the definition of Borel summability of formal series with coefficients in a Banach space, see [27].Definition 4.1.
A formal series(4.1)X^(t)=∑j=0∞‍ajj!tj∈𝔼[[t]]
with coefficients in a Banach space (𝔼,∥·∥𝔼) is said to be 1-summable with respect to t in the direction d∈[0,2π) if (i)
there existsρ∈ℝ+ such that the following formal series, called formal Borel transform of X^ of order 1,
(4.2)ℬ(X^)(τ)=∑j=0∞‍ajτj(j!)2∈𝔼[[τ]]
is absolutely convergent for |τ|<ρ;(ii)
there existsδ>0 such that the series ℬ(X^)(τ) can be analytically continued with respect to τ in a sector Sd,δ={τ∈ℂ*:|d-arg(τ)|<δ}. Moreover, there exist C>0 and K>0 such that
(4.3)∥ℬ(X^)(τ)∥𝔼≤CeK|τ|
for all τ∈Sd,δ. We say that ℬ(X^)(τ) has exponential growth of order 1 on Sd,δ.If this is so, the vector valued Laplace transform of order1 of ℬ(X^)(τ) in the direction d is defined by
(4.4)ℒd(ℬ(X^))(t)=t-1∫Lγ‍ℬ(X^)(τ)e-(τ/t)dτ
along a half-line Lγ=ℝ+eiγ⊂Sd,δ∪{0}, where γ depends on t and is chosen in such a way that cos(γ-arg(t))≥δ1>0, for some fixed δ1, for all t in a sector
(4.5)Sd,θ,R={t∈ℂ*:|t|<R,|d-arg(t)|<θ2},
where π<θ<π+2δ and 0<R<δ1/K. The function ℒd(ℬ(X^))(t) is called the 1-sum of the formal series X^(t) in the direction d. The function ℒd(ℬ(X^))(t) is a holomorphic and a bounded function on the sector Sd,θ,R. Moreover, the function ℒd(ℬ(X^))(t) has the formal series X^(t) as Gevrey asymptotic expansion of order 1 with respect to t on Sd,θ,R. This means that for all 0<θ1<θ, there exist C,M>0 such that
(4.6)∥ℒd(ℬ(X^))(t)-∑p=0n-1‍app!tp∥𝔼≤CMnn!|t|n
for all n≥1, all t∈Sd,θ1,R.In the next proposition, we recall some well-known identities for the Borel transform that will be useful in the sequel.Proposition 4.2.
LetX^(t)=∑n≥0‍antn/n! and G^(t)=∑n≥0‍bntn/n! be formal series in 𝔼[[t]]. One has the following equalities as formal series in 𝔼[[τ]]:
(4.7)(τ∂τ2+∂τ)(ℬ(X^)(τ))=ℬ(∂tX^(t))(τ),∂τ-1(ℬ(X^))(τ)=ℬ(tX^(t))(τ),τℬ(X^)(τ)=ℬ((t2∂t+t)X^(t))(τ).


### 4.2. Formal Transseries Solutions for an Auxiliary Singular Cauchy Problem

LetS≥1 be an integer. Let 𝒮 be a finite subset of ℕ3 and let
(4.8)bs,k0,k1(z,ϵ)=∑β≥0bs,k0,k1,β(ϵ)zββ!
be holomorphic and bounded functions on a polydisc D(0,ρ)×D(0,ϵ0), for some ρ,ϵ0>0, with ϵ0<1, for all (s,k0,k1)∈𝒮. We consider the following singular Cauchy problems:
(4.9)T2∂T∂zSY^(T,z,ϵ)+(T+1)∂zSY^(T,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)ϵk0-sTs(∂Tk0∂zk1Y^)(T,z,ϵ),
for given formal transseries initial conditions
(4.10)(∂zjY^)(T,0,ϵ)=∑h≥0‍exp(-hλ/T)h!φ^h,j(T,ϵ),0≤j≤S-1,
where φ^h,j(T,ϵ)=∑m≥0‍φh,j,m(ϵ)Tm/m!∈ℂ[[T]] for all ϵ∈ℰ and λ∈ℂ*.Proposition 4.3.
The problem (4.9), (4.10) has a formal transseries solutions
(4.11)Y^(T,z,ϵ)=∑h≥0‍exp(-hλ/T)h!Y^h(T,z,ϵ),
where the formal series Y^h(T,z,ϵ)∈ℂ[[T,z]], for all ϵ∈ℰ, all h≥0, satisfy the following singular Cauchy problems:
(4.12)T2∂T∂zSY^h(T,z,ϵ)+(T+1+λh)∂zSY^h(T,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)(∑q=1k01‍ϵk0-sTs(∂Tk0∂zk1Y^h)(T,z,ϵ)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍cqk01(hλ)qϵk0-sTs-(k01+q)∂Tk02∂zk1Y^h(T,z,ϵ))
with initial conditions
(4.13)(∂zjY^h)(T,0,ϵ)=φ^h,j(T,ϵ),0≤j≤S-1,
for some real numbers cqk01, for 1≤q≤k01 and 1≤k01≤k0.Proof.
We have that(4.14)∂T(exp(-hλT)Y^h(T,z,ϵ))=exp(-hλT)(hλT2Y^h(T,z,ϵ)+∂TY^h(T,z,ϵ)),
and from the Leibniz rule we also have
(4.15)∂Tk0(exp(-hλT)Y^h(T,z,ϵ))=∑k01+k02=k0‍k0!k01!k02!∂Tk01(exp(-hλT))∂Tk02Y^h(T,z,ϵ).
On the other hand, by the Faa Di Bruno formula we have, for all k01≥1, that
(4.16)∂Tk01(exp(-hλT))=∑q=1k01‍exp(-λhT)∑(λ1,…,λk01)∈Aq,k01‍k01!∏i=1k01‍((-1)i+1(hλ/Ti+1))λiλi!=exp(-hλT)(∑q=1k01‍cqk01(hλ)qTk01+q),
where Aq,k01={(λ1,…,λk01)∈ℕk01/∑i=1k01‍λi=q,∑i=1k01‍iλi=k01} and cqk01∈ℝ, for all q=1,…,k01.
Using the expressions (4.14), (4.15), (4.16), by plugging the formal expansion Y^(T,z,ϵ) into the problem (4.9), (4.10) and by identification of the coefficients of exp(-hλ/T) we get that Y^h satisfies the problem (4.12), (4.13).


### 4.3. Formal Solutions to a Sequence of Regular Cauchy Problems

Proposition 4.4.
One makes the assumption that(4.17)S>k1,s≥2k0
for all (s,k0,k1)∈𝒮. Then, the problem (4.12), (4.13) has a unique formal solution Y^h(T,z,ϵ)∈ℂ[[T,z]] for all ϵ∈ℰ. Let
(4.18)Y^h(T,z,ϵ)=∑m≥0‍Yh,m(z,ϵ)Tmm!,
where Yh,m(z,ϵ)∈ℂ[[z]], be the formal solution of (4.12), (4.13) for all ϵ∈ℰ. One denotes by
(4.19)Vh(τ,z,ϵ)=∑m≥0‍Yh,m(z,ϵ)τm(m!)2
the formal Borel transform of Y^h with respect to T. Then, for all h≥0, Vh(τ,z,ϵ) satisfies the problem
(4.20)(τ+1+λh)∂zSVh(τ,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)(ϵk0-s∑(r,p)∈𝒪s-k01‍αr,p1τr∂τ-p∂zk1Vh(τ,z,ϵ)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍cqk01(hλ)qϵk0-s×∑(r,p)∈𝒪s-k0-q2‍αr,p2,qτr∂τ-p∂zk1Vh(τ,z,ϵ))
with initial data
(4.21)(∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ)=∑m≥0‍φh,j,m(ϵ)τm(m!)2∈ℂ[[τ]],0≤j≤S-1,
where 𝒪s-k01 is a finite subset of ℕ2 such that (r,p)∈𝒪s-k01 implies r+p=s-k0 and 𝒪s-k0-q2 is a finite subset of ℕ2 such that (r,p)∈𝒪s-k0-q2 implies r+p=s-k0-q, and αr,p1, αr,p2,q are integers.Proof.
The proof follows by direct computation on the problems (4.12) and (4.13), using Proposition 4.2 and the following two lemmas from [2].
Lemma  4.5.For allk0≥1, there exist constantsak,k0∈ℕ,k0≤k≤2k0, such that(4.22)(τ∂τ2+∂τ)k0u(τ)=∑k=k02k0‍ak,k0τk-k0∂τku(τ)for all holomorphic functionsu:Ω→ℂon an open setΩ⊂ℂ.
Lemma  4.6.Leta,b,c≥0be positive integers such thata≥banda≥c. We putδ=a+b-c. Then, for all holomorphic functionsu:Ω→ℂ, the function∂τ-a(τb∂τcu(τ))can be written in the form(4.23)∂τ-a(τb∂τcu(τ))=∑(b′,c′)∈𝒪δ‍αb′,c′τb′∂τc′u(τ),where𝒪δis a finite subset ofℤ2such that for all(b′,c′)∈𝒪δ,b′-c′=δ,b′≥0,c′≤0, andαb′,c′∈ℤ.


### 4.4. An Auxiliary Cauchy Problem

We denote byΩ1 an open star-shaped domain in ℂ (meaning that Ω1 is an open subset of ℂ such that for all x∈Ω1, the segment [0,x] belongs to Ω1). Let Ω2 be an open set in ℂ* contained in the disc D(0,ϵ0). We denote by Ω=Ω1×Ω2. For any open set 𝒟⊂ℂ, we denote by 𝒪(𝒟) the vector space of holomorphic functions on 𝒟.Definition 4.7.
Letb>1 a real number and let rb(β)=∑n=0β‍1/(n+1)b for all integers β≥0. Let ϵ∈Ω2 and σ>0 be a real number. We denote by Eβ,ϵ,σ,Ω the vector space of all functions v∈𝒪(Ω1) such that
(4.24)∥v(τ)∥β,ϵ,σ,Ω∶=supτ∈Ω1|v(τ)|(1+|τ|2|ϵ|2)exp(-σ2|ϵ|rb(β)|τ|)
is finite.Proposition 4.8.
One makes the assumption that(4.25)S>k1,s≥2k0
for all (s,k0,k1)∈𝒮. Moreover, one makes the assumption that there exists c′,δ′>0 such that
(4.26)|τ+1+hλ|≥c′|τ+1|>δ′,∀τ∈Ω1,∀h∈ℕ.
For all h≥0, all ϵ∈Ω2, the problem (4.20) with initial conditions
(4.27)(∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ)∈𝒪(Ω1),0≤j≤S-1
has a unique formal series
(4.28)Vh(τ,z,ϵ)=∑β≥0‍vh,β(τ,ϵ)zββ!∈𝒪(Ω1)[[z]],
where vh,β(τ,ϵ) satisfies the following recursion:
(4.29)(τ+1+hλ)vh,β+S(τ,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!ϵk0-s(∑(r,p)∈𝒪s-k01‍αr,p1τr∂τ-pvh,β2+k1(τ,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-s(∑(r,p)∈𝒪s-k0-q2‍αr,p2,qτr∂τ-pvh,β2+k1(τ,ϵ)β2!)
for all τ∈Ω1, all ϵ∈Ω2.Proposition 4.9.
One makes the assumption that(4.30)S>k1,s≥2k0
for all (s,k0,k1)∈𝒮. Let also the assumption (4.26) holds. Let us assume that
(4.31)vh,j(τ,ϵ)∈Ej,ϵ,σ,Ω,∀h≥0,∀0≤j≤S-1,∀ϵ∈Ω2.
Then, one has that vh,β(τ,ϵ)∈Eβ,ϵ,σ,Ω for all β≥0, all h≥0, all ϵ∈Ω2. We put vh,β(ϵ)=||vh,β(τ,ϵ)||β,ϵ,σ,Ω, for all h≥0, all β≥0, and all ϵ∈Ω2. Then, the following inequalities hold: there exist two constants C181,C182>0 (depending on 𝒮,σ,S) such that
(4.32)vh,β+S(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!|bs,k0,k1,β1(ϵ)|β1!×C181((β+S+1)b(s-k0)+(β+S+1)b(s-k0+2))vh,β2+k1(ϵ)β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!|bs,k0,k1,β1(ϵ)|β1!|cqk01|hq|λ|q×|ϵ|-qC182((β+S+1)b(s-k0-q)+(β+S+1)b(s-k0-q+2))vh,β2+k1(ϵ)β2!
for all h≥0, all β≥0.Proof.
The proof follows by direct computation using the recursion (4.29) and the next lemma. We keep the notations of Proposition 4.8.
Lemma  4.10.There exists a constantC18>0(depending ons,σ,S,k0,k1) such that(4.33)∥τr∂τ-pvh,β2+k1(τ,ϵ)∥β+S,ϵ,σ,Ω≤|ϵ|r+pC18((β+S+1)b(r+p)+(β+S+1)b(r+p+2))∥vh,β2+k1(τ,ϵ)∥β2+k1,ϵ,σ,Ωfor allh≥0, and allβ≥0,0≤β2≤β, all(r,p)∈ℕ2withr+p≤s-k0.
Proof. We follow the proof of Lemma 1 from [2]. By definition, we have that ∂τ-1vh,β2+k1(τ,ϵ)=∫0τ‍vh,β2+k1(τ1,ϵ)dτ1 for all τ∈Ω1. Using the parametrization τ1=h1τ with 0≤h1≤1, we get that
(4.34)∂τ-1vh,β2+k1(τ,ϵ)=τ∫01‍vh,β2+k1(h1τ,ϵ)M1(h1)dh1,
where M1(h1)=1. More generally, for all p≥2, we have by definition:
(4.35)∂τ-pvh,β2+k1(τ,ϵ)=∫0τ‍∫0τ1‍⋯∫0τp-1‍vh,β2+k1(τp,ϵ)dτpdτp-1⋯dτ1
for all τ∈Ω1. Using the parametrization τj=hjτj-1, τ1=h1τ, with 0≤hj≤1, for 2≤j≤p, we can write
(4.36)∂τ-pvh,β2+k1(τ,ϵ)=τp∫01‍⋯∫01‍vh,β2+k1(hp⋯h1τ,ϵ)Mp(h1,…,hp)dhpdhp-1⋯dh1,
where Mp(h1,…,hp) is a monomial in h1,…,hp whose coefficient is equal to 1. Using these latter expressions, we now write
(4.37)|τr∂τ-pvh,β2+k1(τ,ϵ)|=|τr+p∫01‍⋯∫01‍vh,β2+k1(hp⋯h1τ,ϵ)(1+|hp⋯h1τ|2|ϵ|2)exp(-σ2|ϵ|rb(β2+k1)|hp⋯h1τ|)×exp((σ/2|ϵ|)rb(β2+k1)|hp⋯h1τ|)1+|hp⋯h1τ|2/|ϵ|2Mp(h1,…,hp)dhp…dh1|.
Therefore,
(4.38)|τr∂τ-pvh,β2+k1(τ,ϵ)|(1+|τ|2|ϵ|2)exp(-σ2|ϵ|rb(β+S)|τ|)≤∥vh,β2+k1(τ,ϵ)∥β2+k1,ϵ,σ,Ω|τ|r+p(1+|τ|2|ϵ|2)exp(-σ2|ϵ|(rb(β+S)-rb(β2+k1))|τ|).
By construction of rb(β), we have
(4.39)rb(β+S)-rb(β2+k1)=∑n=β2+k1+1β+S‍1(n+1)b≥β-β2+S-k1(β+S+1)b≥S-k1(β+S+1)b
for all β≥0. From (4.38) and (4.39), we get that
(4.40)|τr∂τ-pvh,β2+k1(τ,ϵ)|(1+|τ|2|ϵ|2)exp(-σ2|ϵ|rb(β+S)|τ|)≤∥vh,β2+k1(τ,ϵ)∥β2+k1,ϵ,σ,Ω|τ|r+p(1+|τ|2|ϵ|2)exp(-σ2|ϵ|S-k1(β+S+1)b|τ|)
for all β≥0. From (2.41), we deduce that
(4.41)|τ|r+p(1+|τ|2|ϵ|2)exp(-σ2|ϵ|S-k1(β+S+1)b|τ|)≤|ϵ|r+p((2(r+p+2)e-1σ(S-k1))r+p+2(2(r+p)e-1σ(S-k1))r+p(β+S+1)b(r+p)+(2(r+p+2)e-1σ(S-k1))r+p+2(β+S+1)b(r+p+2))
for all τ∈Ω1. From the estimates (4.40) and (4.41), we deduce the inequality (4.33).Proposition 4.11.
Assume that the conditions (4.26) and (4.31) hold. Assume moreover, that
(4.42)S≥b(s-k0+2)+k1,s≥2k0
for all (s,k0,k1)∈𝒮 and that the following sums converge near the origin in ℂ,
(4.43)Wj(u):=∑h≥0‍supϵ∈Ω2∥vh,j(τ,ϵ)∥j,ϵ,σ,Ωuhh!∈ℂ{u},0≤j≤S-1.
One make also the hypothesis that for all (s,k0,k1)∈𝒮, one can write
(4.44)bs,k0,k1(z,ϵ)=ϵk0b~s,k0,k1(z,ϵ),
where b~s,k0,k1(z,ϵ)=∑β≥0‍b~s,k0,k1,β(ϵ)zβ/β! is holomorphic for all ϵ∈D(0,ϵ0) on D(0,ρ). Then, the problem (4.20) with initial data
(4.45)(∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ),0≤j≤S-1
has a unique solution Vh(τ,z,ϵ) which is holomorphic with respect to (τ,z)∈Ω1×D(0,x1/2) for all ϵ∈Ω2.
The constantx1 is such that 0<x1<ρ and depends on S, u0 (which denotes a common radius of absolute convergence of the series (4.43)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ and |b|s,k0,k1, |b~|s,k0,k1 are defined below.
Moreover, the following estimates hold: there exists a constantu1 such that 0<u1<u0 (depending on u0, 𝒮, and b, σ) and a constant C19>0 (depending on max0≤j≤S-1Wj(u0) (where Wj are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.46)|Vh(τ,z,ϵ)|≤C191-2|z|/x1h!(2u1)h(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|ζ(b)|τ|)
for all (τ,z)∈Ω1×D(0,x1/2), all ϵ∈Ω2, and all h≥0.Proof.
We consider the following Cauchy problem(4.47)∂xSW(u,x)=∑(s,k0,k1)∈𝒮C181((x∂x+S+1)b(s-k0)‍+(x∂x+S+1)b(s-k0+2))(|b|s,k0,k1(x)∂xk1W(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C182|cqk01||λ|q×((x∂x+S+1)b(s-k0-q)+(x∂x+S+1)b(s-k0-q+2))×(|b~|s,k0,k1(x)(u∂u)q∂xk1W(u,x))
for given initial data
(4.48)(∂xjW)(u,0)=Wj(u)=∑h≥0‍supϵ∈Ω2|vh,j(ϵ)|uhh!∈ℂ{u},0≤j≤S-1,
where
(4.49)|b|s,k0,k1(x)=∑β≥0‍supϵ∈D(0,ϵ0)|bs,k0,k1,β(ϵ)|xββ!,|b~|s,k0,k1(x)=∑β≥0‍supϵ∈D(0,ϵ0)|b~s,k0,k1,β(ϵ)|xββ!
are convergent series near the origin in ℂ with respect to x. From the assumption (4.42) and the fact that b>1, we also deduce that
(4.50)S≥b(s-k0-q+2)+q+k1
for all (s,k0,k1)∈𝒮 and all 0≤q≤k0. Since the initial data (4.48) and the coefficients (4.47) are analytic near the origin, we get that all the hypotheses of the classical Cauchy Kowalevski theorem from Proposition 2.22 are fulfilled. We deduce the existence of U1 with 0<U1<U0, where U0 denotes a common radius of absolute convergence for the series (4.48), which depends on U0, 𝒮 and b, and X1 with 0<X1<ρ (depending on S, U0, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(X0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(X0), where X0<ρ) such that there exist a unique formal series W(u,x)∈G(U1,X1) which solves the problem (4.47), (4.48).
Now, letW(u,x)=∑h,β≥0‍wh,β(uh/h!)(xβ/β!) be its Taylor expansion at (0,0). Then, by construction the sequence wh,β satisfies the following equalities:
(4.51)wh,β+S=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!supϵ∈D(0,ϵ0)|bs,k0,k1,β1(ϵ)|β1!C181×((β+S+1)b(s-k0)+(β+S+1)b(s-k0+2))wh,β2+k1β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!supϵ∈D(0,ϵ0)|b~s,k0,k1,β1(ϵ)|β1!|cqk01|hq|λ|q×C182((β+S+1)b(s-k0-q)+(β+S+1)b(s-k0-q+2))wh,β2+k1β2!
for all h≥0 and all β≥0, with
(4.52)wh,j=supϵ∈Ω2|vh,j(ϵ)|,∀h≥0,∀0≤j≤S-1.
Using the inequality (4.32) and the equality (4.51), with the initial conditions (4.52), one gets that
(4.53)supϵ∈Ω2|vh,β(ϵ)|≤wh,β
for all h≥0, all β≥0. Using the fact that W(u,x)∈G(U1,X1) and the estimates (2.111), we deduce from (4.53) that there exist a constant C19>0 (depending on max0≤j≤S-1Wj(U0),|λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(X0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(X0), S, U0, X0, 𝒮, b, σ) such that
(4.54)|vh,β(τ,ϵ)|≤C19(h+β)!(1U1)h(1X1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)≤C19h!β!(2U1)h(2X1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)
for all τ∈Ω1, all ϵ∈Ω2, all h≥0, and all β≥0.


### 4.5. Analytic Solutions for a Sequence of Singular Cauchy Problems

Assume that the conditions (4.42) and (4.44) hold. We consider the following problem:
(4.55)T2∂T∂zSYh,Sd,ℰ(T,z,ϵ)+(T+1+λh)∂zSYh,Sd,ℰ(T,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)(∑q=1k01‍ϵk0-sTs(∂Tk0∂zk1Yh,Sd,ℰ)(T,z,ϵ)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍cqk01(hλ)qϵk0-sTs-(k01+q)∂Tk02∂zk1Yh,Sd,ℰ(T,z,ϵ))
with initial conditions
(4.56)(∂zjYh,Sd,ℰ)(T,0,ϵ)=φh,j,Sd,ℰ(T,ϵ),0≤j≤S-1.
The initial conditions φh,j,Sd,ℰ(T,ϵ), 0≤j≤S-1 are defined as follows. Let Sd be an open sector centered at 0, with infinite radius and bisecting direction d∈[0,2π), D(0,τ0) an open disc centered at 0 with radius τ0>0, and ℰ an open sector centered at 0 contained in the disc D(0,ϵ0). We make the assumption that the condition (4.26) holds for the set Ω1=(Sd∪D(0,τ0)). We consider a set of functions vh,j(τ,ϵ)∈Ej,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0}) for all ϵ∈D(0,ϵ0)∖{0} such that
(4.57)Wj,τ0,ϵ0(u)∶=∑h≥0‍supϵ∈D(0,ϵ0)∖{0}∥vh,j(τ,ϵ)∥j,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0})uhh!∈ℂ{u},0≤j≤S-1.
We also assume that for all h≥0 and all 0≤j≤S-1, vh,j(τ,ϵ) has an analytic continuation denoted by vh,j,Sd,ℰ(τ,ϵ)∈Ej,ϵ,σ,(Sd∪D(0,τ0))×ℰ for all ϵ∈ℰ such that
(4.58)Wj,Sd,ℰ(u)∶=∑h≥0‍supϵ∈ℰ∥vh,j,Sd,ℰ(τ,ϵ)∥j,ϵ,σ,(Sd∪D(0,τ0))×ℰuhh!∈ℂ{u},0≤j≤S-1.
Let
(4.59)vh,j(τ,ϵ)=∑m≥0‍φh,j,m(ϵ)τm(m!)2
be the convergent Taylor expansion of vh,j with respect to τ on D(0,τ0) for all ϵ∈D(0,ϵ0)∖{0}. We consider the formal series
(4.60)φ^h,j(T,ϵ)=∑m≥0‍φh,j,m(ϵ)Tmm!
for all ϵ∈D(0,ϵ0)∖{0}. We define φh,j,Sd,ℰ(T,ϵ) as the 1-sum (in the sense of Definition 4.1) of φ^j,h(T,ϵ) in the direction d. From the hypotheses, we deduce that T↦φh,j,Sd,ℰ(T,ϵ) defines a holomorphic function for all T∈Ud,θ,ι|ϵ|, for all ϵ∈ℰ, where
(4.61)Ud,θ,ι|ϵ|={T∈ℂ*:|T|<ι|ϵ|,|d-arg(T)|<θ2}
for some θ>π and some constant ι>0 (independent of ϵ) for all 0≤j≤S-1.Proposition 4.12.
Assume that the conditions (4.26), (4.31), (4.42), and (4.44) hold.
Then, the problem (4.55), (4.56) has a solution (T,z)↦Yh,Sd,ℰ(T,z,ϵ) which is holomorphic and bounded on the set Ud,θ,ι′|ϵ|×D(0,x1/4), for some ι′>0 (independent of ϵ), for all ϵ∈ℰ, where 0<x1<ρ depends on S, u0 (which denotes a common radius of absolute convergence of the series (4.57), (4.58)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ.
The functionYh,Sd,ℰ(T,z,ϵ) can be written as the Laplace transform of order 1 in the direction d (in the sense of Definition 4.1) of a function Vh,Sd,ℰ(τ,z,ϵ), which is holomorphic on the domain (Sd∪D(0,τ0))×D(0,x1/2)×ℰ and satisfies the following estimates.
There exists a constantu1 such that 0<u1<u0 (depending on u0, 𝒮 and b,σ) and a constant CΩ(d,ℰ)>0 (depending on max0≤j≤S-1Wj,Sd,ℰ(u0) (where Wj,Sd,ℰ are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.62)|Vh,Sd,ℰ(τ,z,ϵ)|≤CΩ(d,ℰ)1-2|z|/x1h!(2u1)h(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|ζ(b)|τ|)
for all (τ,z,ϵ)∈(Sd∪D(0,τ0))×D(0,x1/2)×ℰ, all h≥0.
Moreover, the functionVh,Sd,ℰ(τ,z,ϵ) is the analytic continuation of a function Vh(τ,z,ϵ) which is holomorphic on the punctured polydisc D(0,τ0)×D(0,x1/2)×(D(0,ϵ0)∖{0}) and verifies the following estimates.
There exists a constantCΩτ0,ϵ0>0 (depending on max0≤j≤S-1Wj,τ0,ϵ0(u0) (where Wj,τ0,ϵ0 are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.63)|Vh(τ,z,ϵ)|≤CΩτ0,ϵ01-2|z|/x1h!(2u1)h(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|ζ(b)|τ|)
for all τ∈D(0,τ0), all z∈D(0,x1/2), all ϵ∈D(0,ϵ0)∖{0}, and all h≥0.Proof.
From the hypotheses of Proposition4.12, we deduce from Proposition 4.11 applied to the situation Ω=D(0,τ0)×(D(0,ϵ0)∖{0}) the existence of a holomorphic function Vh(τ,z,ϵ) satisfying the estimates (4.63), which is the solution of the problem (4.20) with initial conditions (∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ), 0≤j≤S-1, on the domain D(0,τ0)×D(0,x1/2)×(D(0,ϵ0)∖{0}). Likewise, from Proposition 4.11 applied to the situation Ω=(Sd∪D(0,τ0))×ℰ, we get the existence of a holomorphic function Vh,Sd,ℰ(τ,z,ϵ) satisfying (4.62), which is the solution of the problem (4.20) with initial conditions (∂zjVh)(τ,0,ϵ)=vh,j,Sd,ℰ(τ,ϵ), 0≤j≤S-1 on the domain (Sd∪D(0,τ0))×D(0,x1/2)×ℰ.
With Proposition4.3, we deduce that the formal solution Y^h(T,z,ϵ) of the problem (4.12), (4.13) is 1-summable with respect to T in the direction d as series in the Banach space 𝒪(D(0,x1/4)), for all ϵ∈ℰ. We denote by Yh,Sd,ℰ(T,z,ϵ) its 1-sum which is holomorphic with respect to T on a domain Ud,θ,ι′|ϵ| due to Definition 4.1 and the estimates (4.62). Moreover, from the algebraic properties of the κ-summability procedure, see [27, Section 6.3], we deduce that Yh,Sd,ℰ(T,z,ϵ) is a solution of the problem (4.55), (4.56).


### 4.6. Summability in a Complex Parameter

We recall the definition of a good covering.Definition 4.13.
Letν≥2 be an integer. For all 0≤i≤ν-1, we consider open sectors ℰi centered at 0, with radius ϵ0, bisecting direction κi∈[0,2π) and opening π+δi, with δi>0, such that ℰi∩ℰi+1≠∅ for all 0≤i≤ν-1 (with the convention that ℰν=ℰ0) and such that ∪i=0ν-1ℰi=𝒰∖{0}, where 𝒰 is some neighborhood of 0 in ℂ. Such a set of sectors {ℰi}0≤i≤ν-1 is called a good covering in ℂ*.Definition 4.14.
Let{ℰi}0≤i≤ν-1 be a good covering in ℂ*. Let 𝒯 be an open sector centered at 0 with radius r𝒯 and consider a family of open sectors
(4.64)Udi,θ,ϵ0r𝒯:={t∈ℂ:|t|<ϵ0r𝒯,|di-arg(t)|<θ2},
where di∈[0,2π), for 0≤i≤ν-1, where θ>π, which satisfy the following properties:
(1) For all0≤i≤ν-1, all h∈ℕ, arg(di)≠arg(-1-λh).
(2) For all0≤i≤ν-1, for all t∈𝒯, and all ϵ∈ℰi, we have that ϵt∈Udi,θ,ϵ0r𝒯.
(3) (3.1) We assume thatd0<arg(λ)<d1. We consider the two closed sectors
(4.65)ℳd0={τ∈ℂ*arg(τ)∈[d0,arg(λ)]},ℳd1={τ∈ℂ*arg(τ)∈[arg(λ),d1]}.
We make the assumption that there exist two constants c′,δ′>0 with
(4.66)|τ+1+λh|≥c′|τ+1|>δ′
for all τ∈ℳd0∪ℳd1∪D(0,τ0) and all h≥0.
(3.2) There exists0<δ𝒯<π/2 such that arg(λ/(ϵt))∈(-π/2+δ𝒯,π/2-δ𝒯) for all ϵ∈ℰ0∩ℰ1 and all t∈𝒯.
We say that the family{{Udi,θ,ϵ0r𝒯}0≤i≤ν-1,𝒯,λ} is associated to the good covering {ℰi}0≤i≤ν-1.Now, we consider a set of functionsφh,i,j(T,ϵ) for 0≤i≤ν-1, 0≤j≤S-1, h≥0, constructed as follows. For all 0≤i≤ν-1, let Sdi be an open sector of infinite radius centered at 0, with bisecting direction di and with opening ni>θ-π. The numbers θ>π and ni>0 are chosen in such a way that -1-λh∉Sdi for all 0≤i≤ν-1 and all h≥0. Now, we put
(4.67)φh,i,j(T,ϵ):=φh,j,Sdi,ℰi(T,ϵ)
for all T∈Udi,θ,ι|ϵ| and all ϵ∈ℰi, where φh,j,Sdi,ℰi(T,ϵ) is given by the formula (4.56). Recalling how these functions are constructed, we consider a set of functions
(4.68)vh,j(τ,ϵ)∈Ej,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0})
for all ϵ∈D(0,ϵ0)∖{0} such that
(4.69)Wj,τ0,ϵ0(u)∶=∑h≥0‍supϵ∈D(0,ϵ0)∖{0}∥vh,j(τ,ϵ)∥j,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0})uhh!∈ℂ{u},0≤j≤S-1.
We also assume that for all h≥0, all 0≤j≤S-1, vh,j(τ,ϵ) has an analytic continuation denoted by vh,j,Sdi,ℰi(τ,ϵ)∈Ej,ϵ,σ,(Sdi∪D(0,τ0))×ℰi for all ϵ∈ℰi such that
(4.70)Wj,Sdi,ℰi(u):=∑h≥0‍supϵ∈ℰi∥vh,j,Sdi,ℰi(τ,ϵ)∥j,ϵ,σ,(Sdi∪D(0,τ0)×ℰiuhh!∈ℂ{u},0≤j≤S-1.
Let
(4.71)vh,j(τ,ϵ)=∑m≥0‍φh,j,m(ϵ)τm(m!)2
be the convergent Taylor expansion of vh,j with respect to τ on D(0,τ0) for all ϵ∈D(0,ϵ0)∖{0}. We consider the formal series
(4.72)φ^h,j(T,ϵ)=∑m≥0‍φh,j,m(ϵ)Tmm!
for all ϵ∈D(0,ϵ0)∖{0}. We define φh,j,Sdi,ℰi(T,ϵ) as the 1-sum (in the sense of Definition 4.1) of φ^j,h(T,ϵ) in the direction di. We deduce that T↦φh,j,Sdi,ℰi(T,ϵ) defines a holomorphic function for all T∈Udi,θ,ι|ϵ| and for all ϵ∈ℰi, where
(4.73)Udi,θ,ι|ϵ|={T∈ℂ*:|T|<ι|ϵ|,|di-arg(T)|<θ2}
for some θ>π and some constant ι>0 (independent of ϵ) for all 0≤j≤S-1.From Proposition4.12, for all 0≤i≤ν-1, we consider the solution Yh,Sdi,ℰi(T,z,ϵ) of the problem (4.55) with the initial conditions
(4.74)(∂zjYh,Sdi,ℰi)(T,0,ϵ)=φh,i,j(T,ϵ),0≤j≤S-1,h≥0,
which defines a bounded and holomorphic function on Udi,θ,ι′|ϵ|×D(0,x1/4)×ℰi.Proposition 4.15.
The function defined by(4.75)Xh,i(t,z,ϵ)=Yh,Sdi,ℰi(ϵt,z,ϵ)
is holomorphic and bounded on (𝒯∩D(0,ι′′))×D(0,x1/4)×ℰi, for all h≥0, all 0≤i≤ν-1, and for some 0<ι′′<ι′.
Moreover, the functionsGh,i:ϵ↦Xh,i(t,z,ϵ) from ℰi into the Banach space 𝒪((𝒯∩D(0,ι′′))×D(0,x1/4)) are the 1-sums on ℰi of a formal series G^h(ϵ)∈𝒪((𝒯∩D(0,ι′′))×D(0,x1/4))[[ϵ]]. In other words, for all h≥0, there exists a function gh(s,t,z) which is holomorphic on D(0,sh)×(𝒯∩D(0,ι′′))×D(0,x1/4) which admits for all 0≤i≤ν-1, an analytic continuation gh,i(s,t,z) which is holomorphic on (𝒢κi∪D(0,sh))×(𝒯∩D(0,ι′′))×D(0,x1/4), where 𝒢κi is an open sector centered at 0 with infinite radius and bisecting direction κi such that
(4.76)Xh,i(t,z,ϵ)=ϵ-1∫Lκi‍gh,i(s,t,z)e-s/ϵds
along a half-line Lκi=ℝ+eiκi⊂𝒢κi∪{0}.Proof.
The proof is based on a cohomological criterion for summability of formal series with coefficients in a Banach space, see [27, page 121], which is known as the Ramis-Sibuya theorem in the literature.
Theorem (RS).Let(𝔼,||·||𝔼)be a Banach space overℂand{ℰi}0≤i≤ν-1a good covering inℂ*. For all0≤i≤ν-1, letGibe a holomorphic function fromℰiinto the Banach space(𝔼,||·||𝔼)and let the cocycleΔi(ϵ)=Gi+1(ϵ)-Gi(ϵ)be a holomorphic function from the sectorZi=ℰi+1∩ℰiinto𝔼(with the convention thatℰν=ℰ0andGν=G0). We make the following assumptions.(1)
The functionsGi(ϵ)are bounded asϵ∈ℰitends to the origin inℂfor all0≤i≤ν-1.(2)
The functionsΔi(ϵ)are exponentially flat of order 1 onZifor all0≤i≤ν-1. This means that there exist constantsCi,Ai>0such that(4.77)∥Δi(ϵ)∥𝔼≤Cie-Ai/|ϵ|for allϵ∈Ziall0≤i≤ν-1.
Then, for all
0
≤
i
≤
ν
-
1
, the functions
G
i
(
ϵ
)
are the
1
-sums onℰiof a
1
-summable formal seriesG^(ϵ)inϵwith coefficients in the Banach space𝔼.
By Definition4.14 and the construction of Yh,Sdi,ℰi(T,z,ϵ) in Proposition 4.12, we get that the function Xh,i(t,z,ϵ)=Yh,Sdi,ℰi(ϵt,z,ϵ) defines a bounded and holomorphic function on the domain (𝒯∩D(0,ι′))×D(0,x1/4)×ℰi for all h≥0 all 0≤i≤ν-1, where 0<x1<ρ depends on S, u0>0 (which denotes a common radius of absolute convergence of the series (4.69), (4.70)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ. More precisely, we have the following.
Lemma  4.16.Consider the following:(1)
There exist a constant0<ι′′<ι′, a constantu1such that0<u1<u0(depending onu0,𝒮andb,σ), a constantx1such that0<x1<ρ(depending onS,u0,𝒮,b,σ,|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), wherex0<ρ), and a constantC~i>0(depending onmax0≤j≤S-1Wj,Sdi,ℰi(u0)(whereWj,Sdi,ℰiare defined above),|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0),S,u0,x0,𝒮,b) such that(4.78)supt∈𝒯∩D(0,ι′′),z∈D(0,x1/4)|Xh,i(t,z,ϵ)|≤2C~ih!(2u1)hfor allϵ∈ℰi, for all0≤i≤ν-1, and allh≥0.(2)
There exist a constant0<ι′′≤ι′, a constantu1such that0<u1<u0(depending onu0,𝒮andb,σ), a constantx1such that0<x1<ρ(depending onS,u0,𝒮,b,σ,|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), wherex0<ρ), a constantMi>0, a constantKi>0(depending onmax0≤j≤S-1Wj,Sdq,ℰq(u0), forq=i,i+1(whereWj,Sdq,ℰqare defined above),max0≤j≤S-1Wj,τ0,ϵ0(u0),|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0),S,u0,x0,𝒮,b) such that(4.79)supt∈𝒯∩D(0,ι′′),z∈D(0,x1/4)|Xh,i+1(t,z,ϵ)-Xh,i(t,z,ϵ)|≤h!(2u1)h2Kie-Mi/|ϵ|for allϵ∈ℰi∩ℰi+1, for all0≤i≤ν-1, and allh≥0(where by conventionXh,ν=Xh,0).
Proof. (1) Let i be an integer such that 0≤i≤ν-1. From Proposition 4.12, we can write
(4.80)Xh,i(t,z,ϵ)=(ϵt)-1∫Lγi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ,
where Lγi=ℝ+e-1γi⊂Sdi∪{0} and Vh,Sdi,ℰi is a holomorphic function on (Sdi∪D(0,τ0))×D(0,x1/4)×ℰi for which the estimates (4.62) hold. By construction, the direction γi (which depends on ϵt) is chosen in such a way that cos(γi-arg(ϵt))≥δ1, for all ϵ∈ℰi, all t∈𝒯∩D(0,ι′), and for some fixed δ1>0. From the estimates (4.62), we get
(4.81)|Xh,i(t,z,ϵ)|≤|ϵt|-1∫0+∞‍CΩ(di,ℰi)1-2|z|/x1h!(2u1)h(1+r2|ϵ|2)-1eσζ(b)r/2|ϵ|e-r/|ϵ||t|cos(γi-arg(ϵt))dr≤|ϵt|-1∫0+∞‍CΩ(di,ℰi)1-2|z|/x1h!(2u1)he(σζ(b)/2-δ1/|t|)(r/|ϵ|)dr=CΩ(di,ℰi)1-2|z|/x1h!(2u1)h1δ1-(σζ(b)/2)|t|≤CΩ(di,ℰi)δ2(1-2|z|/x1)h!(2u1)h
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi.
(2) Leti an integer such that 0≤i≤ν-1. From Proposition 4.12, we can write again
(4.82)Xh,i(t,z,ϵ)=(ϵt)-1∫Lγi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ,Xi+1(t,z,ϵ)=(ϵt)-1∫Lγi+1‍Vh,Sdi+1,ℰi+1(τ,z,ϵ)e-τ/ϵtdτ,
where Lγi=ℝ+e-1γi⊂Sdi∪{0}, Lγi+1=ℝ+e-1γi+1⊂Sdi+1∪{0}, and Vh,Sdi,ℰi (resp., Vh,Sdi+1,ℰi+1) is a holomorphic function on (Sdi∪D(0,τ0))×D(0,x1/4)×ℰi (resp., on (Sdi+1∪D(0,τ0))×D(0,x1/4)×ℰi+1) for which the estimates (4.62) hold and which is moreover an analytic continuation of a function Vh(τ,z,ϵ) which satisfies the estimates (4.63).
From the fact thatτ↦Vh(τ,z,ϵ) is holomorphic on D(0,τ0) for all (z,ϵ)∈D(0,x1/4)×(D(0,ϵ0)∖{0}), the integral of τ↦Vh(τ,z,ϵ) along the union of a segment starting from 0 to (τ0/2)e-1γi+1, an arc of circle with radius τ0/2 connecting (τ0/2)e-1γi+1 and (τ0/2)e-1γi and a segment starting from (τ0/2)e-1γi to 0 is equal to zero. Therefore, we can rewrite the difference Xh,i+1-Xh,i as a sum of three integrals:
(4.83)Xh,i+1(t,z,ϵ)-Xh,i(t,z,ϵ)=(ϵt)-1(∫Lτ0/2,γi+1‍Vh,Sdi+1,ℰi+1(τ,z,ϵ)e-τ/ϵtdτ-∫Lτ0/2,γi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ+∫C(τ0/2,γi,γi+1)‍Vh(τ,z,ϵ)e-τ/ϵtdτ),
where Lτ0/2,γi=[τ0/2,+∞)e-1γi, Lτ0/2,γi+1=[τ0/2,+∞)e-1γi+1, and C(τ0/2,γi,γi+1) is an arc of circle with radius τ0/2 connecting (τ0/2)e-1γi with (τ0/2)e-1γi+1 with a well-chosen orientation.
We give estimates forI1=|(ϵt)-1∫Lτ0/2,γi+1‍Vh,Sdi+1,ℰi+1(τ,z,ϵ)e-τ/ϵtdτ|. By construction, the direction γi+1 (which depends on ϵt) is chosen in such a way that cos(γi+1-arg(ϵt))≥δ1, for all ϵ∈ℰi+1∩ℰi, all t∈𝒯∩D(0,ι′), and for some fixed δ1>0. From the estimates (4.62), we get
(4.84)I1≤|ϵt|-1∫τ0/2+∞‍CΩ(di+1,ℰi+1)1-2|z|/x1h!(2u1)h(1+r2|ϵ|2)-1eσζ(b)r/2|ϵ|e-(r/|ϵ||t|)cos(γi+1-arg(ϵt))dr≤|ϵt|-1∫τ0/2+∞‍CΩ(di+1,ℰi+1)1-2|z|/x1h!(2u1)he(σζ(b)/2-δ1/|t|)(r/|ϵ|)dr=CΩ(di+1,ℰi+1)1-2|z|/x1h!(2u1)he-((δ1/|t|-σζ(b)/2)(τ0/2))(1/|ϵ|)δ1-(σζ(b)/2)|t|≤CΩ(di+1,ℰi+1)δ2(1-2|z|/x1)h!(2u1)he-(δ2τ0/2)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi.
We give estimates forI2=|(ϵt)-1∫Lτ0/2,γi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ|. By construction, the direction γi (which depends on ϵt) is chosen in such a way that there exists a fixed δ1>0 with cos(γi-arg(ϵt))≥δ1, for all ϵ∈ℰi+1∩ℰi and all t∈𝒯∩D(0,ι′). From the estimates (4.62), we deduce as before that
(4.85)I2≤CΩ(di,ℰi)δ2(1-2|z|/x1)h!(2u1)he-(δ2τ0/2)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi.
Finally, we get estimates forI3=|ϵt|-1|∫C(τ0/2,γi,γi+1)‍Vh(τ,z,ϵ)e-τ/ϵtdτ|. From the estimates (4.63), we have
(4.86)I3≤|ϵt|-1|∫γiγi+1‍CΩτ0,ϵ01-2|z|/x1h!(2u1)h(1+(τ0/2)2|ϵ|2)-1eσζ(b)τ0/4|ϵ|e-(τ0/2|ϵ||t|)cos(θ-arg(ϵt))τ02dθ|.
By construction, the arc of circle C(τ0/2,γi,γi+1) is chosen in such a way that that cos(θ-arg(ϵt))≥δ1 for all θ∈[γi,γi+1] (if γi<γi+1), θ∈[γi+1,γi] (if γi+1<γi) for all t∈𝒯, all ϵ∈ℰi∩ℰi+1. From (4.86), we deduce that
(4.87)I3≤|γi+1-γi|CΩτ0,ϵ01-2|z|/x1h!(2u1)hτ021|ϵt|e-((δ1/|t|-σζ(b)/2)(τ0/2))(1/|ϵ|)≤|γi+1-γi|CΩτ0,ϵ01-2|z|/x1h!(2u1)hτ021|ϵt|e-(δ2τ0/4)/|ϵt|e-(δ2τ0/4)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi. Using the inequality (4.87) and the estimates (2.41), we deduce that
(4.88)I3≤|γi+1-γi|CΩτ0,ϵ01-2|z|/x1h!(2u1)h2e-1δ2e-(δ2τ0/4)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), and for all ϵ∈ℰi+1∩ℰi.
Finally, collecting the inequalities (4.84), (4.85), and (4.88), we deduce from (4.83), that
(4.89)|Xi+1(t,z,ϵ)-Xi(t,z,ϵ)|≤h!(2/u1)h1-2|z|/x1(CΩ(di+1,ℰi)+CΩ(di,ℰi)δ2e-(δ2τ0/2)/|ϵ|ι′+|γi+1-γi|CΩτ0,ϵ02e-1δ2e-(δ2τ0/4)/|ϵ|ι′)
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, for all ϵ∈ℰi+1∩ℰi, and for all 0≤i≤ν-1. Hence, the estimates (4.79) hold.
Now, let us fixh≥0. For all 0≤i≤ν-1, we define Gh,i(ϵ):=(t,z)↦Xh,i(t,z,ϵ), which is, by Lemma  4.16, a holomorphic and bounded function from ℰi into the Banach space 𝔼=𝒪((𝒯∩D(0,ι′′))×D(0,x1/4)) of holomorphic and bounded functions on the set (𝒯∩D(0,ι′′))×D(0,x1/4) equipped with the supremum norm. Therefore, the property (1) of Theorem (RS) is satisfied for the functions Gh,i, 0≤i≤ν-1. From the estimates (4.79), we get that the cocycle Δi=Gh,i+1(ϵ)-Gh,i(ϵ) is exponentially flat of order 1 on Zi=ℰi+1∩ℰi for all 0≤i≤ν-1. We deduce that the property (2) of Theorem (RS) is fulfilled for the functions Gh,i, 0≤i≤ν-1. From Theorem (RS), we get that Gh,i(ϵ) are the 1-sums of a formal series G^h(ϵ) with coefficients in 𝔼. In particular, from Definition 4.1, we deduce the existence of the functions gh,i(s,t,z) which satisfy the expression (4.76).


### 4.7. Analytic Transseries Solutions for a Singularly Perturbed Cauchy Problem

We keep the notations of the previous section.Proposition 4.17.
The following singularly perturbed Cauchy problem(4.90)ϵt2∂t∂zSZ0(t,z,ϵ)+(ϵt+1)∂zSZ0(t,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)ts(∂tk0∂zk1Z0)(t,z,ϵ)
for given initial data
(4.91)(∂zjZ0)(t,0,ϵ)=γ0,j(t,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!φh,0,j(ϵt,ϵ),0≤j≤S-1,
which are holomorphic and bounded functions on (𝒯∩D(0,ι′′))×(ℰ0∩ℰ1), has a solution
(4.92)Z0(t,z,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!Xh,0(t,z,ϵ),
which defines a holomorphic and bounded function on (𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1), for some ι′′,δZ0>0.Proof.
Leth≥0 and 0≤j≤S-1. By construction, we have that φh,0,j(ϵt,ϵ)=(∂zjXh,0)(t,0,ϵ) for all t∈𝒯 and all ϵ∈ℰ0. From Lemma  4.16, (1), we get that there exist a constant ι′′>0, a constant u1 such that 0<u1<u0 (depending on u0, 𝒮 and b, σ), and a constant Cˇ0>0 (depending on max0≤j≤S-1Wj,Sd0,ℰ0(u0) (where Wj,Sd0,ℰ0 are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.93)supt∈𝒯∩D(0,ι′′)|φh,0,j(ϵt,ϵ)|≤h!(2u1)hCˇ0
for all ϵ∈ℰ0, all 0≤j≤S-1, all h≥0. From (4.93) and from the property (3) of Definition 4.14, we deduce the estimates
(4.94)supt∈𝒯∩D(0,ι′′)|γ0,j(t,ϵ)|≤Cˇ0∑h≥0‍(2exp(-(|λ|/ϵ0ι′′)cos(π/2-δ𝒯))u1)h,
for all ϵ∈ℰ0∩ℰ1. This latter sum converges provided that ϵ0 is small enough. We deduce that γ0,j(t,ϵ) defines a holomorphic and bounded function on (𝒯∩D(0,ι′′))×(ℰ0∩ℰ1).
Likewise, from (4.78) and from the property (3) of Definition 4.14, we deduce that there exist a constant ι′′>0, a constant u1 such that 0<u1<u0 (depending on u0, 𝒮 and b, σ), a constant x1 such that 0<x1<ρ (depending on S, u0, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ) and a constant C~0>0 (depending on max0≤j≤S-1Wj,Sd0,ℰ0(u0) (where Wj,Sd0,ℰ0 are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.95)supt∈𝒯∩D(0,ι′′),z∈D(0,δZ0)|Z0(t,z,ϵ)|≤C~01-2δZ0/x1∑h≥0‍(2exp(-(|λ|/ϵ0ι′′)cos(π/2-δ𝒯))u1)h
for all ϵ∈ℰ0∩ℰ1. Again, this latter sum converges if ϵ0 is small enough and if 0<δZ0≤x1/4. We get that Z0(t,z,ϵ) defines a holomorphic and bounded function on (𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1). By construction, we have that (∂zjZ0)(t,0,ϵ)=γ0,j(t,ϵ), for 0≤j≤S-1. Finally, from Proposition 4.3, we deduce that Z0(t,z,ϵ) solves (4.90).


## 4.1. Laplace Transform and Asymptotic Expansions

We recall the definition of Borel summability of formal series with coefficients in a Banach space, see [27].Definition 4.1.
A formal series(4.1)X^(t)=∑j=0∞‍ajj!tj∈𝔼[[t]]
with coefficients in a Banach space (𝔼,∥·∥𝔼) is said to be 1-summable with respect to t in the direction d∈[0,2π) if (i)
there existsρ∈ℝ+ such that the following formal series, called formal Borel transform of X^ of order 1,
(4.2)ℬ(X^)(τ)=∑j=0∞‍ajτj(j!)2∈𝔼[[τ]]
is absolutely convergent for |τ|<ρ;(ii)
there existsδ>0 such that the series ℬ(X^)(τ) can be analytically continued with respect to τ in a sector Sd,δ={τ∈ℂ*:|d-arg(τ)|<δ}. Moreover, there exist C>0 and K>0 such that
(4.3)∥ℬ(X^)(τ)∥𝔼≤CeK|τ|
for all τ∈Sd,δ. We say that ℬ(X^)(τ) has exponential growth of order 1 on Sd,δ.If this is so, the vector valued Laplace transform of order1 of ℬ(X^)(τ) in the direction d is defined by
(4.4)ℒd(ℬ(X^))(t)=t-1∫Lγ‍ℬ(X^)(τ)e-(τ/t)dτ
along a half-line Lγ=ℝ+eiγ⊂Sd,δ∪{0}, where γ depends on t and is chosen in such a way that cos(γ-arg(t))≥δ1>0, for some fixed δ1, for all t in a sector
(4.5)Sd,θ,R={t∈ℂ*:|t|<R,|d-arg(t)|<θ2},
where π<θ<π+2δ and 0<R<δ1/K. The function ℒd(ℬ(X^))(t) is called the 1-sum of the formal series X^(t) in the direction d. The function ℒd(ℬ(X^))(t) is a holomorphic and a bounded function on the sector Sd,θ,R. Moreover, the function ℒd(ℬ(X^))(t) has the formal series X^(t) as Gevrey asymptotic expansion of order 1 with respect to t on Sd,θ,R. This means that for all 0<θ1<θ, there exist C,M>0 such that
(4.6)∥ℒd(ℬ(X^))(t)-∑p=0n-1‍app!tp∥𝔼≤CMnn!|t|n
for all n≥1, all t∈Sd,θ1,R.In the next proposition, we recall some well-known identities for the Borel transform that will be useful in the sequel.Proposition 4.2.
LetX^(t)=∑n≥0‍antn/n! and G^(t)=∑n≥0‍bntn/n! be formal series in 𝔼[[t]]. One has the following equalities as formal series in 𝔼[[τ]]:
(4.7)(τ∂τ2+∂τ)(ℬ(X^)(τ))=ℬ(∂tX^(t))(τ),∂τ-1(ℬ(X^))(τ)=ℬ(tX^(t))(τ),τℬ(X^)(τ)=ℬ((t2∂t+t)X^(t))(τ).


## 4.2. Formal Transseries Solutions for an Auxiliary Singular Cauchy Problem

LetS≥1 be an integer. Let 𝒮 be a finite subset of ℕ3 and let
(4.8)bs,k0,k1(z,ϵ)=∑β≥0bs,k0,k1,β(ϵ)zββ!
be holomorphic and bounded functions on a polydisc D(0,ρ)×D(0,ϵ0), for some ρ,ϵ0>0, with ϵ0<1, for all (s,k0,k1)∈𝒮. We consider the following singular Cauchy problems:
(4.9)T2∂T∂zSY^(T,z,ϵ)+(T+1)∂zSY^(T,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)ϵk0-sTs(∂Tk0∂zk1Y^)(T,z,ϵ),
for given formal transseries initial conditions
(4.10)(∂zjY^)(T,0,ϵ)=∑h≥0‍exp(-hλ/T)h!φ^h,j(T,ϵ),0≤j≤S-1,
where φ^h,j(T,ϵ)=∑m≥0‍φh,j,m(ϵ)Tm/m!∈ℂ[[T]] for all ϵ∈ℰ and λ∈ℂ*.Proposition 4.3.
The problem (4.9), (4.10) has a formal transseries solutions
(4.11)Y^(T,z,ϵ)=∑h≥0‍exp(-hλ/T)h!Y^h(T,z,ϵ),
where the formal series Y^h(T,z,ϵ)∈ℂ[[T,z]], for all ϵ∈ℰ, all h≥0, satisfy the following singular Cauchy problems:
(4.12)T2∂T∂zSY^h(T,z,ϵ)+(T+1+λh)∂zSY^h(T,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)(∑q=1k01‍ϵk0-sTs(∂Tk0∂zk1Y^h)(T,z,ϵ)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍cqk01(hλ)qϵk0-sTs-(k01+q)∂Tk02∂zk1Y^h(T,z,ϵ))
with initial conditions
(4.13)(∂zjY^h)(T,0,ϵ)=φ^h,j(T,ϵ),0≤j≤S-1,
for some real numbers cqk01, for 1≤q≤k01 and 1≤k01≤k0.Proof.
We have that(4.14)∂T(exp(-hλT)Y^h(T,z,ϵ))=exp(-hλT)(hλT2Y^h(T,z,ϵ)+∂TY^h(T,z,ϵ)),
and from the Leibniz rule we also have
(4.15)∂Tk0(exp(-hλT)Y^h(T,z,ϵ))=∑k01+k02=k0‍k0!k01!k02!∂Tk01(exp(-hλT))∂Tk02Y^h(T,z,ϵ).
On the other hand, by the Faa Di Bruno formula we have, for all k01≥1, that
(4.16)∂Tk01(exp(-hλT))=∑q=1k01‍exp(-λhT)∑(λ1,…,λk01)∈Aq,k01‍k01!∏i=1k01‍((-1)i+1(hλ/Ti+1))λiλi!=exp(-hλT)(∑q=1k01‍cqk01(hλ)qTk01+q),
where Aq,k01={(λ1,…,λk01)∈ℕk01/∑i=1k01‍λi=q,∑i=1k01‍iλi=k01} and cqk01∈ℝ, for all q=1,…,k01.
Using the expressions (4.14), (4.15), (4.16), by plugging the formal expansion Y^(T,z,ϵ) into the problem (4.9), (4.10) and by identification of the coefficients of exp(-hλ/T) we get that Y^h satisfies the problem (4.12), (4.13).


## 4.3. Formal Solutions to a Sequence of Regular Cauchy Problems

Proposition 4.4.
One makes the assumption that(4.17)S>k1,s≥2k0
for all (s,k0,k1)∈𝒮. Then, the problem (4.12), (4.13) has a unique formal solution Y^h(T,z,ϵ)∈ℂ[[T,z]] for all ϵ∈ℰ. Let
(4.18)Y^h(T,z,ϵ)=∑m≥0‍Yh,m(z,ϵ)Tmm!,
where Yh,m(z,ϵ)∈ℂ[[z]], be the formal solution of (4.12), (4.13) for all ϵ∈ℰ. One denotes by
(4.19)Vh(τ,z,ϵ)=∑m≥0‍Yh,m(z,ϵ)τm(m!)2
the formal Borel transform of Y^h with respect to T. Then, for all h≥0, Vh(τ,z,ϵ) satisfies the problem
(4.20)(τ+1+λh)∂zSVh(τ,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)(ϵk0-s∑(r,p)∈𝒪s-k01‍αr,p1τr∂τ-p∂zk1Vh(τ,z,ϵ)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍cqk01(hλ)qϵk0-s×∑(r,p)∈𝒪s-k0-q2‍αr,p2,qτr∂τ-p∂zk1Vh(τ,z,ϵ))
with initial data
(4.21)(∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ)=∑m≥0‍φh,j,m(ϵ)τm(m!)2∈ℂ[[τ]],0≤j≤S-1,
where 𝒪s-k01 is a finite subset of ℕ2 such that (r,p)∈𝒪s-k01 implies r+p=s-k0 and 𝒪s-k0-q2 is a finite subset of ℕ2 such that (r,p)∈𝒪s-k0-q2 implies r+p=s-k0-q, and αr,p1, αr,p2,q are integers.Proof.
The proof follows by direct computation on the problems (4.12) and (4.13), using Proposition 4.2 and the following two lemmas from [2].
Lemma  4.5.For allk0≥1, there exist constantsak,k0∈ℕ,k0≤k≤2k0, such that(4.22)(τ∂τ2+∂τ)k0u(τ)=∑k=k02k0‍ak,k0τk-k0∂τku(τ)for all holomorphic functionsu:Ω→ℂon an open setΩ⊂ℂ.
Lemma  4.6.Leta,b,c≥0be positive integers such thata≥banda≥c. We putδ=a+b-c. Then, for all holomorphic functionsu:Ω→ℂ, the function∂τ-a(τb∂τcu(τ))can be written in the form(4.23)∂τ-a(τb∂τcu(τ))=∑(b′,c′)∈𝒪δ‍αb′,c′τb′∂τc′u(τ),where𝒪δis a finite subset ofℤ2such that for all(b′,c′)∈𝒪δ,b′-c′=δ,b′≥0,c′≤0, andαb′,c′∈ℤ.


## 4.4. An Auxiliary Cauchy Problem

We denote byΩ1 an open star-shaped domain in ℂ (meaning that Ω1 is an open subset of ℂ such that for all x∈Ω1, the segment [0,x] belongs to Ω1). Let Ω2 be an open set in ℂ* contained in the disc D(0,ϵ0). We denote by Ω=Ω1×Ω2. For any open set 𝒟⊂ℂ, we denote by 𝒪(𝒟) the vector space of holomorphic functions on 𝒟.Definition 4.7.
Letb>1 a real number and let rb(β)=∑n=0β‍1/(n+1)b for all integers β≥0. Let ϵ∈Ω2 and σ>0 be a real number. We denote by Eβ,ϵ,σ,Ω the vector space of all functions v∈𝒪(Ω1) such that
(4.24)∥v(τ)∥β,ϵ,σ,Ω∶=supτ∈Ω1|v(τ)|(1+|τ|2|ϵ|2)exp(-σ2|ϵ|rb(β)|τ|)
is finite.Proposition 4.8.
One makes the assumption that(4.25)S>k1,s≥2k0
for all (s,k0,k1)∈𝒮. Moreover, one makes the assumption that there exists c′,δ′>0 such that
(4.26)|τ+1+hλ|≥c′|τ+1|>δ′,∀τ∈Ω1,∀h∈ℕ.
For all h≥0, all ϵ∈Ω2, the problem (4.20) with initial conditions
(4.27)(∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ)∈𝒪(Ω1),0≤j≤S-1
has a unique formal series
(4.28)Vh(τ,z,ϵ)=∑β≥0‍vh,β(τ,ϵ)zββ!∈𝒪(Ω1)[[z]],
where vh,β(τ,ϵ) satisfies the following recursion:
(4.29)(τ+1+hλ)vh,β+S(τ,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!ϵk0-s(∑(r,p)∈𝒪s-k01‍αr,p1τr∂τ-pvh,β2+k1(τ,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-s(∑(r,p)∈𝒪s-k0-q2‍αr,p2,qτr∂τ-pvh,β2+k1(τ,ϵ)β2!)
for all τ∈Ω1, all ϵ∈Ω2.Proposition 4.9.
One makes the assumption that(4.30)S>k1,s≥2k0
for all (s,k0,k1)∈𝒮. Let also the assumption (4.26) holds. Let us assume that
(4.31)vh,j(τ,ϵ)∈Ej,ϵ,σ,Ω,∀h≥0,∀0≤j≤S-1,∀ϵ∈Ω2.
Then, one has that vh,β(τ,ϵ)∈Eβ,ϵ,σ,Ω for all β≥0, all h≥0, all ϵ∈Ω2. We put vh,β(ϵ)=||vh,β(τ,ϵ)||β,ϵ,σ,Ω, for all h≥0, all β≥0, and all ϵ∈Ω2. Then, the following inequalities hold: there exist two constants C181,C182>0 (depending on 𝒮,σ,S) such that
(4.32)vh,β+S(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!|bs,k0,k1,β1(ϵ)|β1!×C181((β+S+1)b(s-k0)+(β+S+1)b(s-k0+2))vh,β2+k1(ϵ)β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!|bs,k0,k1,β1(ϵ)|β1!|cqk01|hq|λ|q×|ϵ|-qC182((β+S+1)b(s-k0-q)+(β+S+1)b(s-k0-q+2))vh,β2+k1(ϵ)β2!
for all h≥0, all β≥0.Proof.
The proof follows by direct computation using the recursion (4.29) and the next lemma. We keep the notations of Proposition 4.8.
Lemma  4.10.There exists a constantC18>0(depending ons,σ,S,k0,k1) such that(4.33)∥τr∂τ-pvh,β2+k1(τ,ϵ)∥β+S,ϵ,σ,Ω≤|ϵ|r+pC18((β+S+1)b(r+p)+(β+S+1)b(r+p+2))∥vh,β2+k1(τ,ϵ)∥β2+k1,ϵ,σ,Ωfor allh≥0, and allβ≥0,0≤β2≤β, all(r,p)∈ℕ2withr+p≤s-k0.
Proof. We follow the proof of Lemma 1 from [2]. By definition, we have that ∂τ-1vh,β2+k1(τ,ϵ)=∫0τ‍vh,β2+k1(τ1,ϵ)dτ1 for all τ∈Ω1. Using the parametrization τ1=h1τ with 0≤h1≤1, we get that
(4.34)∂τ-1vh,β2+k1(τ,ϵ)=τ∫01‍vh,β2+k1(h1τ,ϵ)M1(h1)dh1,
where M1(h1)=1. More generally, for all p≥2, we have by definition:
(4.35)∂τ-pvh,β2+k1(τ,ϵ)=∫0τ‍∫0τ1‍⋯∫0τp-1‍vh,β2+k1(τp,ϵ)dτpdτp-1⋯dτ1
for all τ∈Ω1. Using the parametrization τj=hjτj-1, τ1=h1τ, with 0≤hj≤1, for 2≤j≤p, we can write
(4.36)∂τ-pvh,β2+k1(τ,ϵ)=τp∫01‍⋯∫01‍vh,β2+k1(hp⋯h1τ,ϵ)Mp(h1,…,hp)dhpdhp-1⋯dh1,
where Mp(h1,…,hp) is a monomial in h1,…,hp whose coefficient is equal to 1. Using these latter expressions, we now write
(4.37)|τr∂τ-pvh,β2+k1(τ,ϵ)|=|τr+p∫01‍⋯∫01‍vh,β2+k1(hp⋯h1τ,ϵ)(1+|hp⋯h1τ|2|ϵ|2)exp(-σ2|ϵ|rb(β2+k1)|hp⋯h1τ|)×exp((σ/2|ϵ|)rb(β2+k1)|hp⋯h1τ|)1+|hp⋯h1τ|2/|ϵ|2Mp(h1,…,hp)dhp…dh1|.
Therefore,
(4.38)|τr∂τ-pvh,β2+k1(τ,ϵ)|(1+|τ|2|ϵ|2)exp(-σ2|ϵ|rb(β+S)|τ|)≤∥vh,β2+k1(τ,ϵ)∥β2+k1,ϵ,σ,Ω|τ|r+p(1+|τ|2|ϵ|2)exp(-σ2|ϵ|(rb(β+S)-rb(β2+k1))|τ|).
By construction of rb(β), we have
(4.39)rb(β+S)-rb(β2+k1)=∑n=β2+k1+1β+S‍1(n+1)b≥β-β2+S-k1(β+S+1)b≥S-k1(β+S+1)b
for all β≥0. From (4.38) and (4.39), we get that
(4.40)|τr∂τ-pvh,β2+k1(τ,ϵ)|(1+|τ|2|ϵ|2)exp(-σ2|ϵ|rb(β+S)|τ|)≤∥vh,β2+k1(τ,ϵ)∥β2+k1,ϵ,σ,Ω|τ|r+p(1+|τ|2|ϵ|2)exp(-σ2|ϵ|S-k1(β+S+1)b|τ|)
for all β≥0. From (2.41), we deduce that
(4.41)|τ|r+p(1+|τ|2|ϵ|2)exp(-σ2|ϵ|S-k1(β+S+1)b|τ|)≤|ϵ|r+p((2(r+p+2)e-1σ(S-k1))r+p+2(2(r+p)e-1σ(S-k1))r+p(β+S+1)b(r+p)+(2(r+p+2)e-1σ(S-k1))r+p+2(β+S+1)b(r+p+2))
for all τ∈Ω1. From the estimates (4.40) and (4.41), we deduce the inequality (4.33).Proposition 4.11.
Assume that the conditions (4.26) and (4.31) hold. Assume moreover, that
(4.42)S≥b(s-k0+2)+k1,s≥2k0
for all (s,k0,k1)∈𝒮 and that the following sums converge near the origin in ℂ,
(4.43)Wj(u):=∑h≥0‍supϵ∈Ω2∥vh,j(τ,ϵ)∥j,ϵ,σ,Ωuhh!∈ℂ{u},0≤j≤S-1.
One make also the hypothesis that for all (s,k0,k1)∈𝒮, one can write
(4.44)bs,k0,k1(z,ϵ)=ϵk0b~s,k0,k1(z,ϵ),
where b~s,k0,k1(z,ϵ)=∑β≥0‍b~s,k0,k1,β(ϵ)zβ/β! is holomorphic for all ϵ∈D(0,ϵ0) on D(0,ρ). Then, the problem (4.20) with initial data
(4.45)(∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ),0≤j≤S-1
has a unique solution Vh(τ,z,ϵ) which is holomorphic with respect to (τ,z)∈Ω1×D(0,x1/2) for all ϵ∈Ω2.
The constantx1 is such that 0<x1<ρ and depends on S, u0 (which denotes a common radius of absolute convergence of the series (4.43)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ and |b|s,k0,k1, |b~|s,k0,k1 are defined below.
Moreover, the following estimates hold: there exists a constantu1 such that 0<u1<u0 (depending on u0, 𝒮, and b, σ) and a constant C19>0 (depending on max0≤j≤S-1Wj(u0) (where Wj are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.46)|Vh(τ,z,ϵ)|≤C191-2|z|/x1h!(2u1)h(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|ζ(b)|τ|)
for all (τ,z)∈Ω1×D(0,x1/2), all ϵ∈Ω2, and all h≥0.Proof.
We consider the following Cauchy problem(4.47)∂xSW(u,x)=∑(s,k0,k1)∈𝒮C181((x∂x+S+1)b(s-k0)‍+(x∂x+S+1)b(s-k0+2))(|b|s,k0,k1(x)∂xk1W(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C182|cqk01||λ|q×((x∂x+S+1)b(s-k0-q)+(x∂x+S+1)b(s-k0-q+2))×(|b~|s,k0,k1(x)(u∂u)q∂xk1W(u,x))
for given initial data
(4.48)(∂xjW)(u,0)=Wj(u)=∑h≥0‍supϵ∈Ω2|vh,j(ϵ)|uhh!∈ℂ{u},0≤j≤S-1,
where
(4.49)|b|s,k0,k1(x)=∑β≥0‍supϵ∈D(0,ϵ0)|bs,k0,k1,β(ϵ)|xββ!,|b~|s,k0,k1(x)=∑β≥0‍supϵ∈D(0,ϵ0)|b~s,k0,k1,β(ϵ)|xββ!
are convergent series near the origin in ℂ with respect to x. From the assumption (4.42) and the fact that b>1, we also deduce that
(4.50)S≥b(s-k0-q+2)+q+k1
for all (s,k0,k1)∈𝒮 and all 0≤q≤k0. Since the initial data (4.48) and the coefficients (4.47) are analytic near the origin, we get that all the hypotheses of the classical Cauchy Kowalevski theorem from Proposition 2.22 are fulfilled. We deduce the existence of U1 with 0<U1<U0, where U0 denotes a common radius of absolute convergence for the series (4.48), which depends on U0, 𝒮 and b, and X1 with 0<X1<ρ (depending on S, U0, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(X0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(X0), where X0<ρ) such that there exist a unique formal series W(u,x)∈G(U1,X1) which solves the problem (4.47), (4.48).
Now, letW(u,x)=∑h,β≥0‍wh,β(uh/h!)(xβ/β!) be its Taylor expansion at (0,0). Then, by construction the sequence wh,β satisfies the following equalities:
(4.51)wh,β+S=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!supϵ∈D(0,ϵ0)|bs,k0,k1,β1(ϵ)|β1!C181×((β+S+1)b(s-k0)+(β+S+1)b(s-k0+2))wh,β2+k1β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!supϵ∈D(0,ϵ0)|b~s,k0,k1,β1(ϵ)|β1!|cqk01|hq|λ|q×C182((β+S+1)b(s-k0-q)+(β+S+1)b(s-k0-q+2))wh,β2+k1β2!
for all h≥0 and all β≥0, with
(4.52)wh,j=supϵ∈Ω2|vh,j(ϵ)|,∀h≥0,∀0≤j≤S-1.
Using the inequality (4.32) and the equality (4.51), with the initial conditions (4.52), one gets that
(4.53)supϵ∈Ω2|vh,β(ϵ)|≤wh,β
for all h≥0, all β≥0. Using the fact that W(u,x)∈G(U1,X1) and the estimates (2.111), we deduce from (4.53) that there exist a constant C19>0 (depending on max0≤j≤S-1Wj(U0),|λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(X0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(X0), S, U0, X0, 𝒮, b, σ) such that
(4.54)|vh,β(τ,ϵ)|≤C19(h+β)!(1U1)h(1X1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)≤C19h!β!(2U1)h(2X1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)
for all τ∈Ω1, all ϵ∈Ω2, all h≥0, and all β≥0.


## 4.5. Analytic Solutions for a Sequence of Singular Cauchy Problems

Assume that the conditions (4.42) and (4.44) hold. We consider the following problem:
(4.55)T2∂T∂zSYh,Sd,ℰ(T,z,ϵ)+(T+1+λh)∂zSYh,Sd,ℰ(T,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)(∑q=1k01‍ϵk0-sTs(∂Tk0∂zk1Yh,Sd,ℰ)(T,z,ϵ)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍cqk01(hλ)qϵk0-sTs-(k01+q)∂Tk02∂zk1Yh,Sd,ℰ(T,z,ϵ))
with initial conditions
(4.56)(∂zjYh,Sd,ℰ)(T,0,ϵ)=φh,j,Sd,ℰ(T,ϵ),0≤j≤S-1.
The initial conditions φh,j,Sd,ℰ(T,ϵ), 0≤j≤S-1 are defined as follows. Let Sd be an open sector centered at 0, with infinite radius and bisecting direction d∈[0,2π), D(0,τ0) an open disc centered at 0 with radius τ0>0, and ℰ an open sector centered at 0 contained in the disc D(0,ϵ0). We make the assumption that the condition (4.26) holds for the set Ω1=(Sd∪D(0,τ0)). We consider a set of functions vh,j(τ,ϵ)∈Ej,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0}) for all ϵ∈D(0,ϵ0)∖{0} such that
(4.57)Wj,τ0,ϵ0(u)∶=∑h≥0‍supϵ∈D(0,ϵ0)∖{0}∥vh,j(τ,ϵ)∥j,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0})uhh!∈ℂ{u},0≤j≤S-1.
We also assume that for all h≥0 and all 0≤j≤S-1, vh,j(τ,ϵ) has an analytic continuation denoted by vh,j,Sd,ℰ(τ,ϵ)∈Ej,ϵ,σ,(Sd∪D(0,τ0))×ℰ for all ϵ∈ℰ such that
(4.58)Wj,Sd,ℰ(u)∶=∑h≥0‍supϵ∈ℰ∥vh,j,Sd,ℰ(τ,ϵ)∥j,ϵ,σ,(Sd∪D(0,τ0))×ℰuhh!∈ℂ{u},0≤j≤S-1.
Let
(4.59)vh,j(τ,ϵ)=∑m≥0‍φh,j,m(ϵ)τm(m!)2
be the convergent Taylor expansion of vh,j with respect to τ on D(0,τ0) for all ϵ∈D(0,ϵ0)∖{0}. We consider the formal series
(4.60)φ^h,j(T,ϵ)=∑m≥0‍φh,j,m(ϵ)Tmm!
for all ϵ∈D(0,ϵ0)∖{0}. We define φh,j,Sd,ℰ(T,ϵ) as the 1-sum (in the sense of Definition 4.1) of φ^j,h(T,ϵ) in the direction d. From the hypotheses, we deduce that T↦φh,j,Sd,ℰ(T,ϵ) defines a holomorphic function for all T∈Ud,θ,ι|ϵ|, for all ϵ∈ℰ, where
(4.61)Ud,θ,ι|ϵ|={T∈ℂ*:|T|<ι|ϵ|,|d-arg(T)|<θ2}
for some θ>π and some constant ι>0 (independent of ϵ) for all 0≤j≤S-1.Proposition 4.12.
Assume that the conditions (4.26), (4.31), (4.42), and (4.44) hold.
Then, the problem (4.55), (4.56) has a solution (T,z)↦Yh,Sd,ℰ(T,z,ϵ) which is holomorphic and bounded on the set Ud,θ,ι′|ϵ|×D(0,x1/4), for some ι′>0 (independent of ϵ), for all ϵ∈ℰ, where 0<x1<ρ depends on S, u0 (which denotes a common radius of absolute convergence of the series (4.57), (4.58)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ.
The functionYh,Sd,ℰ(T,z,ϵ) can be written as the Laplace transform of order 1 in the direction d (in the sense of Definition 4.1) of a function Vh,Sd,ℰ(τ,z,ϵ), which is holomorphic on the domain (Sd∪D(0,τ0))×D(0,x1/2)×ℰ and satisfies the following estimates.
There exists a constantu1 such that 0<u1<u0 (depending on u0, 𝒮 and b,σ) and a constant CΩ(d,ℰ)>0 (depending on max0≤j≤S-1Wj,Sd,ℰ(u0) (where Wj,Sd,ℰ are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.62)|Vh,Sd,ℰ(τ,z,ϵ)|≤CΩ(d,ℰ)1-2|z|/x1h!(2u1)h(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|ζ(b)|τ|)
for all (τ,z,ϵ)∈(Sd∪D(0,τ0))×D(0,x1/2)×ℰ, all h≥0.
Moreover, the functionVh,Sd,ℰ(τ,z,ϵ) is the analytic continuation of a function Vh(τ,z,ϵ) which is holomorphic on the punctured polydisc D(0,τ0)×D(0,x1/2)×(D(0,ϵ0)∖{0}) and verifies the following estimates.
There exists a constantCΩτ0,ϵ0>0 (depending on max0≤j≤S-1Wj,τ0,ϵ0(u0) (where Wj,τ0,ϵ0 are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.63)|Vh(τ,z,ϵ)|≤CΩτ0,ϵ01-2|z|/x1h!(2u1)h(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|ζ(b)|τ|)
for all τ∈D(0,τ0), all z∈D(0,x1/2), all ϵ∈D(0,ϵ0)∖{0}, and all h≥0.Proof.
From the hypotheses of Proposition4.12, we deduce from Proposition 4.11 applied to the situation Ω=D(0,τ0)×(D(0,ϵ0)∖{0}) the existence of a holomorphic function Vh(τ,z,ϵ) satisfying the estimates (4.63), which is the solution of the problem (4.20) with initial conditions (∂zjVh)(τ,0,ϵ)=vh,j(τ,ϵ), 0≤j≤S-1, on the domain D(0,τ0)×D(0,x1/2)×(D(0,ϵ0)∖{0}). Likewise, from Proposition 4.11 applied to the situation Ω=(Sd∪D(0,τ0))×ℰ, we get the existence of a holomorphic function Vh,Sd,ℰ(τ,z,ϵ) satisfying (4.62), which is the solution of the problem (4.20) with initial conditions (∂zjVh)(τ,0,ϵ)=vh,j,Sd,ℰ(τ,ϵ), 0≤j≤S-1 on the domain (Sd∪D(0,τ0))×D(0,x1/2)×ℰ.
With Proposition4.3, we deduce that the formal solution Y^h(T,z,ϵ) of the problem (4.12), (4.13) is 1-summable with respect to T in the direction d as series in the Banach space 𝒪(D(0,x1/4)), for all ϵ∈ℰ. We denote by Yh,Sd,ℰ(T,z,ϵ) its 1-sum which is holomorphic with respect to T on a domain Ud,θ,ι′|ϵ| due to Definition 4.1 and the estimates (4.62). Moreover, from the algebraic properties of the κ-summability procedure, see [27, Section 6.3], we deduce that Yh,Sd,ℰ(T,z,ϵ) is a solution of the problem (4.55), (4.56).


## 4.6. Summability in a Complex Parameter

We recall the definition of a good covering.Definition 4.13.
Letν≥2 be an integer. For all 0≤i≤ν-1, we consider open sectors ℰi centered at 0, with radius ϵ0, bisecting direction κi∈[0,2π) and opening π+δi, with δi>0, such that ℰi∩ℰi+1≠∅ for all 0≤i≤ν-1 (with the convention that ℰν=ℰ0) and such that ∪i=0ν-1ℰi=𝒰∖{0}, where 𝒰 is some neighborhood of 0 in ℂ. Such a set of sectors {ℰi}0≤i≤ν-1 is called a good covering in ℂ*.Definition 4.14.
Let{ℰi}0≤i≤ν-1 be a good covering in ℂ*. Let 𝒯 be an open sector centered at 0 with radius r𝒯 and consider a family of open sectors
(4.64)Udi,θ,ϵ0r𝒯:={t∈ℂ:|t|<ϵ0r𝒯,|di-arg(t)|<θ2},
where di∈[0,2π), for 0≤i≤ν-1, where θ>π, which satisfy the following properties:
(1) For all0≤i≤ν-1, all h∈ℕ, arg(di)≠arg(-1-λh).
(2) For all0≤i≤ν-1, for all t∈𝒯, and all ϵ∈ℰi, we have that ϵt∈Udi,θ,ϵ0r𝒯.
(3) (3.1) We assume thatd0<arg(λ)<d1. We consider the two closed sectors
(4.65)ℳd0={τ∈ℂ*arg(τ)∈[d0,arg(λ)]},ℳd1={τ∈ℂ*arg(τ)∈[arg(λ),d1]}.
We make the assumption that there exist two constants c′,δ′>0 with
(4.66)|τ+1+λh|≥c′|τ+1|>δ′
for all τ∈ℳd0∪ℳd1∪D(0,τ0) and all h≥0.
(3.2) There exists0<δ𝒯<π/2 such that arg(λ/(ϵt))∈(-π/2+δ𝒯,π/2-δ𝒯) for all ϵ∈ℰ0∩ℰ1 and all t∈𝒯.
We say that the family{{Udi,θ,ϵ0r𝒯}0≤i≤ν-1,𝒯,λ} is associated to the good covering {ℰi}0≤i≤ν-1.Now, we consider a set of functionsφh,i,j(T,ϵ) for 0≤i≤ν-1, 0≤j≤S-1, h≥0, constructed as follows. For all 0≤i≤ν-1, let Sdi be an open sector of infinite radius centered at 0, with bisecting direction di and with opening ni>θ-π. The numbers θ>π and ni>0 are chosen in such a way that -1-λh∉Sdi for all 0≤i≤ν-1 and all h≥0. Now, we put
(4.67)φh,i,j(T,ϵ):=φh,j,Sdi,ℰi(T,ϵ)
for all T∈Udi,θ,ι|ϵ| and all ϵ∈ℰi, where φh,j,Sdi,ℰi(T,ϵ) is given by the formula (4.56). Recalling how these functions are constructed, we consider a set of functions
(4.68)vh,j(τ,ϵ)∈Ej,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0})
for all ϵ∈D(0,ϵ0)∖{0} such that
(4.69)Wj,τ0,ϵ0(u)∶=∑h≥0‍supϵ∈D(0,ϵ0)∖{0}∥vh,j(τ,ϵ)∥j,ϵ,σ,D(0,τ0)×(D(0,ϵ0)∖{0})uhh!∈ℂ{u},0≤j≤S-1.
We also assume that for all h≥0, all 0≤j≤S-1, vh,j(τ,ϵ) has an analytic continuation denoted by vh,j,Sdi,ℰi(τ,ϵ)∈Ej,ϵ,σ,(Sdi∪D(0,τ0))×ℰi for all ϵ∈ℰi such that
(4.70)Wj,Sdi,ℰi(u):=∑h≥0‍supϵ∈ℰi∥vh,j,Sdi,ℰi(τ,ϵ)∥j,ϵ,σ,(Sdi∪D(0,τ0)×ℰiuhh!∈ℂ{u},0≤j≤S-1.
Let
(4.71)vh,j(τ,ϵ)=∑m≥0‍φh,j,m(ϵ)τm(m!)2
be the convergent Taylor expansion of vh,j with respect to τ on D(0,τ0) for all ϵ∈D(0,ϵ0)∖{0}. We consider the formal series
(4.72)φ^h,j(T,ϵ)=∑m≥0‍φh,j,m(ϵ)Tmm!
for all ϵ∈D(0,ϵ0)∖{0}. We define φh,j,Sdi,ℰi(T,ϵ) as the 1-sum (in the sense of Definition 4.1) of φ^j,h(T,ϵ) in the direction di. We deduce that T↦φh,j,Sdi,ℰi(T,ϵ) defines a holomorphic function for all T∈Udi,θ,ι|ϵ| and for all ϵ∈ℰi, where
(4.73)Udi,θ,ι|ϵ|={T∈ℂ*:|T|<ι|ϵ|,|di-arg(T)|<θ2}
for some θ>π and some constant ι>0 (independent of ϵ) for all 0≤j≤S-1.From Proposition4.12, for all 0≤i≤ν-1, we consider the solution Yh,Sdi,ℰi(T,z,ϵ) of the problem (4.55) with the initial conditions
(4.74)(∂zjYh,Sdi,ℰi)(T,0,ϵ)=φh,i,j(T,ϵ),0≤j≤S-1,h≥0,
which defines a bounded and holomorphic function on Udi,θ,ι′|ϵ|×D(0,x1/4)×ℰi.Proposition 4.15.
The function defined by(4.75)Xh,i(t,z,ϵ)=Yh,Sdi,ℰi(ϵt,z,ϵ)
is holomorphic and bounded on (𝒯∩D(0,ι′′))×D(0,x1/4)×ℰi, for all h≥0, all 0≤i≤ν-1, and for some 0<ι′′<ι′.
Moreover, the functionsGh,i:ϵ↦Xh,i(t,z,ϵ) from ℰi into the Banach space 𝒪((𝒯∩D(0,ι′′))×D(0,x1/4)) are the 1-sums on ℰi of a formal series G^h(ϵ)∈𝒪((𝒯∩D(0,ι′′))×D(0,x1/4))[[ϵ]]. In other words, for all h≥0, there exists a function gh(s,t,z) which is holomorphic on D(0,sh)×(𝒯∩D(0,ι′′))×D(0,x1/4) which admits for all 0≤i≤ν-1, an analytic continuation gh,i(s,t,z) which is holomorphic on (𝒢κi∪D(0,sh))×(𝒯∩D(0,ι′′))×D(0,x1/4), where 𝒢κi is an open sector centered at 0 with infinite radius and bisecting direction κi such that
(4.76)Xh,i(t,z,ϵ)=ϵ-1∫Lκi‍gh,i(s,t,z)e-s/ϵds
along a half-line Lκi=ℝ+eiκi⊂𝒢κi∪{0}.Proof.
The proof is based on a cohomological criterion for summability of formal series with coefficients in a Banach space, see [27, page 121], which is known as the Ramis-Sibuya theorem in the literature.
Theorem (RS).Let(𝔼,||·||𝔼)be a Banach space overℂand{ℰi}0≤i≤ν-1a good covering inℂ*. For all0≤i≤ν-1, letGibe a holomorphic function fromℰiinto the Banach space(𝔼,||·||𝔼)and let the cocycleΔi(ϵ)=Gi+1(ϵ)-Gi(ϵ)be a holomorphic function from the sectorZi=ℰi+1∩ℰiinto𝔼(with the convention thatℰν=ℰ0andGν=G0). We make the following assumptions.(1)
The functionsGi(ϵ)are bounded asϵ∈ℰitends to the origin inℂfor all0≤i≤ν-1.(2)
The functionsΔi(ϵ)are exponentially flat of order 1 onZifor all0≤i≤ν-1. This means that there exist constantsCi,Ai>0such that(4.77)∥Δi(ϵ)∥𝔼≤Cie-Ai/|ϵ|for allϵ∈Ziall0≤i≤ν-1.
Then, for all
0
≤
i
≤
ν
-
1
, the functions
G
i
(
ϵ
)
are the
1
-sums onℰiof a
1
-summable formal seriesG^(ϵ)inϵwith coefficients in the Banach space𝔼.
By Definition4.14 and the construction of Yh,Sdi,ℰi(T,z,ϵ) in Proposition 4.12, we get that the function Xh,i(t,z,ϵ)=Yh,Sdi,ℰi(ϵt,z,ϵ) defines a bounded and holomorphic function on the domain (𝒯∩D(0,ι′))×D(0,x1/4)×ℰi for all h≥0 all 0≤i≤ν-1, where 0<x1<ρ depends on S, u0>0 (which denotes a common radius of absolute convergence of the series (4.69), (4.70)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ. More precisely, we have the following.
Lemma  4.16.Consider the following:(1)
There exist a constant0<ι′′<ι′, a constantu1such that0<u1<u0(depending onu0,𝒮andb,σ), a constantx1such that0<x1<ρ(depending onS,u0,𝒮,b,σ,|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), wherex0<ρ), and a constantC~i>0(depending onmax0≤j≤S-1Wj,Sdi,ℰi(u0)(whereWj,Sdi,ℰiare defined above),|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0),S,u0,x0,𝒮,b) such that(4.78)supt∈𝒯∩D(0,ι′′),z∈D(0,x1/4)|Xh,i(t,z,ϵ)|≤2C~ih!(2u1)hfor allϵ∈ℰi, for all0≤i≤ν-1, and allh≥0.(2)
There exist a constant0<ι′′≤ι′, a constantu1such that0<u1<u0(depending onu0,𝒮andb,σ), a constantx1such that0<x1<ρ(depending onS,u0,𝒮,b,σ,|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), wherex0<ρ), a constantMi>0, a constantKi>0(depending onmax0≤j≤S-1Wj,Sdq,ℰq(u0), forq=i,i+1(whereWj,Sdq,ℰqare defined above),max0≤j≤S-1Wj,τ0,ϵ0(u0),|λ|,max(s,k0,k1)∈𝒮|b|s,k0,k1(x0),max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0),S,u0,x0,𝒮,b) such that(4.79)supt∈𝒯∩D(0,ι′′),z∈D(0,x1/4)|Xh,i+1(t,z,ϵ)-Xh,i(t,z,ϵ)|≤h!(2u1)h2Kie-Mi/|ϵ|for allϵ∈ℰi∩ℰi+1, for all0≤i≤ν-1, and allh≥0(where by conventionXh,ν=Xh,0).
Proof. (1) Let i be an integer such that 0≤i≤ν-1. From Proposition 4.12, we can write
(4.80)Xh,i(t,z,ϵ)=(ϵt)-1∫Lγi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ,
where Lγi=ℝ+e-1γi⊂Sdi∪{0} and Vh,Sdi,ℰi is a holomorphic function on (Sdi∪D(0,τ0))×D(0,x1/4)×ℰi for which the estimates (4.62) hold. By construction, the direction γi (which depends on ϵt) is chosen in such a way that cos(γi-arg(ϵt))≥δ1, for all ϵ∈ℰi, all t∈𝒯∩D(0,ι′), and for some fixed δ1>0. From the estimates (4.62), we get
(4.81)|Xh,i(t,z,ϵ)|≤|ϵt|-1∫0+∞‍CΩ(di,ℰi)1-2|z|/x1h!(2u1)h(1+r2|ϵ|2)-1eσζ(b)r/2|ϵ|e-r/|ϵ||t|cos(γi-arg(ϵt))dr≤|ϵt|-1∫0+∞‍CΩ(di,ℰi)1-2|z|/x1h!(2u1)he(σζ(b)/2-δ1/|t|)(r/|ϵ|)dr=CΩ(di,ℰi)1-2|z|/x1h!(2u1)h1δ1-(σζ(b)/2)|t|≤CΩ(di,ℰi)δ2(1-2|z|/x1)h!(2u1)h
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi.
(2) Leti an integer such that 0≤i≤ν-1. From Proposition 4.12, we can write again
(4.82)Xh,i(t,z,ϵ)=(ϵt)-1∫Lγi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ,Xi+1(t,z,ϵ)=(ϵt)-1∫Lγi+1‍Vh,Sdi+1,ℰi+1(τ,z,ϵ)e-τ/ϵtdτ,
where Lγi=ℝ+e-1γi⊂Sdi∪{0}, Lγi+1=ℝ+e-1γi+1⊂Sdi+1∪{0}, and Vh,Sdi,ℰi (resp., Vh,Sdi+1,ℰi+1) is a holomorphic function on (Sdi∪D(0,τ0))×D(0,x1/4)×ℰi (resp., on (Sdi+1∪D(0,τ0))×D(0,x1/4)×ℰi+1) for which the estimates (4.62) hold and which is moreover an analytic continuation of a function Vh(τ,z,ϵ) which satisfies the estimates (4.63).
From the fact thatτ↦Vh(τ,z,ϵ) is holomorphic on D(0,τ0) for all (z,ϵ)∈D(0,x1/4)×(D(0,ϵ0)∖{0}), the integral of τ↦Vh(τ,z,ϵ) along the union of a segment starting from 0 to (τ0/2)e-1γi+1, an arc of circle with radius τ0/2 connecting (τ0/2)e-1γi+1 and (τ0/2)e-1γi and a segment starting from (τ0/2)e-1γi to 0 is equal to zero. Therefore, we can rewrite the difference Xh,i+1-Xh,i as a sum of three integrals:
(4.83)Xh,i+1(t,z,ϵ)-Xh,i(t,z,ϵ)=(ϵt)-1(∫Lτ0/2,γi+1‍Vh,Sdi+1,ℰi+1(τ,z,ϵ)e-τ/ϵtdτ-∫Lτ0/2,γi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ+∫C(τ0/2,γi,γi+1)‍Vh(τ,z,ϵ)e-τ/ϵtdτ),
where Lτ0/2,γi=[τ0/2,+∞)e-1γi, Lτ0/2,γi+1=[τ0/2,+∞)e-1γi+1, and C(τ0/2,γi,γi+1) is an arc of circle with radius τ0/2 connecting (τ0/2)e-1γi with (τ0/2)e-1γi+1 with a well-chosen orientation.
We give estimates forI1=|(ϵt)-1∫Lτ0/2,γi+1‍Vh,Sdi+1,ℰi+1(τ,z,ϵ)e-τ/ϵtdτ|. By construction, the direction γi+1 (which depends on ϵt) is chosen in such a way that cos(γi+1-arg(ϵt))≥δ1, for all ϵ∈ℰi+1∩ℰi, all t∈𝒯∩D(0,ι′), and for some fixed δ1>0. From the estimates (4.62), we get
(4.84)I1≤|ϵt|-1∫τ0/2+∞‍CΩ(di+1,ℰi+1)1-2|z|/x1h!(2u1)h(1+r2|ϵ|2)-1eσζ(b)r/2|ϵ|e-(r/|ϵ||t|)cos(γi+1-arg(ϵt))dr≤|ϵt|-1∫τ0/2+∞‍CΩ(di+1,ℰi+1)1-2|z|/x1h!(2u1)he(σζ(b)/2-δ1/|t|)(r/|ϵ|)dr=CΩ(di+1,ℰi+1)1-2|z|/x1h!(2u1)he-((δ1/|t|-σζ(b)/2)(τ0/2))(1/|ϵ|)δ1-(σζ(b)/2)|t|≤CΩ(di+1,ℰi+1)δ2(1-2|z|/x1)h!(2u1)he-(δ2τ0/2)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi.
We give estimates forI2=|(ϵt)-1∫Lτ0/2,γi‍Vh,Sdi,ℰi(τ,z,ϵ)e-τ/ϵtdτ|. By construction, the direction γi (which depends on ϵt) is chosen in such a way that there exists a fixed δ1>0 with cos(γi-arg(ϵt))≥δ1, for all ϵ∈ℰi+1∩ℰi and all t∈𝒯∩D(0,ι′). From the estimates (4.62), we deduce as before that
(4.85)I2≤CΩ(di,ℰi)δ2(1-2|z|/x1)h!(2u1)he-(δ2τ0/2)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi.
Finally, we get estimates forI3=|ϵt|-1|∫C(τ0/2,γi,γi+1)‍Vh(τ,z,ϵ)e-τ/ϵtdτ|. From the estimates (4.63), we have
(4.86)I3≤|ϵt|-1|∫γiγi+1‍CΩτ0,ϵ01-2|z|/x1h!(2u1)h(1+(τ0/2)2|ϵ|2)-1eσζ(b)τ0/4|ϵ|e-(τ0/2|ϵ||t|)cos(θ-arg(ϵt))τ02dθ|.
By construction, the arc of circle C(τ0/2,γi,γi+1) is chosen in such a way that that cos(θ-arg(ϵt))≥δ1 for all θ∈[γi,γi+1] (if γi<γi+1), θ∈[γi+1,γi] (if γi+1<γi) for all t∈𝒯, all ϵ∈ℰi∩ℰi+1. From (4.86), we deduce that
(4.87)I3≤|γi+1-γi|CΩτ0,ϵ01-2|z|/x1h!(2u1)hτ021|ϵt|e-((δ1/|t|-σζ(b)/2)(τ0/2))(1/|ϵ|)≤|γi+1-γi|CΩτ0,ϵ01-2|z|/x1h!(2u1)hτ021|ϵt|e-(δ2τ0/4)/|ϵt|e-(δ2τ0/4)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, and for all ϵ∈ℰi+1∩ℰi. Using the inequality (4.87) and the estimates (2.41), we deduce that
(4.88)I3≤|γi+1-γi|CΩτ0,ϵ01-2|z|/x1h!(2u1)h2e-1δ2e-(δ2τ0/4)/|ϵ|ι′
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), and for all ϵ∈ℰi+1∩ℰi.
Finally, collecting the inequalities (4.84), (4.85), and (4.88), we deduce from (4.83), that
(4.89)|Xi+1(t,z,ϵ)-Xi(t,z,ϵ)|≤h!(2/u1)h1-2|z|/x1(CΩ(di+1,ℰi)+CΩ(di,ℰi)δ2e-(δ2τ0/2)/|ϵ|ι′+|γi+1-γi|CΩτ0,ϵ02e-1δ2e-(δ2τ0/4)/|ϵ|ι′)
for all t∈𝒯∩D(0,ι′), with |t|<2(δ1-δ2)/(σζ(b)), for some 0<δ2<δ1, for all ϵ∈ℰi+1∩ℰi, and for all 0≤i≤ν-1. Hence, the estimates (4.79) hold.
Now, let us fixh≥0. For all 0≤i≤ν-1, we define Gh,i(ϵ):=(t,z)↦Xh,i(t,z,ϵ), which is, by Lemma  4.16, a holomorphic and bounded function from ℰi into the Banach space 𝔼=𝒪((𝒯∩D(0,ι′′))×D(0,x1/4)) of holomorphic and bounded functions on the set (𝒯∩D(0,ι′′))×D(0,x1/4) equipped with the supremum norm. Therefore, the property (1) of Theorem (RS) is satisfied for the functions Gh,i, 0≤i≤ν-1. From the estimates (4.79), we get that the cocycle Δi=Gh,i+1(ϵ)-Gh,i(ϵ) is exponentially flat of order 1 on Zi=ℰi+1∩ℰi for all 0≤i≤ν-1. We deduce that the property (2) of Theorem (RS) is fulfilled for the functions Gh,i, 0≤i≤ν-1. From Theorem (RS), we get that Gh,i(ϵ) are the 1-sums of a formal series G^h(ϵ) with coefficients in 𝔼. In particular, from Definition 4.1, we deduce the existence of the functions gh,i(s,t,z) which satisfy the expression (4.76).


## 4.7. Analytic Transseries Solutions for a Singularly Perturbed Cauchy Problem

We keep the notations of the previous section.Proposition 4.17.
The following singularly perturbed Cauchy problem(4.90)ϵt2∂t∂zSZ0(t,z,ϵ)+(ϵt+1)∂zSZ0(t,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)ts(∂tk0∂zk1Z0)(t,z,ϵ)
for given initial data
(4.91)(∂zjZ0)(t,0,ϵ)=γ0,j(t,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!φh,0,j(ϵt,ϵ),0≤j≤S-1,
which are holomorphic and bounded functions on (𝒯∩D(0,ι′′))×(ℰ0∩ℰ1), has a solution
(4.92)Z0(t,z,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!Xh,0(t,z,ϵ),
which defines a holomorphic and bounded function on (𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1), for some ι′′,δZ0>0.Proof.
Leth≥0 and 0≤j≤S-1. By construction, we have that φh,0,j(ϵt,ϵ)=(∂zjXh,0)(t,0,ϵ) for all t∈𝒯 and all ϵ∈ℰ0. From Lemma  4.16, (1), we get that there exist a constant ι′′>0, a constant u1 such that 0<u1<u0 (depending on u0, 𝒮 and b, σ), and a constant Cˇ0>0 (depending on max0≤j≤S-1Wj,Sd0,ℰ0(u0) (where Wj,Sd0,ℰ0 are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.93)supt∈𝒯∩D(0,ι′′)|φh,0,j(ϵt,ϵ)|≤h!(2u1)hCˇ0
for all ϵ∈ℰ0, all 0≤j≤S-1, all h≥0. From (4.93) and from the property (3) of Definition 4.14, we deduce the estimates
(4.94)supt∈𝒯∩D(0,ι′′)|γ0,j(t,ϵ)|≤Cˇ0∑h≥0‍(2exp(-(|λ|/ϵ0ι′′)cos(π/2-δ𝒯))u1)h,
for all ϵ∈ℰ0∩ℰ1. This latter sum converges provided that ϵ0 is small enough. We deduce that γ0,j(t,ϵ) defines a holomorphic and bounded function on (𝒯∩D(0,ι′′))×(ℰ0∩ℰ1).
Likewise, from (4.78) and from the property (3) of Definition 4.14, we deduce that there exist a constant ι′′>0, a constant u1 such that 0<u1<u0 (depending on u0, 𝒮 and b, σ), a constant x1 such that 0<x1<ρ (depending on S, u0, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ) and a constant C~0>0 (depending on max0≤j≤S-1Wj,Sd0,ℰ0(u0) (where Wj,Sd0,ℰ0 are defined above), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, u0, x0, 𝒮, b) such that
(4.95)supt∈𝒯∩D(0,ι′′),z∈D(0,δZ0)|Z0(t,z,ϵ)|≤C~01-2δZ0/x1∑h≥0‍(2exp(-(|λ|/ϵ0ι′′)cos(π/2-δ𝒯))u1)h
for all ϵ∈ℰ0∩ℰ1. Again, this latter sum converges if ϵ0 is small enough and if 0<δZ0≤x1/4. We get that Z0(t,z,ϵ) defines a holomorphic and bounded function on (𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1). By construction, we have that (∂zjZ0)(t,0,ϵ)=γ0,j(t,ϵ), for 0≤j≤S-1. Finally, from Proposition 4.3, we deduce that Z0(t,z,ϵ) solves (4.90).


## 5. Parametric Stokes Relations and Analytic Continuation of the Borel Transform in the Perturbation Parameter

### 5.1. Assumptions on the Initial Data

We keep the notations of the previous section. Now, we make the following additional assumption that there exists a sequence of unbounded open sectorsSd0,ϑn such that
(5.1)Sd0⊂Sd0,ϑn⊂ℳd0∪Sd0
for all n≥0 and a sequence of real numbers ζn, n≥0 such that
(5.2)eiζn∈Sd0,ϑn,limn→+∞ζn=arg(λ)
with the property that arg(eiζn/ϵt)∈(-π/2+δ𝒯,π/2-δ𝒯) for all ϵ∈ℰ0∩ℰ1, all t∈𝒯, and for all n≥0 (where 𝒯 and δ𝒯 were introduced in Definition 4.14). We also make the assumption that for all n≥0, the function vh,j,Sd0,ℰ0(τ,ϵ) can be analytically continued to a holomorphic function τ↦vh,j,Sd0,ϑn,ℰ0(τ,ϵ) on Sd0,ϑn for all ϵ∈ℰ0 such that
(5.3)vh,j,Sd0,ϑn,ℰ0(τ,ϵ)∈Ej,ϵ,σ,(Sd0,ϑn∪D(0,τ0))×ℰ0
with the property that
(5.4)Wj,Sd0,ϑn,ℰ0(u)∶=∑h≥0‍supϵ∈ℰ0∥vh,j,Sd0,ϑn,ℰ0(τ,ϵ)∥j,ϵ,σ,(Sd0,ϑn∪D(0,τ0))×ℰ0uhh!∈ℂ{u},0≤j≤S-1
and has a common radius of absolute convergence (denoted by uℰ0>0) for all n≥0. From the assumption (5.4), we get a constant u0,j>0 (depending on j∈{0,…,S-1}) and a constant Cn,j>0 (depending on n and j∈{0,…,S-1}) such that
(5.5)supϵ∈ℰ0∥vh,j,Sd0,ϑn,ℰ0(τ,ϵ)∥j,ϵ,σ,(Sd0,ϑn∪D(0,τ0))×ℰ0≤Cn,j(1u0,j)hh!
for all h≥0. We deduce that
(5.6)|vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)|≤Cn,j(1u0,j)hh!exp(σ2|ϵ|rb(j)r)
for all r≥0, all ϵ∈ℰ0, all 0≤j≤S-1, and all h≥0. In particular, we have that r↦vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to the space L0,σ~/2,ϵ for σ~>σrb(S-1). Moreover, from Proposition 2.7, we deduce that r↦vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to the space 𝒟0,σ~,ϵ′ and that there exists a universal constant C1>0 such that
(5.7)∥vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥0,σ~,ϵ,d≤C1∥vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥0,σ~/2,ϵ≤2|ϵ|σ~-σrb(S-1)C1Cn,j(1u0,j)hh!
for all 0≤j≤S-1, all h≥0, and all n≥0, all ϵ∈ℰ0.We make the crucial assumption that for all0≤j≤S-1, there exists a sequence of distributions vh,j,ℳd0,ℰ0(r,ϵ)∈𝒟0,σ~,ϵ′, for h≥0, a constant uj>0 and a sequence In,j>0 with limn→+∞In,j=0 such that
(5.8)supϵ∈ℰ0∥vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)-vh,j,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d≤In,jh!(1uj)h
for all n≥0 and all h≥0. From the estimates (5.7) and (5.8), we deduce that
(5.9)∑h≥0‍supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,duhh!∈ℂ{u},0≤j≤S-1.Lemma 5.1.
Letσ~>σrb(S-1). One can write the initial data γ0,j(t,ϵ) in the form of a Laplace transform in direction arg(λ),
(5.10)γ0,j(t,ϵ)=ℒarg(λ)(𝕍j,arg(λ),Sd0,ℰ0(r,ϵ))(ϵt),
where 𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)∈𝒟0,σ~,ϵ′ and for all 0≤j≤S-1, and all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′).Proof.
For0≤j≤S-1, from the definition of the initial data, we can write
(5.11)γ0,j(t,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!1ϵt∫Lζn‍vh,j,Sd0,ϑn,ℰ0(τ,ϵ)exp(-τϵt)dτ=∑h≥0‍exp(-(h|λ|eiarg(λ))/ϵt)h!eiζnϵt∫0+∞‍vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)exp(-reiζnϵt)dr
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′), and all n≥0. Now, we can write
(5.12)ℒζn(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵt)=ℒarg(λ)(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵtei(arg(λ)-ζn))
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′), all n≥0. From the continuity estimates (3.3) for the Laplace transform, we deduce that for given t∈𝒯∩D(0,ι′), ϵ∈ℰ0∩ℰ1, there exists a constant Cϵ,t (depending on ϵ, t) such that
(5.13)|ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵt)-ℒarg(λ)(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵtei(arg(λ)-ζn))|≤Cϵ,t∥(vh,j,ℳd0,ℰ0(r,ϵ)-vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥0,σ~,ϵ,d+|ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵtei(arg(λ)-ζn))-ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵt)|
for all n≥0. By letting n tend to +∞ in this latter inequality and using the hypothesis (5.8), we get that
(5.14)ℒζn(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵt)=ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵt)
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′), and all n≥0.
On the other hand, from Corollary2.10, we have that for all h≥0, the distribution ∂r-h(vh,j,ℳd0,ℰ0(r,ϵ)) belongs to 𝒟0,σ~,ϵ′ and that there exists a universal constant C3>0 such that
(5.15)∥∂r-h(vh,j,ℳd0,ℰ0(r,ϵ))∥0,σ~,ϵ,d≤C3(|ϵ|σ~)h∥vh,j,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d
for all h≥0, all 0≤j≤S-1. From (5.14) and using Propositions 3.3 and 3.7, we can write
(5.16)exp(-h|λ|eiarg(λ)/ϵt)h!eiζnϵt∫0+∞‍vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)exp(-reiζnϵt)dr=(eiarg(λ)ϵt)hexp(-h|λ|eiarg(λ)/ϵt)h!ℒarg(λ)(∂r-h(vh,j,ℳd0,ℰ0(r,ϵ)))(ϵt)=ℒarg(λ)(𝕍h,j,λ,ℳd0,ℰ0(r,ϵ))(ϵt),
where
(5.17)𝕍h,j,λ,ℳd0,ℰ0(r,ϵ)=(fh,j,λ,ℳd0,ℰ0(r-|λ|h,ϵ)1[|λ|h,+∞)(r))(h)h!∈𝒟0,σ~,ϵ′
with fh,j,λ,ℳd0,ℰ0(r,ϵ)=∂r-h(vh,j,ℳd0,ℰ0(r,ϵ))∈𝒟0,σ~,ϵ′ for all h≥0 and all 0≤j≤S-1. From Proposition 3.6, we have a universal constant A>0 and a constant B(σ~,b,ϵ) (depending on σ~, b, and ϵ, which tend to zero as ϵ→0) such that
(5.18)∥𝕍h,j,λ,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d≤A(B(σ~,b,ϵ))hh!∥fh,j,λ,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d.
From the estimates (5.9) and using (5.15), (5.18), we deduce that the distribution
(5.19)𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)=∑h≥0‍𝕍h,j,λ,ℳd0,ℰ0(r,ϵ)∈𝒟0,σ~,ϵ,d′
for all 0≤j≤S-1, if ϵ0>0 is chosen small enough. Finally, by the continuity estimates (3.3) for the Laplace transform ℒarg(λ) and the formula (5.11), (5.16), we get the expression (5.10).On the other hand, we assume the existence of a sequence of unbounded open sectorsSd1,δn with
(5.20)Sd1⊂Sd1,δn⊂ℳd1∪Sd1
for all n≥0 and a sequence of real numbers ξn, n≥0 such that
(5.21)eiξn∈Sd1,δn,limn→+∞ξn=arg(λ)
with the property that arg(eiξn/ϵt)∈(-π/2+δ𝒯,π/2-δ𝒯) for all ϵ∈ℰ0∩ℰ1, all t∈𝒯, and all n≥0 (where 𝒯 and δ𝒯 are introduced in Definition 4.14). We make the assumption that for all n≥0, the function vh,j,Sd1,ℰ1(τ,ϵ) can be analytically continued to a holomorphic function τ↦vh,j,Sd1,δn,ℰ1(τ,ϵ) on Sd1,δn for all ϵ∈ℰ1 such that
(5.22)vh,j,Sd1,δn,ℰ1(τ,ϵ)∈Ej,ϵ,σ,(Sd1,δn∪D(0,τ0))×ℰ1
with the property that
(5.23)Wj,Sd1,δn,ℰ1(u)∶=∑h≥0‍supϵ∈ℰ1∥vh,j,Sd1,δn,ℰ1(τ,ϵ)∥j,ϵ,σ,(Sd1,δn∪D(0,τ0))×ℰ1uhh!∈ℂ{u},0≤j≤S-1
and has a common radius of absolute convergence (defined by uℰ1>0) for all n≥0. From the assumption (5.23), we get a constant u1,j>0 (depending on j∈{0,…,S-1}) and a constant Cn,1,j>0 (depending on n and j∈{0,…,S-1}) such that
(5.24)supϵ∈ℰ1∥vh,j,Sd1,δn,ℰ1(τ,ϵ)∥j,ϵ,σ,(Sd1,δn∪D(0,τ0))×ℰ1≤Cn,1,j(1u1,j)hh!
for all h≥0. We deduce that
(5.25)|vh,j,Sd1,δn,ℰ1(reiξn,ϵ)|≤Cn,1,j(1u1,j)hh!exp(σ2|ϵ|rb(j)r)
for all r≥0, all ϵ∈ℰ1, all 0≤j≤S-1, and all h≥0. In particular, we have that r↦vh,j,Sd1,δn,ℰ1(reiξn,ϵ) belongs to the space L0,σ~/2,ϵ for σ~>σrb(S-1). Moreover, from Proposition 2.7, we deduce that r↦vh,j,Sd1,δn,ℰ1(reiξn,ϵ) belongs to the space 𝒟0,σ~,ϵ′ and that there exists a universal constant C1>0 such that
(5.26)∥vh,j,Sd1,δn,ℰ1(reiξn,ϵ)∥0,σ~,ϵ,d≤C1∥vh,j,Sd1,δn,ℰ1(reiξn,ϵ)∥0,σ~/2,ϵ≤2|ϵ|σ~-σrb(S-1)C1Cn,1,j(1u1,j)hh!
for all 0≤j≤S-1, all h≥0, all n≥0, and all ϵ∈ℰ1.Now, we make the crucial assumption that for all0≤j≤S-1, there exists a sequence Jn,j>0 with limn→+∞Jn,j=0 such that
(5.27)supϵ∈ℰ0∩ℰ1∥v0,j,Sd1,δn,ℰ1(reiξn,ϵ)-𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d≤Jn,j
for all n≥0, where 𝕍j,arg(λ),Sd0,ℰ0(r,ϵ) are the distributions defined in Lemma 5.1.


### 5.2. The Stokes Relation and the Main Result

In the next proposition, we establish a connection formula for the two holomorphic solutionsX0,0(t,z,ϵ) and X0,1(t,z,ϵ) of (4.90) constructed in Proposition 4.15.Proposition 5.2.
Let the assumptions (5.1), (5.4), (5.8), (5.20), (5.23), and (5.27) hold for the initial data. Then, there exists 0<δD0,1<δZ0 such that one can write the following connection formula:
(5.28)X0,1(t,z,ϵ)=Z0(t,z,ϵ)=X0,0(t,z,ϵ)+∑h≥1‍exp(-hλ/ϵt)h!Xh,0(t,z,ϵ)
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′′), and all z∈D(0,δD0,1).The proof of this proposition will need two long steps and will be the consequence of the formula (5.79) and (5.124) from Lemmas 5.5 and 5.7.Step 1.
In this step, we show that the functionZ0(t,z,ϵ) can be expressed as a Laplace transform of some staircase distribution in direction arg(λ) satisfying the problem (5.80), (5.81).From the assumption (5.4), we deduce from Proposition 4.12 that the function Vh,Sd0,ℰ0(τ,z,ϵ) constructed in (4.80) has an analytic continuation denoted by Vh,Sd0,ϑn,ℰ0(τ,z,ϵ) on the domain (Sd0,ϑn∪D(0,τ0))×D(0,δℰ0)×ℰ0 which satisfies estimates of the form (4.62) for all n≥0, where δℰ0>0 depends on S, uℰ0 (which denotes a common radius of convergence of the series (5.4)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ. This constant δℰ0 is, therefore, independent of n and h. Now, one defines the functions
(5.29)𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ):=Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)
for all r≥0, all z∈D(0,δℰ0), all ϵ∈ℰ0, and all n≥0.Lemma 5.3.
Letσˇ>σ~>σrb(S-1). Then, there exists 0<δD<δℰ0 (depending on S, b, σˇ, |λ|, uj, 0≤j≤S-1 (introduced in (5.8)), 𝒮, uℰ0, ρ, μ, A, B (introduced in Lemma  5.4)), there exist M1>0 (depending on 𝒮, S, σˇ, |λ|, uj, for 0≤j≤S-1, ρ, μ, A, B), M1′>0 (depending on 𝒮, S, σˇ, |λ|, ρ, μ, ρ′, μ′ (introduced in Lemma  5.4), A, B, uj for 0≤j≤S-1) and a constant U1 (depending on 𝒮, S, σˇ, |λ|, ρ, μ, A, B, uℰ0, uj for 0≤j≤S-1) such that for all h≥0 all n≥0, there exists a staircase distribution 𝕍h,ℳd0,ℰ0(r,z,ϵ)∈𝒟'(σˇ,ϵ,δD) with
(5.30)supϵ∈ℰ0∥𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)-𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD)≤(M1max0≤j≤S-1In,j+M1′Dn)h!(2U1)h,
where In,j is a positive sequence (converging to 0 as n tends to ∞) introduced in the assumption (5.8) and Dn is the positive sequence (tending to 0 as n→+∞) introduced in Lemma  5.4. Moreover, one has
(5.31)∑h≥0‍supϵ∈ℰ0∥𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD)uhh!∈ℂ{u}.Proof.
From the estimates (4.54), we can write
(5.32)Vh,Sd0,ϑn,ℰ0(τ,z,ϵ)=∑β≥0‍Vh,β,Sd0,ϑn,ℰ0(τ,ϵ)zββ!,
where Vh,β,Sd0,ϑn,ℰ0(τ,ϵ) are holomorphic functions such that there exists a constant u1 such that 0<u1<uℰ0 (depending on uℰ0, 𝒮, and b, σ), a constant x1 such that 0<x1<ρ (depending on S, uℰ0, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ), and a constant CΩ(d0,ℰ0),n>0 (depending on max0≤j≤S-1Wj,Sd0,ϑn,ℰ0(uℰ0) (where Wj,Sd0,ϑn,ℰ0 are defined in (5.4)), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, uℰ0, x0, 𝒮, b) with
(5.33)|Vh,β,Sd0,ϑn,ℰ0(τ,ϵ)|≤CΩ(d0,ℰ0),nh!β!(2u1)h(2x1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)
for all τ∈Sd0,ϑn∪D(0,τ0), ϵ∈ℰ0, all h≥0, all β≥0, and all n≥0. We deduce that
(5.34)|Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)|≤CΩ(d0,ℰ0),n(2u1)h(2x1)βh!β!exp(σ2|ϵ|rb(β)r)
for all r≥0, all ϵ∈ℰ0, all β≥0, all h≥0, and all n≥0. In particular, r↦Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to Lβ,σˇ/2,ϵ. From Proposition 2.7, we deduce that r↦Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to 𝒟β,σˇ,ϵ′. From Proposition 2.7 and (5.34), we get a universal constant C1>0 such that
(5.35)∥Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)∥β,σˇ,ϵ,d≤C1∥Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)∥β,σˇ/2,ϵ≤C1CΩ(d0,ℰ0),n2|ϵ|σˇ-σ(2u1)h(2x1)βh!β!
for all β≥0, all h≥0, and all n≥0. From (5.35), we deduce that the distribution
(5.36)𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)=∑β≥0‍Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)zββ!∈𝒟′(σˇ,ϵ,δˇ)
for all ϵ∈ℰ0, all δˇ<x1/2, all h≥0, and all n≥0.
One gets from (4.20), (4.21) and the assumption (4.44) that the following problem holds:
(5.37)(reiζn+1+λh)∂zS𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0b~s,k0,k1(z,ϵ)ϵk0-sei(s-k0)ζn(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍ϵk0b~s,k0,k1(z,ϵ)cqk01(hλ)qϵk0-sei(s-k0-q)ζn×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p∂zk1𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ))
with initial data
(5.38)(∂zj𝕍h,Sd0,ϑn,ℰ0)(r,0,ϵ)=vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ),0≤j≤S-1.
On the other hand, we consider the problem(5.39)(reiarg(λ)+1+λh)∂zS𝕍h,ℳd0,ℰ0(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0b~s,k0,k1(z,ϵ)ϵk0-sei(s-k0)arg(λ)(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍h,ℳd0,ℰ0(r,z,ϵ))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍ϵk0b~s,k0,k1(z,ϵ)cqk01(hλ)qϵk0-sei(s-k0-q)arg(λ)×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p∂zk1𝕍h,ℳd0,ℰ0(r,z,ϵ))
with initial data
(5.40)(∂zj𝕍h,ℳd0,ℰ0)(r,0,ϵ)=vh,j,ℳd0,ℰ0(r,ϵ),0≤j≤S-1.
In the next lemma, we give estimates for the coefficients of (5.37) and (5.39).
Lemma  5.4.Let(5.41)b~s,k0,k1(z,ϵ)=∑β≥0‍b~s,k0,k1,β(ϵ)zββ!the convergent Taylor expansion ofb~s,k0,k1with respect toznear 0. Letα∈ℝbe a real number. Then, there exist positive constantsA,B,ρ,ρ′,μ,μ′and a sequenceDn>0such thatlimn→+∞Dn=0with(5.42)|∂rq(b~s,k0,k1,β(ϵ)eiαarg(λ)reiarg(λ)+1+λh)|≤AB-ββ!q!(ρ(r+μ))q+1,|∂rq(b~s,k0,k1,β(ϵ)eiαζnreiζn+1+λh)|≤AB-ββ!q!(ρ(r+μ))q+1,(5.43)|∂rq(b~s,k0,k1,β(ϵ)eiαarg(λ)reiarg(λ)+1+λh)-∂rq(b~s,k0,k1,β(ϵ)eiαζnreiζn+1+λh)|≤DnB-ββ!q!(ρ′(r+μ′))q+1for allq≥0, allβ≥0, alln≥0, allh≥0, allr≥0, and allϵ∈ℰ0.
Proof. We first show (5.42). From the fact that b~s,k0,k1(z,ϵ) is holomorphic near z=0, we get from the Cauchy formula that there exist A,B>0 such that
(5.44)|b~s,k0,k1,β(ϵ)|≤AB-ββ!
for all β≥0, and all ϵ∈ℰ0. On the other hand, from Definition 4.14(3.1), there exist ρ,μ>0 such that |reiζn+1+λh|≥ρ(r+μ) for all r≥0, all h≥0, and all n≥0. Hence,
(5.45)|∂rq(eiαζnreiζn+1+λh)|≤q!|reiζn+1+λh|q+1≤q!(ρ(r+μ))q+1
for all r≥0, and all h≥0, all q≥0, all n≥0. We deduce (5.42) from (5.44) and (5.45).
Now, we show (5.43). Using the classical identities ab-cd=(a-c)b+c(b-d) and bq+1-aq+1=(b-a)×∑s=0q‍asbq-s, we get the estimates
(5.46)|∂rq(eiαarg(λ)reiarg(λ)+1+λh)-∂rq(eiαζnreiζn+1+λh)|≤q!|eiαarg(λ)eiqarg(λ)(reiarg(λ)+1+λh)q+1-eiαζneiqζn(reiζn+1+λh)q+1|≤q!(|eiζn-eiarg(λ)|×(∑s=1q+1‍r|reiarg(λ)+1+λh|q+2-s|reiζn+1+λh|s)+|eiαarg(λ)-eiαζn|+|eiarg(λ)-eiζn|(q+1)|reiarg(λ)+1+λh|q+1).
On the other hand, again from Definition 4.14 (3.1), there exist ρ1,μ1>0 such that
(5.47)|reiarg(λ)+1+λh|≥ρ1(r+μ1),|reiζn+1+λh|≥ρ1(r+μ1)
for all r≥0, all h≥0, and all n≥0. Using (5.46), (5.47) and the fact that q+1≤2q+1 for all q≥0, we deduce the estimates (5.43).
In the first part of the proof of Lemma5.3, we show the existence of a staircase distribution solution of the problem (5.39), (5.40), which satisfies the estimates (5.31). As a starting point, it is easy to check that the problem (5.39), (5.40) has a formal solution of the form
(5.48)𝕍h,ℳd0,ℰ0(r,z,ϵ)=∑β≥0‍𝕍h,β,ℳd0,ℰ0(r,ϵ)zββ!,
where r↦𝕍h,β,ℳd0,ℰ0(r,ϵ) are distributions on ℝ+, for which the next recursion holds:
(5.49)𝕍h,β+S,ℳd0,ℰ0(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!ϵk0-s×ei(s-k0)arg(λ)reiarg(λ)+1+λh(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-sei(s-k0-q)arg(λ)reiarg(λ)+1+λh(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)
for all β≥0, h≥0, with initial conditions
(5.50)𝕍h,j,ℳd0,ℰ0(r,ϵ)=vh,j,ℳd0,ℰ0(r,ϵ),0≤j≤S-1,h≥0.
Using Corollary 2.10, Propositions 2.11 and 2.12, the estimates (5.9), and Remark 2.4, we deduce that 𝕍h,β,ℳd0,ℰ0(r,ϵ)∈𝒟β,σˇ,ϵ′ for all h,β≥0 and that the following inequalities hold for the real numbers 𝕍h,β,ℳd0(ϵ):=||𝕍h,β,ℳd0,ℰ0(r,ϵ)||β,σˇ,ϵ,d: there exist constants C23.01, C23.02 (depending on 𝒮, σˇ, S, ρ, μ) with
(5.51)𝕍h,β+S,ℳd0(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.01β!AB-β1(β+S+1)(s-k0)b𝕍h,β2+k1,ℳd0(ϵ)β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.02β!AB-β1|cqk01||λ|qhq×(β+S+1)(s-k0-q)b𝕍h,β2+k1,ℳd0(ϵ)β2!
for all β,h≥0, where A,B>0 are defined in Lemma  5.4. We define the following Cauchy problem:
(5.52)∂xS𝕎ℳd0(u,x)=∑(s,k0,k1)∈𝒮‍C23.01(x∂x+S+1)b(s-k0)(A1-x/B∂xk1𝕎ℳd0(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C23.02|cqk01||λ|q(x∂x+S+1)b(s-k0-q)×(A1-x/B(u∂u)q∂xk1𝕎ℳd0(u,x))
for given initial data
(5.53)(∂xj𝕎ℳd0)(u,0)=𝕎ℳd0,j(u)=∑h≥0‍supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)∥j,σˇ,ϵ,duhh!∈ℂ{u},0≤j≤S-1.
From the assumption (4.42) and the fact that b>1, we deduce that
(5.54)S>b(s-k0-q)+q+k1
for all (s,k0,k1)∈𝒮 and all 0≤q≤k0. Hence, the assumption (2.108) is satisfied in Proposition 2.22 for the Cauchy problem (5.52), (5.53). Since the initial data 𝕎ℳd0,j(u) is an analytic function on a disc containing some closed disc D(0,U0), for 0≤j≤S-1 and since the coefficients of (5.52) are analytic on ℂ×D(0,B), we deduce that all the hypotheses of Proposition 2.22 are fulfilled for the problem (5.52), (5.53). We deduce the existence of a formal solution 𝕎ℳd0(u,x)∈G(Uℳd0,Xℳd0), where 0<Uℳd0<U0 (depending on 𝒮) and 0<Xℳd0≤B/2 (depending on S, σˇ, |λ|, ρ, μ, U0, 𝒮, A, B).
Now, let𝕎ℳd0(u,x)=∑h,β≥0‍wh,β,ℳd0(uh/h!)(xβ/β!) be its Taylor expansion at the origin. Then, the sequence wh,β,ℳd0 satisfies the next equalities:
(5.55)wh,β+S,ℳd0=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.01β!AB-β1(β+S+1)(s-k0)bwh,β2+k1,ℳd0β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.02β!AB-β1|cqk01||λ|qhq×(β+S+1)(s-k0-q)bwh,β2+k1,ℳd0β2!
for all β,h≥0, with
(5.56)wh,j,ℳd0=supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)∥j,σˇ,ϵ,d,h≥0,0≤j≤S-1.
Gathering the inequalities (5.51), the equalities (5.55) with the initial conditions (5.56), one gets
(5.57)supϵ∈ℰ0|𝕍h,β,ℳd0(ϵ)|≤wh,β,ℳd0
for all h,β≥0. From (5.57) and the fact that 𝕎ℳd0(u,x)∈G(Uℳd0,Xℳ0), we get a constant Cℳd0>0 such that
(5.58)supϵ∈ℰ0∥𝕍h,β,ℳd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d≤Cℳd0(h+β)!(1Uℳd0)h(1Xℳd0)β≤Cℳd0h!β!(2Uℳd0)h(2Xℳd0)β
for all h,β≥0. From this last estimates (5.58), we deduce that for all h≥0, 𝕍h,ℳd0,ℰ0(r,z,ϵ) belongs to 𝒟′(σˇ,ϵ,δℳd0) for 0<δℳd0≤Xℳd0/4 and moreover that
(5.59)∑h≥0‍supϵ∈ℰ0∥𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δℳd0)uhh!∈ℂ{u}
holds. This yields the property (5.31).
In the second part of the proof, we show (5.30). One defines the distribution
(5.60)𝕍h,Sd0,ϑn,ℰ0Δ(r,z,ϵ):=𝕍h,ℳd0,ℰ0(r,z,ϵ)-𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)
for all r≥0, all z∈D(0,δℳd0)∩D(0,δˇ), with 0<δˇ<x1/2, all ϵ∈ℰ0. If one writes the  Taylor expansion
(5.61)𝕍h,Sd0,ϑn,ℰ0Δ(r,z,ϵ)=∑β≥0‍𝕍h,β,Sd0,ϑn,ℰ0Δ(r,ϵ)zββ!
for z∈D(0,δℳd0)∩D(0,δˇ), then the coefficients 𝕍h,β,Sd0,ϑn,ℰ0Δ(r,ϵ) satisfy the following recursion:
(5.62)𝕍h,β+S,Sd0,ϑn,ℰ0Δ(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!ϵk0-s(ei(s-k0)ζnreiζn+1+λh)×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍h,β2+k1,Sd0,ϑn,ℰ0Δ(r,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-sei(s-k0-q)ζnreiζn+1+λh×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p𝕍h,β2+k1,Sd0,ϑn,ℰ0Δ(r,ϵ)β2!)+𝔹h,β,n(r,ϵ),
where
(5.63)𝔹h,β,n(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!ϵk0-s(ei(s-k0)arg(λ)reiarg(λ)+1+λh-ei(s-k0)ζnreiζn+1+λh)×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-s(ei(s-k0-q)arg(λ)reiarg(λ)+1+λh-ei(s-k0-q)ζnreiζn+1+λh)×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)
for all h≥0 and all β≥0. Now, we put 𝕍h,β,nΔ(ϵ)=||𝕍h,β,Sd0,ϑn,ℰ0Δ(r,ϵ)||β,σˇ,ϵ,d. Using Corollary 2.10, Propositions 2.11 and 2.12, and Lemma  5.4, we get that there exist constants C23.11,C23.12 (depending on 𝒮, σˇ, S, ρ, μ) such that the following inequalities:
(5.64)𝕍h,β+S,nΔ(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.11β!AB-β1(β+S+1)(s-k0)b𝕍h,β2+k1,nΔ(ϵ)β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.12β!AB-β1×|cqk01||λ|qhq(β+S+1)(s-k0-q)b𝕍h,β2+k1,nΔ(ϵ)β2!+𝔹h,β,n(ϵ)
hold for all h,β≥0, where A,B>0 are defined in Lemma  5.4 and 𝔹h,β,n(ϵ) is a sequence which satisfies the next estimates: there exist constants C23.13,C23.14>0 (depending on 𝒮,σˇ,S,ρ′,μ′) with
(5.65)𝔹h,β,n(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.13β!DnB-β1×(β+S+1)(s-k0)b∥𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)∥β2+k1,σˇ,ϵ,dβ2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.14β!DnB-β1×|cqk01||λ|qhq(β+S+1)(s-k0-q)b∥𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)∥β2+k1,σˇ,ϵ,dβ2!
for all h,β,n≥0, where Dn, n≥0 is the sequence defined in Lemma  5.4.
We consider the following sequence of Cauchy problem:(5.66)∂xS𝕎nΔ(u,x)=∑(s,k0,k1)∈𝒮‍C23.11(x∂x+S+1)b(s-k0)(A1-x/B∂xk1𝕎nΔ(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C23.12|cqk01||λ|q(x∂x+S+1)b(s-k0-q)×(A1-x/B(u∂u)q∂xk1𝕎nΔ(u,x))+𝔻n(u,x),
where
(5.67)𝔻n(u,x)=∑(s,k0,k1)∈𝒮‍C23.13(x∂x+S+1)b(s-k0)(Dn1-x/B∂xk1𝕎ℳd0(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C23.14|cqk01||λ|q(x∂x+S+1)b(s-k0-q)×(Dn1-x/B(u∂u)q∂xk1𝕎ℳd0(u,x))
and 𝕎ℳd0(u,x) is already defined as the solution of the problem (5.52), (5.53), for given initial data
(5.68)(∂xj𝕎nΔ)(u,0)=𝕎j,nΔ(u)=∑h≥0‍supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)-vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥j,σˇ,ϵ,duhh!∈ℂ{u},0≤j≤S-1,
which are convergent near the origin with respect to u due to the assumption (5.8) and Remark 2.4. Moreover, the initial data satisfy the estimates
(5.69)|𝕎j,nΔ(u)|≤In,j1-|u|/uj
for all |u|<uj, 0≤j≤S-1, all n≥0.
From the assumption (4.42) and the fact that b>1, we deduce that
(5.70)S>b(s-k0-q)+q+k1
for all (s,k0,k1)∈𝒮 and all 0≤q≤k0. Therefore, the assumption (2.108) is satisfied in Proposition 2.22 for the problem (5.66), (5.68).
On the other hand, from Lemmas2.20 and 2.21, there exist a constant Dℳd0>0 (depending on 𝒮,σˇ,S,ρ′,μ′,|λ|,B,Uℳd0,Xℳd0), a constant 0<U1,ℳd0<Uℳd0, and a constant 0<X1,ℳd0<Xℳd0 such that
(5.71)∥𝔻n(u,x)∥(U1,ℳd0,X1,ℳd0)≤DnDℳd0∥𝕎ℳd0(u,x)∥(Uℳd0,Xℳd0)≤DnDℳd0Cℳd0
for all n≥0, where the constant Cℳd0 is introduced in (5.58).
Since the initial data𝕎j,nΔ(u) is an analytic function on some disc containing the closed disc D-(0,uj/2), for 0≤j≤S-1 and the coefficients of (5.66) are analytic on ℂ×D(0,B), we deduce that all the hypotheses of Proposition 2.22 for the problem (5.66), (5.68) are fulfilled. We deduce the existence of a formal solution 𝕎nΔ(u,x)∈G(U1,X1) of (5.66), (5.68), where 0<U1<min(U1,ℳd0,min0≤j≤S-1uj/2) (depending on 𝒮) and 0<X1≤min(B/2,X1,ℳd0) (depending on S, σˇ, |λ|, uj, for 0≤j≤S-1, 𝒮, A, B, ρ, μ).
Moreover, from (2.111) and (5.71), there exist constants M1>0 (depending on S, σˇ, λ, uj, for 0≤j≤S-1, 𝒮, A, B, ρ, μ) and M2>0 (depending on S, uj for 0≤j≤S-1, B, 𝒮) such that
(5.72)∥𝕎nΔ(u,x)∥(U1,X1)≤M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0
for all n≥0. Now, let 𝕎nΔ(u,x)=∑h,β≥0‍wh,β,nΔ(uh/h!)(xβ/β!) be its Taylor expansion at the origin. Then, the sequence wh,β,nΔ satisfies the following equalities:
(5.73)wh,β+S,nΔ=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.11β!AB-β1(β+S+1)(s-k0)bwh,β2+k1,nΔβ2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.12β!AB-β1|cqk01||λ|qhq(β+S+1)(s-k0-q)bwh,β2+k1,nΔβ2!+𝔻h,β,n,
where
(5.74)𝔻h,β,n=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.13β!DnB-β1(β+S+1)(s-k0)bwh,β2+k1,ℳd0β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.14β!DnB-β1|cqk01||λ|qhq(β+S+1)(s-k0-q)bwh,β2+k1,ℳd0β2!
for all h,β,n≥0, with
(5.75)wh,j,nΔ=supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)-vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥j,σˇ,ϵ,d,∀h≥0,∀0≤j≤S-1.
Gathering the inequalities (5.64), (5.65) and the equalities (5.73), with the initial conditions (5.75), one gets that
(5.76)supϵ∈ℰ0|𝕍h,β,nΔ(ϵ)|≤wh,β,nΔ
for all h,β≥0 and all n≥0.
From (5.76) and the estimates (5.72), we deduce that
(5.77)supϵ∈ℰ0∥Vh,β,Sd0,ϑn,ℰ0Δ(r,ϵ)∥β,σˇ,ϵ,d≤(M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0)(h+β)!(1U1)h(1X1)β≤(M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0)h!β!(2U1)h(2X1)β
for all h,β≥0, all n≥0. From (5.77), we get that
(5.78)supϵ∈ℰ0∥Vh,Sd0,ϑn,ℰ0(r,z,ϵ)-𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD)≤(M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0)h!(2U1)h
for all h≥0 and all 0<δD≤X1/4. This yields the estimates (5.30).In the next lemma, we expressZ0(t,z,ϵ) as a Laplace transform of a staircase distribution.Lemma 5.5.
Letσˇ>σ~>σrb(S-1). Then, one can write the solution Z0(t,z,ϵ) of (4.90), (4.91) in the form of a Laplace transform in direction arg(λ)(5.79)Z0(t,z,ϵ)=ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δD,Z0)×(ℰ0∩ℰ1), where 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∈𝒟′(σˇ,ϵ,δD,Z0) (with δD,Z0=min(δD,δZ0)) solves the following Cauchy problem:
(5.80)(reiarg(λ)+1)∂zS𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0-sbs,k0,k1(z,ϵ)(ei(s-k0)arg(λ)∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)),
where the sets 𝒪s-k01 and the integers αm,p1 are introduced in (4.20), with initial data
(5.81)(∂zj𝕍arg(λ),Sd0,ℰ0)(r,0,ϵ)=𝕍j,arg(λ),Sd0,ℰ0(r,ϵ),0≤j≤S-1.Proof.
From Proposition4.17, we can write the solution Z0(t,z,ϵ) of (4.90), (4.91) in the form
(5.82)Z0(t,z,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!1ϵt∫Lζn‍Vh,Sd0,ϑn,ℰ0(τ,z,ϵ)exp(-τϵt)dτ=∑h≥0‍exp(-h|λ|eiarg(λ)/ϵt)h!eiζnϵt∫0+∞‍Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)exp(-reiζnϵt)dr
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1) and all n≥0. Now, we write
(5.83)ℒζn(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵt)=ℒarg(λ)(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵtei(arg(λ)-ζn))
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1) and all n≥0. Now, we define δD,Z0=min(δD,δZ0). From the continuity estimates (3.5) for the Laplace transform, we deduce that for given ϵ∈ℰ0∩ℰ1, t∈𝒯∩D(0,ι′′), there exists a constant Cϵ,t (depending on ϵ, t) such that
(5.84)|ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵt)-ℒarg(λ)(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵtei(arg(λ)-ζn))|≤Cϵ,t∥𝕍h,ℳd0,ℰ0(r,z,ϵ)-Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)∥(σˇ,ϵ,d,δD,Z0)+|ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵtei(arg(λ)-ζn))-ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵt)|
for all z∈D(0,δD,Z0), all n≥0. By letting n tend to +∞ in this latter inequality and using the estimates (5.30), we obtain
(5.85)ℒζn(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵt)=ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵt)
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δD,Z0)×(ℰ0∩ℰ1) and all n≥0.
On the other hand, from Corollary2.10, we have that for all h≥0, the distribution ∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ)) belongs to 𝒟′(σˇ,ϵ,δD,Z0) and that there exists a universal constant C3>0 such that
(5.86)∥∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ))∥σˇ,ϵ,d,δD,Z0≤C3(|ϵ|σˇ)h∥𝕍h,ℳd0,ℰ0(r,z,ϵ)∥σˇ,ϵ,d,δD,Z0
for all h≥0.
From (5.85) and using Propositions 3.3 and 3.7, we can write
(5.87)exp(-h|λ|eiarg(λ)/ϵt)h!eiζnϵt∫0+∞‍Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)exp(-reiζnϵt)dr=(eiarg(λ)ϵt)hexp(-h|λ|eiarg(λ)/ϵt)h!ℒarg(λ)(∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ)))(ϵt)=ℒarg(λ)(𝕍h,λ,ℳd0,ℰ0(r,z,ϵ))(ϵt),
where
(5.88)𝕍h,λ,ℳd0,ℰ0(r,ϵ)=(fh,λ,ℳd0,ℰ0(r-|λ|h,z,ϵ)1[|λ|h,+∞)(r))(h)h!∈𝒟′(σˇ,ϵ,δD,Z0)
with fh,λ,ℳd0,ℰ0(r,z,ϵ)=∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ))∈𝒟'(σˇ,ϵ,δD,Z0), for all h≥0, all 0≤j≤S-1. From Proposition 3.6, we have a universal constant A>0 and a constant B(σˇ,b,ϵ) (depending on σˇ, b, and ϵ, which tend to zero as ϵ→0) such that
(5.89)∥𝕍h,λ,ℳd0,ℰ0(r,z,ϵ)∥σˇ,ϵ,d,δD,Z0≤A(B(σˇ,b,ϵ))hh!∥fh,λ,ℳd0,ℰ0(r,z,ϵ)∥σˇ,ϵ,d,δD,Z0.
From the convergence of the series (5.31) near the origin and using (5.86), (5.89), we deduce the distribution
(5.90)𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)=∑h≥0‍𝕍h,λ,ℳd0,ℰ0(r,z,ϵ)∈𝒟'(σˇ,ϵ,δD,Z0),
if ϵ0>0 is chosen small enough. Finally, by the continuity estimates (3.5) of the Laplace transform ℒarg(λ) and the formula (5.82), (5.87), we get the expression (5.79). Moreover, from the formulas in Proposition 3.3, as Z0(t,z,ϵ) solves the problem (4.90), (4.91), we deduce that the distribution 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ) solves the Cauchy problem (5.80), (5.81).Step 2.
In this step, we show that the functionX0,1(t,z,ϵ) can be expressed as a Laplace transform of some staircase distribution in direction arg(λ), satisfying the problem (5.80), (5.81).From the assumption (5.23), we deduce from Proposition 4.12, that the function V0,Sd1,ℰ1(τ,z,ϵ) constructed in (4.80) has an analytic continuation denoted by V0,Sd1,δn,ℰ1(τ,z,ϵ) on (Sd1,δn∪D(0,τ0))×D(0,δℰ1)×(ℰ0∩ℰ1) and satisfies estimates (4.62) for all n≥0, where δℰ1>0 depends on S,uℰ1 (which denotes a common radius of absolute convergence of the series (5.23), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ. This constant δℰ1 is, therefore, independent of n. Now, one defines the functions
(5.91)𝕍0,Sd1,δn,ℰ1(r,z,ϵ)=V0,Sd1,δn,ℰ1(reiξn,z,ϵ)
for all r≥0, all z∈D(0,δℰ1), and all n≥0.Lemma 5.6.
Letσˇ>σ~>σrb(S-1) as in Lemma 5.3. Then, there exists 0<δD0,1<min(δℰ1,δD,Z0) (depending on 𝒮,S,σˇ,|λ|,A,B,ρ,μ and A~,B~,ρ~,μ~ introduced in Lemma  5.7), there exist M~1,M~1′ (depending on 𝒮,S,σˇ,|λ|,A,B,ρ,μ,A~,B~,ρ~,μ~ and ρ~′,μ~′ introduced in Lemma  5.7) such that
(5.92)supϵ∈ℰ0∩ℰ1∥𝕍0,Sd1,δn,ℰ1(r,z,ϵ)-𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD0,1)≤(M~1max0≤j≤S-1Jn,j+M~1′D~n)
for all n≥0, where 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ) is defined in Lemma 5.5 and solves the problem (5.80), (5.81) and D~n is the sequence (which tends to zero as n→+∞) defined in Lemma  5.7.Proof.
From the estimates (4.54), we can write
(5.93)V0,Sd1,δn,ℰ1(τ,z,ϵ)=∑β≥0‍V0,β,Sd1,δn,ℰ1(τ,ϵ)zββ!,
where V0,β,Sd1,δn,ℰ1(τ,ϵ) are holomorphic functions such that there exist a constant u1 with 0<u1<uℰ1 (depending on uℰ1, 𝒮, and b, σ), a constant x1 such that 0<x1<ρ (depending on S, uℰ1, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ), and a constant CΩ(d1,ℰ1),n>0 (depending on max0≤j≤S-1Wj,Sd1,δn,ℰ1(uℰ1) (where Wj,Sd1,δn,ℰ1 are defined in (5.23)), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, uℰ1, x0, 𝒮, b) with
(5.94)|V0,β,Sd1,δn,ℰ1(τ,ϵ)|≤CΩ(d1,ℰ1),nβ!(2x1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)
for all τ∈Sd1,δn∪D(0,τ0), ϵ∈ℰ1, all β≥0, and all n≥0. We deduce that
(5.95)|V0,β,Sd1,δn,ℰ1(reiξn,ϵ)|≤CΩ(d1,ℰ1),n(2x1)ββ!exp(σ2|ϵ|rb(β)r)
for all r≥0, all ϵ∈ℰ1, all β≥0, and all n≥0. In particular, r↦V0,β,Sd1,δn,ℰ1(reiξn,ϵ) belongs to Lβ,σˇ/2,ϵ. From Proposition 2.7, we deduce that r↦V0,β,Sd1,δn,ℰ1(reiξn,ϵ) belongs to 𝒟β,σˇ,ϵ′. From Proposition 2.7 and (5.95), we get a universal constant C1>0 such that
(5.96)∥V0,β,Sd1,δn,ℰ1(reiξn,ϵ)∥β,σˇ,ϵ,d≤C1∥V0,β,Sd1,δn,ℰ1(reiξn,ϵ)∥β,σˇ/2,ϵ≤C1CΩ(d1,ℰ1),n2|ϵ|σˇ-σ(2x1)ββ!
for all β≥0 and all n≥0. From (5.96), we deduce that the distribution
(5.97)𝕍0,Sd1,δn,ℰ1(r,z,ϵ)=∑β≥0‍V0,β,Sd1,δn,ℰ1(reiξn,ϵ)zββ!∈𝒟'(σˇ,ϵ,δˇ)
for all ϵ∈ℰ0∩ℰ1, all δˇ<x1/2, and all n≥0.
From (4.20), (4.21), we have that the distribution 𝕍0,Sd1,δn,ℰ1(r,z,ϵ) solves the following problem:
(5.98)(reiξn+1)∂zS𝕍0,Sd1,δn,ℰ1(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0-sbs,k0,k1(z,ϵ)(ei(s-k0)ξn∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍0,Sd1,δn,ℰ1(r,z,ϵ)),
where 𝒪s-k01 is the set and αm,p1 are the integers from (5.80), with initial data
(5.99)(∂zj𝕍0,Sd1,δn,ℰ1)(r,0,ϵ)=v0,j,Sd1,δn,ℰ1(reiξn,ϵ),0≤j≤S-1.
In the next lemma, we give estimates for the coefficients of (5.98) and (5.80). The proof is exactly the same as the one described for Lemma  5.4.
Lemma  5.7.Let(5.100)bs,k0,k1(z,ϵ)=∑β≥0‍bs,k0,k1,β(ϵ)zββ!be the convergent Taylor expansion ofbs,k0,k1with respect toznear 0. Then, there exist positive constantsA~,B~,ρ~,ρ~′,μ~,μ~′and a sequenceD~n>0such thatlimn→+∞D~n=0  with(5.101)|∂rq(bs,k0,k1,β(ϵ)ei(s-k0)arg(λ)reiarg(λ)+1)|≤A~B~-ββ!q!(ρ~(r+μ~))q+1,|∂rq(bs,k0,k1,β(ϵ)ei(s-k0)ξnreiξn+1)|≤A~B~-ββ!q!(ρ~(r+μ~))q+1,|∂rq(bs,k0,k1,β(ϵ)ei(s-k0)arg(λ)reiarg(λ)+1)-∂rq(bs,k0,k1,β(ϵ)ei(s-k0)ξnreiξn+1)|≤D~nB~-ββ!q!(ρ~′(r+μ~′))q+1for allq≥0, allβ≥0, alln≥0, allr≥0and allϵ∈ℰ0∩ℰ1.
Now, we consider the distribution(5.102)𝕍0,Sd1,δn,ℰ1Δ(r,z,ϵ):=𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)-𝕍0,Sd1,δn,ℰ1(r,z,ϵ)
for all r≥0, all z∈D(0,δD,Z0)∩D(0,δˇ), with 0<δˇ<x1/2 and δD,Z0 defined in Lemma 5.5, for all ϵ∈ℰ0∩ℰ1. One writes the Taylor expansions as follows:
(5.103)𝕍0,Sd1,δn,ℰ1Δ(r,z,ϵ)=∑β≥0‍𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ)zββ!,𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)=∑β≥0‍𝕍β,arg(λ),Sd0,ℰ0(r,ϵ)zββ!,
for z∈D(0,δD,Z0)∩D(0,δˇ); then the coefficients 𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ) satisfy the next recursion:
(5.104)𝕍0,β+S,Sd1,δn,ℰ1Δ(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!ϵk0-sei(s-k0)ξnreiξn+1×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍0,β2+k1,Sd1,δn,ℰ1Δ(r,ϵ)β2!)+∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!ϵk0-s(ei(s-k0)arg(λ)reiarg(λ)+1-ei(s-k0)ξnreiξn+1)×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍β2+k1,arg(λ),Sd0,ℰ0(r,ϵ)β2!)
for all h≥0, all β≥0. We put 𝕍0,β,n,ℰ1Δ(ϵ)=||𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ)||β,σˇ,ϵ,d. Using Corollary 2.10, Propositions 2.11 and 2.12 and Lemma  5.7, we get a constant C23.21>0 (depending on 𝒮,σˇ,S,ρ~,μ~) and C23.22>0 (depending on 𝒮,σˇ,S,ρ~′,μ~′) such that the next inequalities:
(5.105)𝕍0,β+S,n,ℰ1Δ(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.21β!A~B~-β1(β+S+1)b(s-k0)𝕍0,β2+k1,n,ℰ1Δ(ϵ)β2!+∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.22β!D~nB~-β1(β+S+1)b(s-k0)×∥𝕍β2+k1,arg(λ),Sd0,ℰ0(r,ϵ)∥β2+k1,σˇ,ϵ,dβ2!
hold for all β≥0, where A~,B~>0 and the sequence D~n, n≥0 are defined in Lemma  5.7.
We consider the following sequence of Cauchy problems:(5.106)∂xS𝕎n,ℰ1Δ(x)=∑(s,k0,k1)∈𝒮‍C23.21(x∂x+S+1)b(s-k0)(A~1-x/B~∂xk1𝕎n,ℰ1Δ(x))+𝔻~n(x),
where
(5.107)𝔻~n(x)=∑(s,k0,k1)∈𝒮‍C23.22(x∂x+S+1)b(s-k0)(D~n1-x/B~∂xk1𝕎arg(λ),ℰ0(x))
with
(5.108)𝕎arg(λ),ℰ0(x)=∑β≥0‍supϵ∈ℰ0∩ℰ1∥𝕍β,arg(λ),Sd0,ℰ0(x)∥β,σˇ,ϵ,dxββ!
for given initial data
(5.109)(∂xj𝕎n,ℰ1Δ)(0)=𝕎j,n,ℰ1Δ=supϵ∈ℰ0∩ℰ1∥𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)-v0,j,Sd1,δn,ℰ1(reiξn,ϵ)∥j,σˇ,ϵ,d,0≤j≤S-1,
which are finite positive numbers due to the assumption (5.27) and Remark 2.4. Moreover, the initial data satisfy the estimates
(5.110)|𝕎j,n,ℰ1Δ|≤Jn,j
for all 0≤j≤S-1 and all n≥0.
On the other hand, we have that𝕎arg(λ),ℰ0(x) is convergent for all |x|≤Xℳd0/4 (where Xℳd0 is chosen in (5.58)). Indeed, we know, from (5.90), that
(5.111)𝕍h,λ,ℳd0,ℰ0(r,z,ϵ)=∑β≥0‍𝕍h,β,λ,ℳd0,ℰ0(r,ϵ)zββ!
is convergent for all |z|<δD,Z0, all r>0, and all h≥0. From (5.86) and (5.89), we know that
(5.112)∥𝕍h,β,λ,ℳd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d≤C3A(|ϵ|B(σˇ,b,ϵ)/σˇ)hh!∥𝕍h,β,ℳd0,ℰ0(r,ϵ)∥β,σ˘,ϵ,d
for all h≥0 and all β≥0. From (5.58) and (5.112), we deduce that
(5.113)∥𝕍β,arg(λ),Sd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d=∥∑h≥0‍𝕍h,β,λ,ℳd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d≤C3ACℳd0β!(2Xℳd0)β∑h≥0‍(2|ϵ|B(σˇ,b,ϵ)σˇUℳd0)h
and this last sum is convergent provided that ϵ0 is small enough. We deduce that 𝕎arg(λ),ℰ0(x) belongs to G(U,Xℳd0/4), for any U>0. Let C~ℳd0:=||𝕎arg(λ),ℰ0(x)||(U,Xℳd0/4).
From Lemmas2.20 and 2.21, we get constants D~ℳd0>0 (depending on 𝒮,σˇ,S,ρ~′,μ~′,B~,U,Xℳd0), 0<U~1,ℳd0<U, and 0<X~1,ℳd0<Xℳd0/4 such that
(5.114)∥𝔻~n(x)∥(U~1,ℳd0,X~1,ℳd0)≤D~nD~ℳd0C~ℳd0
for all n≥0.
From the assumption (4.42) and the fact that b>1, we deduce that
(5.115)S>b(s-k0)+k1
for all (s,k0,k1)∈𝒮. Hence, the assumption (2.108) is satisfied in Proposition 2.22 for the problem (5.106), (5.109). Moreover, the initial data 𝕎j,nΔ can be seen as constant functions (therefore analytic) with respect to a variable u on the closed disc D-(0,U) for any given U>0 and the coefficients of (5.106) are analytic with respect to x on D-(0,B~/2) and constant (therefore analytic) with respect to u on D-(0,U). We deduce that all the hypotheses of Proposition 2.22 for the problem (5.106), (5.109) are fulfilled. A direct computation shows that the problem (5.106), (5.109) has a unique formal solution 𝕎n,ℰ1Δ(x)=∑β≥0‍wβ,n,ℰ1Δxβ/β!, with wβ,n,ℰ1Δ∈ℂ. From Proposition 2.22, we deduce that 𝕎n,ℰ1Δ(x)∈G(U~1,X~1) where 0<U~1<U~1,ℳd0 (depending on 𝒮) and 0<X~1<min(B~/2,X~1,ℳd0) (depending on 𝒮,S,σˇ,A~,B~,ρ~,μ~). Moreover, from (2.111) and (5.114), there exist constants M~1>0 (depending on 𝒮,S,σˇ,A~,B~,ρ~,μ~) and M~2>0 (depending on 𝒮,B~,S) such that
(5.116)∥𝕎n,ℰ1Δ(x)∥(U~1,X~1)≤M~1max0≤j≤S-1Jn,j+D~nM~2D~ℳd0C~ℳd0
for all n≥0.
Now, the coefficientswβ,n,ℰ1Δ satisfy the following equalities:
(5.117)wβ+S,n,ℰ1Δ=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.21β!A~B~-β1(β+S+1)b(s-k0)wβ2+k1,n,ℰ1Δβ2!+∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.22β!D~nB~-β1(β+S+1)b(s-k0)×supϵ∈ℰ0∩ℰ1∥𝕍β2+k1,arg(λ),Sd0,ℰ0(r,ϵ)∥β2+k1,σ˘,ϵ,dβ2!
for all β≥0 and all n≥0, with
(5.118)wj,n,ℰ1Δ=𝕎j,n,ℰ1Δ,0≤j≤S-1.
Gathering the inequalities (5.105) and the equalities (5.117), with the initial data (5.118), one gets that
(5.119)supϵ∈ℰ0∩ℰ1|𝕍0,β,n,ℰ1Δ(ϵ)|≤wβ,n,ℰ1Δ
for all β,n≥0.
From (5.119) and the estimates (5.116), we deduce that
(5.120)supϵ∈ℰ0∩ℰ1∥𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ)∥β,σˇ,ϵ,d≤(M~1max0≤j≤S-1Jn,j+D~nM~2D~ℳd0C~ℳd0)β!(1X~1)β
for all β,n≥0. From (5.120), we get that
(5.121)supϵ∈ℰ0∩ℰ1∥𝕍0,Sd1,δn,ℰ1(r,z,ϵ)-𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD0,1)≤2(M~1max0≤j≤S-1Jn,j+D~nM~2D~ℳd0C~ℳd0)
for all n≥0 and for all 0<δD0,1<X~1/2. This implies the estimates (5.92).In the following lemma, we express the functionX0,1(t,z,ϵ) as Laplace transform of a staircase distribution.Lemma 5.7.
Letσˇ>σ~>σrb(S-1) as in Lemma 5.3. Then, one can write the function X0,1(t,z,ϵ), which by construction of Proposition 4.12, solves the singularly perturbed Cauchy problem
(5.122)ϵt2∂t∂zSX0,1(t,z,ϵ)+(ϵt+1)∂zSX0,1(t,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)ts(∂tk0∂zk1X0,1)(t,z,ϵ)
for given initial data
(5.123)(∂zjX0,1)(t,0,ϵ)=φ0,1,j(ϵt,ϵ),0≤j≤S-1
in the form of a Laplace transform in direction arg(λ)(5.124)X0,1(t,z,ϵ)=ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δD0,1)×(ℰ0∩ℰ1), where 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∈𝒟'(σˇ,ϵ,δD0,1) solves the Cauchy problem (5.80), (5.81).Proof.
From Proposition4.12 and the assumption (5.23), we get that the function X0,1(t,z,ϵ) can be expressed as a Laplace transform in the direction ξn,
(5.125)X0,1(t,z,ϵ)=1ϵt∫Lξn‍V0,Sd1,δn,ℰ1(τ,z,ϵ)exp(-τϵt)dτ=eiξnϵt∫0+∞‍V0,Sd1,δn,ℰ1(reiξn,z,ϵ)exp(-reiξnϵt)dr
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δℰ1)×(ℰ0∩ℰ1), all n≥0. Now, let t∈𝒯∩D(0,ι′′), ϵ∈ℰ0∩ℰ1. For all n≥0, we can rewite X0,1(t,z,ϵ) as a Laplace transform in the direction arg(λ) as follows:
(5.126)X0,1(t,z,ϵ)=ℒarg(λ)(𝕍0,Sd1,δn,ℰ1(r,z,ϵ))(ϵtei(arg(λ)-ξn))
for all z∈D(0,δℰ1). Using the expression (5.126), we deduce that from the estimates (3.5), there exists a constant C(t,ϵ)>0 such that
(5.127)|X0,1(t,z,ϵ)-ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)|≤C(t,ϵ)∥𝕍0,Sd1,δn,ℰ1(r,z,ϵ)-𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD0,1)+|ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵtei(arg(λ)-ξn))-ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)|
for all n≥0 and all z∈D(0,δD0,1). By letting n tend to +∞ and using the estimates (5.92), we get the formula (5.124).Now, we are in the position to state the main result of our work.Theorem 5.8.
Let the assumptions (4.42), (4.44), (4.67), (4.69), (4.70), (5.1), (5.4), (5.8), (5.20), (5.23), and (5.27) hold. Then, if one denote by op>(𝒢κ0) (resp. op(𝒢κ1)) the opening of the sector 𝒢κ0 (resp. 𝒢κ1), one has that for all t∈𝒯∩D(0,ι′′), z∈D(0,δD0,1), the function s↦g0(s,t,z) (constructed in Proposition 4.15) can be analytically continued along any path Γ in the punctured sector
(5.128)S˙κ0,κ1,t,λ={s∈ℂ*κ0-op(𝒢κ0)2<arg(s)<κ1+op(𝒢κ1)2}∖⋃k=1∞{λkt},
as a function denoted by g0Γ,t,z(s). Moreover, for all k≥1, and any path Γ0,k⊂S˙κ0,κ1,t,λ from 0 to a neighborhood of λk/t, there exists a constant Ck>0 such that
(5.129)|g0Γ0,k,t,z(s)|≤Ck|log(s-λkt)|
as s tends to λk/t in a sector centered at λk/t.Proof.
The proof is based on the following version of a result on analytic continuation of Borel transforms obtained by Fruchard and Schäfke in [3]. This result extends a former statement obtained by the same authors in [28].
Theorem (FS).Letr>0and letg:D(0,r)→ℂbe a holomorphic function that can be analytically continued as a functiong+(resp.,g-) with exponential growth of order 1 on an unbounded sectorSκ+,δ+(resp.Sκ-,δ-) centered at 0, with bisecting directionκ+(resp.κ-) and openingδ+(resp.δ-). LetC>rbe a real number and letm≥1be an integer. Let{ak∈ℂ*,1≤k≤m}⊂D(0,C)be a set of aligned points and letα>0witharg(ak)=α∈(κ-,κ+), for all1≤k≤m. For all integers1≤k≤m, letSkbe an unbounded open sector centered atak, with bisecting direction which is parallel toκ-, and openingμ>0such that theSk∩D(0,C)do not intersect for all1≤k≤m.
Now, for all
1
≤
k
≤
m
, let
g
k
be a holomorphic and bounded function on a small neighborhood of 0 and with exponential growth of order 1 on the sector
S
k
-
a
k
=
{
s
∈
ℂ
/
s
+
a
k
∈
S
k
}
with bisecting direction
κ
-
. We consider the Laplace transforms
(5.130)
f
+
(
ϵ
)
=
∫
L
κ
+
‍
g
+
(
s
)
e
-
s
/
ϵ
d
s
,
f
-
(
ϵ
)
=
∫
L
κ
-
‍
g
-
(
s
)
e
-
s
/
ϵ
d
s
,
f
k
-
(
ϵ
)
=
∫
L
κ
-
‍
g
k
(
s
)
e
-
s
/
ϵ
d
s
for all
k
≥
1
, where
L
κ
+
is the half-line starting from 0 in the direction
κ
+
and
L
κ
-
is the half-line starting from 0 in the direction
κ
-
. The function
f
+
(resp.
f
-
) defines a holomorphic and bounded function on an open sector
ℰ
+
(resp.
ℰ
-
) with finite radius, with bisecting direction
κ
+
(resp.
κ
-
) and opening
π
+
δ
+
(resp.
π
+
δ
-
). The sectors
ℰ
+
,
ℰ
-
are chosen in such a way that
ℰ
+
∩
ℰ
-
is contained in a sector with direction
α
and with opening less than
π
. Assume that the following Stokes relation
(5.131)
f
+
(
ϵ
)
=
f
-
(
ϵ
)
+
∑
k
=
1
m
‍
e
-
a
k
/
ϵ
k
!
f
k
-
(
ϵ
)
+
𝒪
(
e
-
C
e
i
α
/
ϵ
)
holds for all
ϵ
∈
ℰ
+
∩
ℰ
-
, where
𝒪
(
e
-
C
e
i
α
/
ϵ
)
is a holomorphic function
R
(
ϵ
)
on
ℰ
+
∩
ℰ
-
such that there exists a constant
H
>
0
with
(5.132)
|
R
(
ϵ
)
|
≤
H
|
e
-
C
e
i
α
/
ϵ
|
=
H
e
-
(
C
/
|
ϵ
|
)
cos
(
α
-
arg
(
ϵ
)
)
for all
ϵ
∈
ℰ
+
∩
ℰ
-
.
Then, the function
g
:
D
(
0
,
r
)
→
ℂ
can be analytically continued along any path
Γ
in the punctured sector
(5.133)
S
˙
κ
-
,
κ
+
,
C
=
{
s
∈
ℂ
*
|
s
|
<
C
,
κ
-
-
δ
-
2
<
arg
(
s
)
<
κ
+
+
δ
+
2
}
∖
⋃
k
=
1
m
{
a
k
}
.
Moreover, for all
1
≤
k
≤
m
, and any path
Γ
0
,
k
⊂
S
˙
κ
-
,
κ
+
,
C
from 0 to a neighborhood of
a
k
, if we denote by
g
Γ
0
,
k
(
s
)
the analytic continuation of
g
along
Γ
0
,
k
, then there exists a constant
C
k
>
0
such that
(5.134)
|
g
Γ
0
,
k
(
s
)
|
≤
C
k
|
log
(
s
-
a
k
)
|
as
s
tends to
a
k
in a sector centered at
a
k.
Proof. For the sake of completeness, we give a sketch of proof of this theorem. In the first step, let us consider the following sums of Cauchy integrals
(5.135)h(t)=12iπ∑k=1m‍1k!∫Lak,κ-,C‍gk(τ-ak)τ-tdτ,
where Lak,κ-,C is the segment starting from ak in the direction κ- with length C. The multivalued function h(t) can be analytically continued along any path Γ in ℂ∖{a1,…,am} by deforming the path of integration Lak,κ-,C in the sector Sk and keeping the endpoints of the segment Lak,κ-,C fixed for all 1≤k≤m. Moreover, let 1≤k≤m and t∈Lak,κ-∖{ak}, where Lak,κ- denotes the half-line starting from ak in the direction κ-. We denote by hΓak,t,ρ(t) the analytic continuation of h(t) along a loop Γak,t,ρ around ak constructed as follows: the loop follows a segment starting from t in the direction ak then turns around ak along a circle Γak,ρ of small radius ρ>0 positively oriented and then goes back to t following the same segment. We have that
(5.136)h(t)-hΓak,t,ρ(t)=gk(t-ak)k!.
Indeed, by the Cauchy theorem, one can write h(t)-hΓak,t,ρ(t) as a Cauchy integral
(5.137)Ik=12iπk!∫𝒞ak,C‍gk(τ-ak)τ-tdτ,
where 𝒞ak,C is a positively oriented closed curve enclosing t starting from ak and containing the point ak+Ceiκ-. By the residue theorem, one gets that Ik=gk(t-ak)/k!. From the relation (5.136), we also deduce the existence of a holomorphic function b(t) near ak such that
(5.138)h(t)=-gk(t-ak)2iπk!log(t-ak)+b(t)
for all t near ak, for a well-chosen determination of the logarithm log(x).
In the second step, let us define the truncated Laplace transforms and Laplace transforms(5.139)HC′+(ϵ)=∫Lκ+,C′‍h(s)e-s/ϵds,HC′-(ϵ)=∫Lκ-,C′‍h(s)e-s/ϵds,H+(ϵ)=∫Lκ+‍h(s)e-s/ϵds,H-(ϵ)=∫Lκ-‍h(s)e-s/ϵds,
where Lκ+,C′ is the segment starting from 0 to C′eiκ+ and Lκ-,C′ is the segment starting from 0 to C′eiκ-, for any fixed C′>C. By the Cauchy formula, one can write the difference HC′+(ϵ)-HC′-(ϵ) as the sum
(5.140)HC′+(ϵ)-HC′-(ϵ)=-∑k=1m‍∫Γak,ρ‍h(s)e-s/ϵds+∫Lak,ρ,C′,κ-‍(h(s)-hΓak,s,ρ(s))e-s/ϵds+𝒪(e-Ceiα/ϵ),
where Lak,ρ,C′,κ- is the segment starting from ak+ρeiκ- to ak+C′eiκ- for any ρ>0 small enough. Due to the decomposition (5.138), h(s) is integrable at ak. By letting ρ tending to 0 and C′ tending to infinity, using the relation (5.136) in (5.140), ones gets that
(5.141)H+(ϵ)-H-(ϵ)=∑k=1m‍1k!∫Lak,κ-‍gk(s-ak)e-s/ϵds+𝒪(e-Ceiα/ϵ)=∑k=1m‍e-ak/ϵk!∫Lκ-‍gk(s)e-s/ϵds+𝒪(e-Ceiα/ϵ),
where Lak,κ- is the half-line starting from ak in the direction κ-.
Now, one considers the differencesD+(ϵ)=f+(ϵ)-H+(ϵ) and D-(ϵ)=f-(ϵ)-H-(ϵ). From the Stokes relations (5.131) and (5.141), one deduces that
(5.142)D+(ϵ)-D-(ϵ)=𝒪(e-Ceiα/ϵ)
for all ϵ∈ℰ+∩ℰ-. Using a similar Borel transform integral representation as in the proof of Theorem 1 in [28], one can show that the difference g(s)-h(s), which is by construction analytic near the origin in ℂ, can be analytically continued to a function G(s), which is holomorphic on the sector Sκ-,κ+,C={s∈ℂ*/|s|<C,κ-<arg(s)<κ+}. Since h can be analytically continued along any path in ℂ∖{a1,…,an}, one gets that the function g can be analytically continued along any path in S˙κ-,κ+,C and from the decomposition (5.138) one deduces the estimates (5.134).
Now, we return to the proof of Theorem5.8. From the formula (4.76) and Proposition 5.2, the following equality
(5.143)∫Lκ1‍g0,1(s,t,z)e-s/ϵds=∫Lκ0‍g0,0(s,t,z)e-s/ϵds+∑h≥1‍exp(-hλ/ϵt)h!∫Lκ0‍gh,0(s,t,z)e-s/ϵds
holds for all ϵ∈ℰ0∩ℰ1, and all t∈𝒯∩D(0,ι′′), all z∈D(0,δD0,1). Let t∈𝒯∩D(0,ι′′) and z∈D(0,δD0,1) fixed. Let m≥1 be an integer. From the estimates (4.78), we get that
(5.144)∑h≥m+1‍|exp(-hλ/ϵt)h!∫Lκ0‍gh,0(s,t,z)e-s/ϵds|≤2C~0∑h≥m+1‍|exp(-hλϵt)|(2u1)h≤2C~0(2u1)m+1|exp(-(m+1)λϵt)|11-2|exp(-λ/ϵt)|/u1
for all ϵ∈ℰ0∩ℰ1. From (5.143) and (5.144), we deduce that the following Stokes relation
(5.145)∫Lκ1‍g0,1(s,t,z)e-s/ϵds=∫Lκ0‍g0,0(s,t,z)e-s/ϵds+∑h=1m‍exp(-hλ/ϵt)h!∫Lκ0‍gh,0(s,t,z)e-s/ϵ+𝒪(e-(m+1)λ/(ϵt)).
Holds, where 𝒪(e-(m+1)λ/(ϵt)) is a holomorphic function R(ϵ) on ℰ0∩ℰ1 such that there exists a constant H>0 with
(5.146)|R(ϵ)|≤H|e-(m+1)λ/(ϵt)|
for all ϵ∈ℰ0∩ℰ1. We can apply Theorem (FS) with ak=kλ/t, for 1≤k≤m, C=|λ|(m+1)/|t| to get that the function s↦g0(s,t,z) (constructed in Proposition 4.15) can be analytically continued along any path in the punctured sector
(5.147)S˙κ0,κ1,t,λ,m={s∈ℂ*|s|<|λ|(m+1)|t|,κ0-op(𝒢κ0)2<arg(s)<κ1+op(𝒢κ1)2}∖⋃k=1m{λkt}
as a function denoted by g0Γ,t,z(s). Moreover, for all 1≤k≤m, and any path Γ0,k⊂S˙κ0,κ1,t,λ,m from 0 to a neighborhood of λk/t, there exists a constant Ck>0 such that |g0Γ0,k,t,z(s)|≤Ck|log(s-λk/t)| as s tends to λk/t in a sector centered at λk/t. Since this result is true for all m≥1, Theorem 5.8 follows.In the next result, Onee show that under the additional hypothesis that the coefficients of (4.90) are polynomials in the parameter ϵ, the function g0(s,t,z) solves a singular linear partial differential equation in ℂ3.Corollary 5.9.
Let the assumptions of Theorem5.8 hold. We assume moreover that, for all tuple (s,k0,k1) chosen in the set 𝒮, the coefficients bs,k0,k1(z,ϵ) belong to ℂ{z}[ϵ] with the following expansion in ϵ:
(5.148)bs,k0,k1(z,ϵ)=∑m=k0ds,k0,k1‍bs,k0,k1m(z)ϵm
for some ds,k0,k1≥k0. Then, for all K∈ℕ with K≥1 and K≥max{ds,k0,k1∈ℕ/(s,k0,k1)∈𝒮}, the function g0(u,t,z) (constructed in Proposition 4.15) satisfies the following singular linear partial differential equation
(5.149)t2∂t∂uK-1∂zSg0(u,t,z)+∂uK∂zSg0(u,t,z)=-t∂uK-1∂zSg0(u,t,z)+∑(s,k0,k1)∈𝒮‍∑m=k0ds,k0,k1‍bs,k0,k1m(z)ts(∂uK-m∂tk0∂zk1g0)(u,t,z)
for all (u,t,z)∈D(0,s0)×(𝒯∩D(0,ι′′))×D(0,δD0,1). From Theorem 5.8, for all (t,z)∈(𝒯∩D(0,ι′′))×D(0,δD0,1), this solution g0(u,t,z) can be analytically continued with respect to u along any path in the punctured sector S˙κ0,κ1,t,λ with logarithmic estimates (5.129) near the singular points λk/t for all k≥1.Proof.
From Proposition4.15, we have that the function
(5.150)X0,0(t,z,ϵ)=ϵ-1∫Lκ0‍g0,0(s,t,z)e-s/ϵds
solves (5.122) on (𝒯∩D(0,ι′′))×D(0,δD0,1)×ℰ0. From the formulas in Proposition 4.2, we deduce that the function g0,0(u,t,z) solves the singular integrodifferential equation
(5.151)t2∂t∂u-1∂zSg0,0(u,t,z)+∂zSg0,0(u,t,z)=-t∂u-1∂zSg0,0(u,t,z)+∑(s,k0,k1)∈𝒮‍∑m=k0ds,k0,k1‍bs,k0,k1m(z)ts(∂u-m∂tk0∂zk1g0,0)(u,t,z)
for all (u,t,z)∈(𝒢κ0∪D(0,s0))×(𝒯∩D(0,ι′′))×D(0,δD0,1). Since g0(u,t,z) is holomorphic on D(0,s0)×(𝒯∩D(0,ι′′))×D(0,δD0,1) and has g0,0(u,t,z) as analytic continuation on (𝒢κ0∪D(0,s0))×(𝒯∩D(0,ι′′))×D(0,δD0,1), we get that g0(u,t,z) also solves (5.151). By differentiating K times of each hand side of the equation with respect to u, one gets that g0(u,t,z) solves the partial differential equation (5.149).


## 5.1. Assumptions on the Initial Data

We keep the notations of the previous section. Now, we make the following additional assumption that there exists a sequence of unbounded open sectorsSd0,ϑn such that
(5.1)Sd0⊂Sd0,ϑn⊂ℳd0∪Sd0
for all n≥0 and a sequence of real numbers ζn, n≥0 such that
(5.2)eiζn∈Sd0,ϑn,limn→+∞ζn=arg(λ)
with the property that arg(eiζn/ϵt)∈(-π/2+δ𝒯,π/2-δ𝒯) for all ϵ∈ℰ0∩ℰ1, all t∈𝒯, and for all n≥0 (where 𝒯 and δ𝒯 were introduced in Definition 4.14). We also make the assumption that for all n≥0, the function vh,j,Sd0,ℰ0(τ,ϵ) can be analytically continued to a holomorphic function τ↦vh,j,Sd0,ϑn,ℰ0(τ,ϵ) on Sd0,ϑn for all ϵ∈ℰ0 such that
(5.3)vh,j,Sd0,ϑn,ℰ0(τ,ϵ)∈Ej,ϵ,σ,(Sd0,ϑn∪D(0,τ0))×ℰ0
with the property that
(5.4)Wj,Sd0,ϑn,ℰ0(u)∶=∑h≥0‍supϵ∈ℰ0∥vh,j,Sd0,ϑn,ℰ0(τ,ϵ)∥j,ϵ,σ,(Sd0,ϑn∪D(0,τ0))×ℰ0uhh!∈ℂ{u},0≤j≤S-1
and has a common radius of absolute convergence (denoted by uℰ0>0) for all n≥0. From the assumption (5.4), we get a constant u0,j>0 (depending on j∈{0,…,S-1}) and a constant Cn,j>0 (depending on n and j∈{0,…,S-1}) such that
(5.5)supϵ∈ℰ0∥vh,j,Sd0,ϑn,ℰ0(τ,ϵ)∥j,ϵ,σ,(Sd0,ϑn∪D(0,τ0))×ℰ0≤Cn,j(1u0,j)hh!
for all h≥0. We deduce that
(5.6)|vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)|≤Cn,j(1u0,j)hh!exp(σ2|ϵ|rb(j)r)
for all r≥0, all ϵ∈ℰ0, all 0≤j≤S-1, and all h≥0. In particular, we have that r↦vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to the space L0,σ~/2,ϵ for σ~>σrb(S-1). Moreover, from Proposition 2.7, we deduce that r↦vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to the space 𝒟0,σ~,ϵ′ and that there exists a universal constant C1>0 such that
(5.7)∥vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥0,σ~,ϵ,d≤C1∥vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥0,σ~/2,ϵ≤2|ϵ|σ~-σrb(S-1)C1Cn,j(1u0,j)hh!
for all 0≤j≤S-1, all h≥0, and all n≥0, all ϵ∈ℰ0.We make the crucial assumption that for all0≤j≤S-1, there exists a sequence of distributions vh,j,ℳd0,ℰ0(r,ϵ)∈𝒟0,σ~,ϵ′, for h≥0, a constant uj>0 and a sequence In,j>0 with limn→+∞In,j=0 such that
(5.8)supϵ∈ℰ0∥vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)-vh,j,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d≤In,jh!(1uj)h
for all n≥0 and all h≥0. From the estimates (5.7) and (5.8), we deduce that
(5.9)∑h≥0‍supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,duhh!∈ℂ{u},0≤j≤S-1.Lemma 5.1.
Letσ~>σrb(S-1). One can write the initial data γ0,j(t,ϵ) in the form of a Laplace transform in direction arg(λ),
(5.10)γ0,j(t,ϵ)=ℒarg(λ)(𝕍j,arg(λ),Sd0,ℰ0(r,ϵ))(ϵt),
where 𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)∈𝒟0,σ~,ϵ′ and for all 0≤j≤S-1, and all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′).Proof.
For0≤j≤S-1, from the definition of the initial data, we can write
(5.11)γ0,j(t,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!1ϵt∫Lζn‍vh,j,Sd0,ϑn,ℰ0(τ,ϵ)exp(-τϵt)dτ=∑h≥0‍exp(-(h|λ|eiarg(λ))/ϵt)h!eiζnϵt∫0+∞‍vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)exp(-reiζnϵt)dr
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′), and all n≥0. Now, we can write
(5.12)ℒζn(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵt)=ℒarg(λ)(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵtei(arg(λ)-ζn))
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′), all n≥0. From the continuity estimates (3.3) for the Laplace transform, we deduce that for given t∈𝒯∩D(0,ι′), ϵ∈ℰ0∩ℰ1, there exists a constant Cϵ,t (depending on ϵ, t) such that
(5.13)|ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵt)-ℒarg(λ)(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵtei(arg(λ)-ζn))|≤Cϵ,t∥(vh,j,ℳd0,ℰ0(r,ϵ)-vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥0,σ~,ϵ,d+|ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵtei(arg(λ)-ζn))-ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵt)|
for all n≥0. By letting n tend to +∞ in this latter inequality and using the hypothesis (5.8), we get that
(5.14)ℒζn(vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ))(ϵt)=ℒarg(λ)(vh,j,ℳd0,ℰ0(r,ϵ))(ϵt)
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′), and all n≥0.
On the other hand, from Corollary2.10, we have that for all h≥0, the distribution ∂r-h(vh,j,ℳd0,ℰ0(r,ϵ)) belongs to 𝒟0,σ~,ϵ′ and that there exists a universal constant C3>0 such that
(5.15)∥∂r-h(vh,j,ℳd0,ℰ0(r,ϵ))∥0,σ~,ϵ,d≤C3(|ϵ|σ~)h∥vh,j,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d
for all h≥0, all 0≤j≤S-1. From (5.14) and using Propositions 3.3 and 3.7, we can write
(5.16)exp(-h|λ|eiarg(λ)/ϵt)h!eiζnϵt∫0+∞‍vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)exp(-reiζnϵt)dr=(eiarg(λ)ϵt)hexp(-h|λ|eiarg(λ)/ϵt)h!ℒarg(λ)(∂r-h(vh,j,ℳd0,ℰ0(r,ϵ)))(ϵt)=ℒarg(λ)(𝕍h,j,λ,ℳd0,ℰ0(r,ϵ))(ϵt),
where
(5.17)𝕍h,j,λ,ℳd0,ℰ0(r,ϵ)=(fh,j,λ,ℳd0,ℰ0(r-|λ|h,ϵ)1[|λ|h,+∞)(r))(h)h!∈𝒟0,σ~,ϵ′
with fh,j,λ,ℳd0,ℰ0(r,ϵ)=∂r-h(vh,j,ℳd0,ℰ0(r,ϵ))∈𝒟0,σ~,ϵ′ for all h≥0 and all 0≤j≤S-1. From Proposition 3.6, we have a universal constant A>0 and a constant B(σ~,b,ϵ) (depending on σ~, b, and ϵ, which tend to zero as ϵ→0) such that
(5.18)∥𝕍h,j,λ,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d≤A(B(σ~,b,ϵ))hh!∥fh,j,λ,ℳd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d.
From the estimates (5.9) and using (5.15), (5.18), we deduce that the distribution
(5.19)𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)=∑h≥0‍𝕍h,j,λ,ℳd0,ℰ0(r,ϵ)∈𝒟0,σ~,ϵ,d′
for all 0≤j≤S-1, if ϵ0>0 is chosen small enough. Finally, by the continuity estimates (3.3) for the Laplace transform ℒarg(λ) and the formula (5.11), (5.16), we get the expression (5.10).On the other hand, we assume the existence of a sequence of unbounded open sectorsSd1,δn with
(5.20)Sd1⊂Sd1,δn⊂ℳd1∪Sd1
for all n≥0 and a sequence of real numbers ξn, n≥0 such that
(5.21)eiξn∈Sd1,δn,limn→+∞ξn=arg(λ)
with the property that arg(eiξn/ϵt)∈(-π/2+δ𝒯,π/2-δ𝒯) for all ϵ∈ℰ0∩ℰ1, all t∈𝒯, and all n≥0 (where 𝒯 and δ𝒯 are introduced in Definition 4.14). We make the assumption that for all n≥0, the function vh,j,Sd1,ℰ1(τ,ϵ) can be analytically continued to a holomorphic function τ↦vh,j,Sd1,δn,ℰ1(τ,ϵ) on Sd1,δn for all ϵ∈ℰ1 such that
(5.22)vh,j,Sd1,δn,ℰ1(τ,ϵ)∈Ej,ϵ,σ,(Sd1,δn∪D(0,τ0))×ℰ1
with the property that
(5.23)Wj,Sd1,δn,ℰ1(u)∶=∑h≥0‍supϵ∈ℰ1∥vh,j,Sd1,δn,ℰ1(τ,ϵ)∥j,ϵ,σ,(Sd1,δn∪D(0,τ0))×ℰ1uhh!∈ℂ{u},0≤j≤S-1
and has a common radius of absolute convergence (defined by uℰ1>0) for all n≥0. From the assumption (5.23), we get a constant u1,j>0 (depending on j∈{0,…,S-1}) and a constant Cn,1,j>0 (depending on n and j∈{0,…,S-1}) such that
(5.24)supϵ∈ℰ1∥vh,j,Sd1,δn,ℰ1(τ,ϵ)∥j,ϵ,σ,(Sd1,δn∪D(0,τ0))×ℰ1≤Cn,1,j(1u1,j)hh!
for all h≥0. We deduce that
(5.25)|vh,j,Sd1,δn,ℰ1(reiξn,ϵ)|≤Cn,1,j(1u1,j)hh!exp(σ2|ϵ|rb(j)r)
for all r≥0, all ϵ∈ℰ1, all 0≤j≤S-1, and all h≥0. In particular, we have that r↦vh,j,Sd1,δn,ℰ1(reiξn,ϵ) belongs to the space L0,σ~/2,ϵ for σ~>σrb(S-1). Moreover, from Proposition 2.7, we deduce that r↦vh,j,Sd1,δn,ℰ1(reiξn,ϵ) belongs to the space 𝒟0,σ~,ϵ′ and that there exists a universal constant C1>0 such that
(5.26)∥vh,j,Sd1,δn,ℰ1(reiξn,ϵ)∥0,σ~,ϵ,d≤C1∥vh,j,Sd1,δn,ℰ1(reiξn,ϵ)∥0,σ~/2,ϵ≤2|ϵ|σ~-σrb(S-1)C1Cn,1,j(1u1,j)hh!
for all 0≤j≤S-1, all h≥0, all n≥0, and all ϵ∈ℰ1.Now, we make the crucial assumption that for all0≤j≤S-1, there exists a sequence Jn,j>0 with limn→+∞Jn,j=0 such that
(5.27)supϵ∈ℰ0∩ℰ1∥v0,j,Sd1,δn,ℰ1(reiξn,ϵ)-𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)∥0,σ~,ϵ,d≤Jn,j
for all n≥0, where 𝕍j,arg(λ),Sd0,ℰ0(r,ϵ) are the distributions defined in Lemma 5.1.


## 5.2. The Stokes Relation and the Main Result

In the next proposition, we establish a connection formula for the two holomorphic solutionsX0,0(t,z,ϵ) and X0,1(t,z,ϵ) of (4.90) constructed in Proposition 4.15.Proposition 5.2.
Let the assumptions (5.1), (5.4), (5.8), (5.20), (5.23), and (5.27) hold for the initial data. Then, there exists 0<δD0,1<δZ0 such that one can write the following connection formula:
(5.28)X0,1(t,z,ϵ)=Z0(t,z,ϵ)=X0,0(t,z,ϵ)+∑h≥1‍exp(-hλ/ϵt)h!Xh,0(t,z,ϵ)
for all ϵ∈ℰ0∩ℰ1, all t∈𝒯∩D(0,ι′′), and all z∈D(0,δD0,1).The proof of this proposition will need two long steps and will be the consequence of the formula (5.79) and (5.124) from Lemmas 5.5 and 5.7.Step 1.
In this step, we show that the functionZ0(t,z,ϵ) can be expressed as a Laplace transform of some staircase distribution in direction arg(λ) satisfying the problem (5.80), (5.81).From the assumption (5.4), we deduce from Proposition 4.12 that the function Vh,Sd0,ℰ0(τ,z,ϵ) constructed in (4.80) has an analytic continuation denoted by Vh,Sd0,ϑn,ℰ0(τ,z,ϵ) on the domain (Sd0,ϑn∪D(0,τ0))×D(0,δℰ0)×ℰ0 which satisfies estimates of the form (4.62) for all n≥0, where δℰ0>0 depends on S, uℰ0 (which denotes a common radius of convergence of the series (5.4)), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ. This constant δℰ0 is, therefore, independent of n and h. Now, one defines the functions
(5.29)𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ):=Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)
for all r≥0, all z∈D(0,δℰ0), all ϵ∈ℰ0, and all n≥0.Lemma 5.3.
Letσˇ>σ~>σrb(S-1). Then, there exists 0<δD<δℰ0 (depending on S, b, σˇ, |λ|, uj, 0≤j≤S-1 (introduced in (5.8)), 𝒮, uℰ0, ρ, μ, A, B (introduced in Lemma  5.4)), there exist M1>0 (depending on 𝒮, S, σˇ, |λ|, uj, for 0≤j≤S-1, ρ, μ, A, B), M1′>0 (depending on 𝒮, S, σˇ, |λ|, ρ, μ, ρ′, μ′ (introduced in Lemma  5.4), A, B, uj for 0≤j≤S-1) and a constant U1 (depending on 𝒮, S, σˇ, |λ|, ρ, μ, A, B, uℰ0, uj for 0≤j≤S-1) such that for all h≥0 all n≥0, there exists a staircase distribution 𝕍h,ℳd0,ℰ0(r,z,ϵ)∈𝒟'(σˇ,ϵ,δD) with
(5.30)supϵ∈ℰ0∥𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)-𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD)≤(M1max0≤j≤S-1In,j+M1′Dn)h!(2U1)h,
where In,j is a positive sequence (converging to 0 as n tends to ∞) introduced in the assumption (5.8) and Dn is the positive sequence (tending to 0 as n→+∞) introduced in Lemma  5.4. Moreover, one has
(5.31)∑h≥0‍supϵ∈ℰ0∥𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD)uhh!∈ℂ{u}.Proof.
From the estimates (4.54), we can write
(5.32)Vh,Sd0,ϑn,ℰ0(τ,z,ϵ)=∑β≥0‍Vh,β,Sd0,ϑn,ℰ0(τ,ϵ)zββ!,
where Vh,β,Sd0,ϑn,ℰ0(τ,ϵ) are holomorphic functions such that there exists a constant u1 such that 0<u1<uℰ0 (depending on uℰ0, 𝒮, and b, σ), a constant x1 such that 0<x1<ρ (depending on S, uℰ0, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ), and a constant CΩ(d0,ℰ0),n>0 (depending on max0≤j≤S-1Wj,Sd0,ϑn,ℰ0(uℰ0) (where Wj,Sd0,ϑn,ℰ0 are defined in (5.4)), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, uℰ0, x0, 𝒮, b) with
(5.33)|Vh,β,Sd0,ϑn,ℰ0(τ,ϵ)|≤CΩ(d0,ℰ0),nh!β!(2u1)h(2x1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)
for all τ∈Sd0,ϑn∪D(0,τ0), ϵ∈ℰ0, all h≥0, all β≥0, and all n≥0. We deduce that
(5.34)|Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)|≤CΩ(d0,ℰ0),n(2u1)h(2x1)βh!β!exp(σ2|ϵ|rb(β)r)
for all r≥0, all ϵ∈ℰ0, all β≥0, all h≥0, and all n≥0. In particular, r↦Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to Lβ,σˇ/2,ϵ. From Proposition 2.7, we deduce that r↦Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ) belongs to 𝒟β,σˇ,ϵ′. From Proposition 2.7 and (5.34), we get a universal constant C1>0 such that
(5.35)∥Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)∥β,σˇ,ϵ,d≤C1∥Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)∥β,σˇ/2,ϵ≤C1CΩ(d0,ℰ0),n2|ϵ|σˇ-σ(2u1)h(2x1)βh!β!
for all β≥0, all h≥0, and all n≥0. From (5.35), we deduce that the distribution
(5.36)𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)=∑β≥0‍Vh,β,Sd0,ϑn,ℰ0(reiζn,ϵ)zββ!∈𝒟′(σˇ,ϵ,δˇ)
for all ϵ∈ℰ0, all δˇ<x1/2, all h≥0, and all n≥0.
One gets from (4.20), (4.21) and the assumption (4.44) that the following problem holds:
(5.37)(reiζn+1+λh)∂zS𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0b~s,k0,k1(z,ϵ)ϵk0-sei(s-k0)ζn(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍ϵk0b~s,k0,k1(z,ϵ)cqk01(hλ)qϵk0-sei(s-k0-q)ζn×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p∂zk1𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ))
with initial data
(5.38)(∂zj𝕍h,Sd0,ϑn,ℰ0)(r,0,ϵ)=vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ),0≤j≤S-1.
On the other hand, we consider the problem(5.39)(reiarg(λ)+1+λh)∂zS𝕍h,ℳd0,ℰ0(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0b~s,k0,k1(z,ϵ)ϵk0-sei(s-k0)arg(λ)(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍h,ℳd0,ℰ0(r,z,ϵ))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍ϵk0b~s,k0,k1(z,ϵ)cqk01(hλ)qϵk0-sei(s-k0-q)arg(λ)×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p∂zk1𝕍h,ℳd0,ℰ0(r,z,ϵ))
with initial data
(5.40)(∂zj𝕍h,ℳd0,ℰ0)(r,0,ϵ)=vh,j,ℳd0,ℰ0(r,ϵ),0≤j≤S-1.
In the next lemma, we give estimates for the coefficients of (5.37) and (5.39).
Lemma  5.4.Let(5.41)b~s,k0,k1(z,ϵ)=∑β≥0‍b~s,k0,k1,β(ϵ)zββ!the convergent Taylor expansion ofb~s,k0,k1with respect toznear 0. Letα∈ℝbe a real number. Then, there exist positive constantsA,B,ρ,ρ′,μ,μ′and a sequenceDn>0such thatlimn→+∞Dn=0with(5.42)|∂rq(b~s,k0,k1,β(ϵ)eiαarg(λ)reiarg(λ)+1+λh)|≤AB-ββ!q!(ρ(r+μ))q+1,|∂rq(b~s,k0,k1,β(ϵ)eiαζnreiζn+1+λh)|≤AB-ββ!q!(ρ(r+μ))q+1,(5.43)|∂rq(b~s,k0,k1,β(ϵ)eiαarg(λ)reiarg(λ)+1+λh)-∂rq(b~s,k0,k1,β(ϵ)eiαζnreiζn+1+λh)|≤DnB-ββ!q!(ρ′(r+μ′))q+1for allq≥0, allβ≥0, alln≥0, allh≥0, allr≥0, and allϵ∈ℰ0.
Proof. We first show (5.42). From the fact that b~s,k0,k1(z,ϵ) is holomorphic near z=0, we get from the Cauchy formula that there exist A,B>0 such that
(5.44)|b~s,k0,k1,β(ϵ)|≤AB-ββ!
for all β≥0, and all ϵ∈ℰ0. On the other hand, from Definition 4.14(3.1), there exist ρ,μ>0 such that |reiζn+1+λh|≥ρ(r+μ) for all r≥0, all h≥0, and all n≥0. Hence,
(5.45)|∂rq(eiαζnreiζn+1+λh)|≤q!|reiζn+1+λh|q+1≤q!(ρ(r+μ))q+1
for all r≥0, and all h≥0, all q≥0, all n≥0. We deduce (5.42) from (5.44) and (5.45).
Now, we show (5.43). Using the classical identities ab-cd=(a-c)b+c(b-d) and bq+1-aq+1=(b-a)×∑s=0q‍asbq-s, we get the estimates
(5.46)|∂rq(eiαarg(λ)reiarg(λ)+1+λh)-∂rq(eiαζnreiζn+1+λh)|≤q!|eiαarg(λ)eiqarg(λ)(reiarg(λ)+1+λh)q+1-eiαζneiqζn(reiζn+1+λh)q+1|≤q!(|eiζn-eiarg(λ)|×(∑s=1q+1‍r|reiarg(λ)+1+λh|q+2-s|reiζn+1+λh|s)+|eiαarg(λ)-eiαζn|+|eiarg(λ)-eiζn|(q+1)|reiarg(λ)+1+λh|q+1).
On the other hand, again from Definition 4.14 (3.1), there exist ρ1,μ1>0 such that
(5.47)|reiarg(λ)+1+λh|≥ρ1(r+μ1),|reiζn+1+λh|≥ρ1(r+μ1)
for all r≥0, all h≥0, and all n≥0. Using (5.46), (5.47) and the fact that q+1≤2q+1 for all q≥0, we deduce the estimates (5.43).
In the first part of the proof of Lemma5.3, we show the existence of a staircase distribution solution of the problem (5.39), (5.40), which satisfies the estimates (5.31). As a starting point, it is easy to check that the problem (5.39), (5.40) has a formal solution of the form
(5.48)𝕍h,ℳd0,ℰ0(r,z,ϵ)=∑β≥0‍𝕍h,β,ℳd0,ℰ0(r,ϵ)zββ!,
where r↦𝕍h,β,ℳd0,ℰ0(r,ϵ) are distributions on ℝ+, for which the next recursion holds:
(5.49)𝕍h,β+S,ℳd0,ℰ0(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!ϵk0-s×ei(s-k0)arg(λ)reiarg(λ)+1+λh(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-sei(s-k0-q)arg(λ)reiarg(λ)+1+λh(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)
for all β≥0, h≥0, with initial conditions
(5.50)𝕍h,j,ℳd0,ℰ0(r,ϵ)=vh,j,ℳd0,ℰ0(r,ϵ),0≤j≤S-1,h≥0.
Using Corollary 2.10, Propositions 2.11 and 2.12, the estimates (5.9), and Remark 2.4, we deduce that 𝕍h,β,ℳd0,ℰ0(r,ϵ)∈𝒟β,σˇ,ϵ′ for all h,β≥0 and that the following inequalities hold for the real numbers 𝕍h,β,ℳd0(ϵ):=||𝕍h,β,ℳd0,ℰ0(r,ϵ)||β,σˇ,ϵ,d: there exist constants C23.01, C23.02 (depending on 𝒮, σˇ, S, ρ, μ) with
(5.51)𝕍h,β+S,ℳd0(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.01β!AB-β1(β+S+1)(s-k0)b𝕍h,β2+k1,ℳd0(ϵ)β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.02β!AB-β1|cqk01||λ|qhq×(β+S+1)(s-k0-q)b𝕍h,β2+k1,ℳd0(ϵ)β2!
for all β,h≥0, where A,B>0 are defined in Lemma  5.4. We define the following Cauchy problem:
(5.52)∂xS𝕎ℳd0(u,x)=∑(s,k0,k1)∈𝒮‍C23.01(x∂x+S+1)b(s-k0)(A1-x/B∂xk1𝕎ℳd0(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C23.02|cqk01||λ|q(x∂x+S+1)b(s-k0-q)×(A1-x/B(u∂u)q∂xk1𝕎ℳd0(u,x))
for given initial data
(5.53)(∂xj𝕎ℳd0)(u,0)=𝕎ℳd0,j(u)=∑h≥0‍supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)∥j,σˇ,ϵ,duhh!∈ℂ{u},0≤j≤S-1.
From the assumption (4.42) and the fact that b>1, we deduce that
(5.54)S>b(s-k0-q)+q+k1
for all (s,k0,k1)∈𝒮 and all 0≤q≤k0. Hence, the assumption (2.108) is satisfied in Proposition 2.22 for the Cauchy problem (5.52), (5.53). Since the initial data 𝕎ℳd0,j(u) is an analytic function on a disc containing some closed disc D(0,U0), for 0≤j≤S-1 and since the coefficients of (5.52) are analytic on ℂ×D(0,B), we deduce that all the hypotheses of Proposition 2.22 are fulfilled for the problem (5.52), (5.53). We deduce the existence of a formal solution 𝕎ℳd0(u,x)∈G(Uℳd0,Xℳd0), where 0<Uℳd0<U0 (depending on 𝒮) and 0<Xℳd0≤B/2 (depending on S, σˇ, |λ|, ρ, μ, U0, 𝒮, A, B).
Now, let𝕎ℳd0(u,x)=∑h,β≥0‍wh,β,ℳd0(uh/h!)(xβ/β!) be its Taylor expansion at the origin. Then, the sequence wh,β,ℳd0 satisfies the next equalities:
(5.55)wh,β+S,ℳd0=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.01β!AB-β1(β+S+1)(s-k0)bwh,β2+k1,ℳd0β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.02β!AB-β1|cqk01||λ|qhq×(β+S+1)(s-k0-q)bwh,β2+k1,ℳd0β2!
for all β,h≥0, with
(5.56)wh,j,ℳd0=supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)∥j,σˇ,ϵ,d,h≥0,0≤j≤S-1.
Gathering the inequalities (5.51), the equalities (5.55) with the initial conditions (5.56), one gets
(5.57)supϵ∈ℰ0|𝕍h,β,ℳd0(ϵ)|≤wh,β,ℳd0
for all h,β≥0. From (5.57) and the fact that 𝕎ℳd0(u,x)∈G(Uℳd0,Xℳ0), we get a constant Cℳd0>0 such that
(5.58)supϵ∈ℰ0∥𝕍h,β,ℳd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d≤Cℳd0(h+β)!(1Uℳd0)h(1Xℳd0)β≤Cℳd0h!β!(2Uℳd0)h(2Xℳd0)β
for all h,β≥0. From this last estimates (5.58), we deduce that for all h≥0, 𝕍h,ℳd0,ℰ0(r,z,ϵ) belongs to 𝒟′(σˇ,ϵ,δℳd0) for 0<δℳd0≤Xℳd0/4 and moreover that
(5.59)∑h≥0‍supϵ∈ℰ0∥𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δℳd0)uhh!∈ℂ{u}
holds. This yields the property (5.31).
In the second part of the proof, we show (5.30). One defines the distribution
(5.60)𝕍h,Sd0,ϑn,ℰ0Δ(r,z,ϵ):=𝕍h,ℳd0,ℰ0(r,z,ϵ)-𝕍h,Sd0,ϑn,ℰ0(r,z,ϵ)
for all r≥0, all z∈D(0,δℳd0)∩D(0,δˇ), with 0<δˇ<x1/2, all ϵ∈ℰ0. If one writes the  Taylor expansion
(5.61)𝕍h,Sd0,ϑn,ℰ0Δ(r,z,ϵ)=∑β≥0‍𝕍h,β,Sd0,ϑn,ℰ0Δ(r,ϵ)zββ!
for z∈D(0,δℳd0)∩D(0,δˇ), then the coefficients 𝕍h,β,Sd0,ϑn,ℰ0Δ(r,ϵ) satisfy the following recursion:
(5.62)𝕍h,β+S,Sd0,ϑn,ℰ0Δ(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!ϵk0-s(ei(s-k0)ζnreiζn+1+λh)×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍h,β2+k1,Sd0,ϑn,ℰ0Δ(r,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-sei(s-k0-q)ζnreiζn+1+λh×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p𝕍h,β2+k1,Sd0,ϑn,ℰ0Δ(r,ϵ)β2!)+𝔹h,β,n(r,ϵ),
where
(5.63)𝔹h,β,n(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!ϵk0-s(ei(s-k0)arg(λ)reiarg(λ)+1+λh-ei(s-k0)ζnreiζn+1+λh)×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍β!ϵk0b~s,k0,k1,β1(ϵ)β1!cqk01(hλ)q×ϵk0-s(ei(s-k0-q)arg(λ)reiarg(λ)+1+λh-ei(s-k0-q)ζnreiζn+1+λh)×(∑(m,p)∈𝒪s-k0-q2‍αm,p2,qrm∂r-p𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)β2!)
for all h≥0 and all β≥0. Now, we put 𝕍h,β,nΔ(ϵ)=||𝕍h,β,Sd0,ϑn,ℰ0Δ(r,ϵ)||β,σˇ,ϵ,d. Using Corollary 2.10, Propositions 2.11 and 2.12, and Lemma  5.4, we get that there exist constants C23.11,C23.12 (depending on 𝒮, σˇ, S, ρ, μ) such that the following inequalities:
(5.64)𝕍h,β+S,nΔ(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.11β!AB-β1(β+S+1)(s-k0)b𝕍h,β2+k1,nΔ(ϵ)β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.12β!AB-β1×|cqk01||λ|qhq(β+S+1)(s-k0-q)b𝕍h,β2+k1,nΔ(ϵ)β2!+𝔹h,β,n(ϵ)
hold for all h,β≥0, where A,B>0 are defined in Lemma  5.4 and 𝔹h,β,n(ϵ) is a sequence which satisfies the next estimates: there exist constants C23.13,C23.14>0 (depending on 𝒮,σˇ,S,ρ′,μ′) with
(5.65)𝔹h,β,n(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.13β!DnB-β1×(β+S+1)(s-k0)b∥𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)∥β2+k1,σˇ,ϵ,dβ2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.14β!DnB-β1×|cqk01||λ|qhq(β+S+1)(s-k0-q)b∥𝕍h,β2+k1,ℳd0,ℰ0(r,ϵ)∥β2+k1,σˇ,ϵ,dβ2!
for all h,β,n≥0, where Dn, n≥0 is the sequence defined in Lemma  5.4.
We consider the following sequence of Cauchy problem:(5.66)∂xS𝕎nΔ(u,x)=∑(s,k0,k1)∈𝒮‍C23.11(x∂x+S+1)b(s-k0)(A1-x/B∂xk1𝕎nΔ(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C23.12|cqk01||λ|q(x∂x+S+1)b(s-k0-q)×(A1-x/B(u∂u)q∂xk1𝕎nΔ(u,x))+𝔻n(u,x),
where
(5.67)𝔻n(u,x)=∑(s,k0,k1)∈𝒮‍C23.13(x∂x+S+1)b(s-k0)(Dn1-x/B∂xk1𝕎ℳd0(u,x))+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍C23.14|cqk01||λ|q(x∂x+S+1)b(s-k0-q)×(Dn1-x/B(u∂u)q∂xk1𝕎ℳd0(u,x))
and 𝕎ℳd0(u,x) is already defined as the solution of the problem (5.52), (5.53), for given initial data
(5.68)(∂xj𝕎nΔ)(u,0)=𝕎j,nΔ(u)=∑h≥0‍supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)-vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥j,σˇ,ϵ,duhh!∈ℂ{u},0≤j≤S-1,
which are convergent near the origin with respect to u due to the assumption (5.8) and Remark 2.4. Moreover, the initial data satisfy the estimates
(5.69)|𝕎j,nΔ(u)|≤In,j1-|u|/uj
for all |u|<uj, 0≤j≤S-1, all n≥0.
From the assumption (4.42) and the fact that b>1, we deduce that
(5.70)S>b(s-k0-q)+q+k1
for all (s,k0,k1)∈𝒮 and all 0≤q≤k0. Therefore, the assumption (2.108) is satisfied in Proposition 2.22 for the problem (5.66), (5.68).
On the other hand, from Lemmas2.20 and 2.21, there exist a constant Dℳd0>0 (depending on 𝒮,σˇ,S,ρ′,μ′,|λ|,B,Uℳd0,Xℳd0), a constant 0<U1,ℳd0<Uℳd0, and a constant 0<X1,ℳd0<Xℳd0 such that
(5.71)∥𝔻n(u,x)∥(U1,ℳd0,X1,ℳd0)≤DnDℳd0∥𝕎ℳd0(u,x)∥(Uℳd0,Xℳd0)≤DnDℳd0Cℳd0
for all n≥0, where the constant Cℳd0 is introduced in (5.58).
Since the initial data𝕎j,nΔ(u) is an analytic function on some disc containing the closed disc D-(0,uj/2), for 0≤j≤S-1 and the coefficients of (5.66) are analytic on ℂ×D(0,B), we deduce that all the hypotheses of Proposition 2.22 for the problem (5.66), (5.68) are fulfilled. We deduce the existence of a formal solution 𝕎nΔ(u,x)∈G(U1,X1) of (5.66), (5.68), where 0<U1<min(U1,ℳd0,min0≤j≤S-1uj/2) (depending on 𝒮) and 0<X1≤min(B/2,X1,ℳd0) (depending on S, σˇ, |λ|, uj, for 0≤j≤S-1, 𝒮, A, B, ρ, μ).
Moreover, from (2.111) and (5.71), there exist constants M1>0 (depending on S, σˇ, λ, uj, for 0≤j≤S-1, 𝒮, A, B, ρ, μ) and M2>0 (depending on S, uj for 0≤j≤S-1, B, 𝒮) such that
(5.72)∥𝕎nΔ(u,x)∥(U1,X1)≤M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0
for all n≥0. Now, let 𝕎nΔ(u,x)=∑h,β≥0‍wh,β,nΔ(uh/h!)(xβ/β!) be its Taylor expansion at the origin. Then, the sequence wh,β,nΔ satisfies the following equalities:
(5.73)wh,β+S,nΔ=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.11β!AB-β1(β+S+1)(s-k0)bwh,β2+k1,nΔβ2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.12β!AB-β1|cqk01||λ|qhq(β+S+1)(s-k0-q)bwh,β2+k1,nΔβ2!+𝔻h,β,n,
where
(5.74)𝔻h,β,n=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.13β!DnB-β1(β+S+1)(s-k0)bwh,β2+k1,ℳd0β2!+∑k01+k02=k0,k01≥1‍k0!k01!k02!∑q=1k01‍∑β1+β2=β‍C23.14β!DnB-β1|cqk01||λ|qhq(β+S+1)(s-k0-q)bwh,β2+k1,ℳd0β2!
for all h,β,n≥0, with
(5.75)wh,j,nΔ=supϵ∈ℰ0∥vh,j,ℳd0,ℰ0(r,ϵ)-vh,j,Sd0,ϑn,ℰ0(reiζn,ϵ)∥j,σˇ,ϵ,d,∀h≥0,∀0≤j≤S-1.
Gathering the inequalities (5.64), (5.65) and the equalities (5.73), with the initial conditions (5.75), one gets that
(5.76)supϵ∈ℰ0|𝕍h,β,nΔ(ϵ)|≤wh,β,nΔ
for all h,β≥0 and all n≥0.
From (5.76) and the estimates (5.72), we deduce that
(5.77)supϵ∈ℰ0∥Vh,β,Sd0,ϑn,ℰ0Δ(r,ϵ)∥β,σˇ,ϵ,d≤(M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0)(h+β)!(1U1)h(1X1)β≤(M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0)h!β!(2U1)h(2X1)β
for all h,β≥0, all n≥0. From (5.77), we get that
(5.78)supϵ∈ℰ0∥Vh,Sd0,ϑn,ℰ0(r,z,ϵ)-𝕍h,ℳd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD)≤(M1max0≤j≤S-1In,j+DnM2Dℳd0Cℳd0)h!(2U1)h
for all h≥0 and all 0<δD≤X1/4. This yields the estimates (5.30).In the next lemma, we expressZ0(t,z,ϵ) as a Laplace transform of a staircase distribution.Lemma 5.5.
Letσˇ>σ~>σrb(S-1). Then, one can write the solution Z0(t,z,ϵ) of (4.90), (4.91) in the form of a Laplace transform in direction arg(λ)(5.79)Z0(t,z,ϵ)=ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δD,Z0)×(ℰ0∩ℰ1), where 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∈𝒟′(σˇ,ϵ,δD,Z0) (with δD,Z0=min(δD,δZ0)) solves the following Cauchy problem:
(5.80)(reiarg(λ)+1)∂zS𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0-sbs,k0,k1(z,ϵ)(ei(s-k0)arg(λ)∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)),
where the sets 𝒪s-k01 and the integers αm,p1 are introduced in (4.20), with initial data
(5.81)(∂zj𝕍arg(λ),Sd0,ℰ0)(r,0,ϵ)=𝕍j,arg(λ),Sd0,ℰ0(r,ϵ),0≤j≤S-1.Proof.
From Proposition4.17, we can write the solution Z0(t,z,ϵ) of (4.90), (4.91) in the form
(5.82)Z0(t,z,ϵ)=∑h≥0‍exp(-hλ/ϵt)h!1ϵt∫Lζn‍Vh,Sd0,ϑn,ℰ0(τ,z,ϵ)exp(-τϵt)dτ=∑h≥0‍exp(-h|λ|eiarg(λ)/ϵt)h!eiζnϵt∫0+∞‍Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)exp(-reiζnϵt)dr
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1) and all n≥0. Now, we write
(5.83)ℒζn(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵt)=ℒarg(λ)(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵtei(arg(λ)-ζn))
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δZ0)×(ℰ0∩ℰ1) and all n≥0. Now, we define δD,Z0=min(δD,δZ0). From the continuity estimates (3.5) for the Laplace transform, we deduce that for given ϵ∈ℰ0∩ℰ1, t∈𝒯∩D(0,ι′′), there exists a constant Cϵ,t (depending on ϵ, t) such that
(5.84)|ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵt)-ℒarg(λ)(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵtei(arg(λ)-ζn))|≤Cϵ,t∥𝕍h,ℳd0,ℰ0(r,z,ϵ)-Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)∥(σˇ,ϵ,d,δD,Z0)+|ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵtei(arg(λ)-ζn))-ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵt)|
for all z∈D(0,δD,Z0), all n≥0. By letting n tend to +∞ in this latter inequality and using the estimates (5.30), we obtain
(5.85)ℒζn(Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ))(ϵt)=ℒarg(λ)(𝕍h,ℳd0,ℰ0(r,z,ϵ))(ϵt)
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δD,Z0)×(ℰ0∩ℰ1) and all n≥0.
On the other hand, from Corollary2.10, we have that for all h≥0, the distribution ∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ)) belongs to 𝒟′(σˇ,ϵ,δD,Z0) and that there exists a universal constant C3>0 such that
(5.86)∥∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ))∥σˇ,ϵ,d,δD,Z0≤C3(|ϵ|σˇ)h∥𝕍h,ℳd0,ℰ0(r,z,ϵ)∥σˇ,ϵ,d,δD,Z0
for all h≥0.
From (5.85) and using Propositions 3.3 and 3.7, we can write
(5.87)exp(-h|λ|eiarg(λ)/ϵt)h!eiζnϵt∫0+∞‍Vh,Sd0,ϑn,ℰ0(reiζn,z,ϵ)exp(-reiζnϵt)dr=(eiarg(λ)ϵt)hexp(-h|λ|eiarg(λ)/ϵt)h!ℒarg(λ)(∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ)))(ϵt)=ℒarg(λ)(𝕍h,λ,ℳd0,ℰ0(r,z,ϵ))(ϵt),
where
(5.88)𝕍h,λ,ℳd0,ℰ0(r,ϵ)=(fh,λ,ℳd0,ℰ0(r-|λ|h,z,ϵ)1[|λ|h,+∞)(r))(h)h!∈𝒟′(σˇ,ϵ,δD,Z0)
with fh,λ,ℳd0,ℰ0(r,z,ϵ)=∂r-h(𝕍h,ℳd0,ℰ0(r,z,ϵ))∈𝒟'(σˇ,ϵ,δD,Z0), for all h≥0, all 0≤j≤S-1. From Proposition 3.6, we have a universal constant A>0 and a constant B(σˇ,b,ϵ) (depending on σˇ, b, and ϵ, which tend to zero as ϵ→0) such that
(5.89)∥𝕍h,λ,ℳd0,ℰ0(r,z,ϵ)∥σˇ,ϵ,d,δD,Z0≤A(B(σˇ,b,ϵ))hh!∥fh,λ,ℳd0,ℰ0(r,z,ϵ)∥σˇ,ϵ,d,δD,Z0.
From the convergence of the series (5.31) near the origin and using (5.86), (5.89), we deduce the distribution
(5.90)𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)=∑h≥0‍𝕍h,λ,ℳd0,ℰ0(r,z,ϵ)∈𝒟'(σˇ,ϵ,δD,Z0),
if ϵ0>0 is chosen small enough. Finally, by the continuity estimates (3.5) of the Laplace transform ℒarg(λ) and the formula (5.82), (5.87), we get the expression (5.79). Moreover, from the formulas in Proposition 3.3, as Z0(t,z,ϵ) solves the problem (4.90), (4.91), we deduce that the distribution 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ) solves the Cauchy problem (5.80), (5.81).Step 2.
In this step, we show that the functionX0,1(t,z,ϵ) can be expressed as a Laplace transform of some staircase distribution in direction arg(λ), satisfying the problem (5.80), (5.81).From the assumption (5.23), we deduce from Proposition 4.12, that the function V0,Sd1,ℰ1(τ,z,ϵ) constructed in (4.80) has an analytic continuation denoted by V0,Sd1,δn,ℰ1(τ,z,ϵ) on (Sd1,δn∪D(0,τ0))×D(0,δℰ1)×(ℰ0∩ℰ1) and satisfies estimates (4.62) for all n≥0, where δℰ1>0 depends on S,uℰ1 (which denotes a common radius of absolute convergence of the series (5.23), 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ. This constant δℰ1 is, therefore, independent of n. Now, one defines the functions
(5.91)𝕍0,Sd1,δn,ℰ1(r,z,ϵ)=V0,Sd1,δn,ℰ1(reiξn,z,ϵ)
for all r≥0, all z∈D(0,δℰ1), and all n≥0.Lemma 5.6.
Letσˇ>σ~>σrb(S-1) as in Lemma 5.3. Then, there exists 0<δD0,1<min(δℰ1,δD,Z0) (depending on 𝒮,S,σˇ,|λ|,A,B,ρ,μ and A~,B~,ρ~,μ~ introduced in Lemma  5.7), there exist M~1,M~1′ (depending on 𝒮,S,σˇ,|λ|,A,B,ρ,μ,A~,B~,ρ~,μ~ and ρ~′,μ~′ introduced in Lemma  5.7) such that
(5.92)supϵ∈ℰ0∩ℰ1∥𝕍0,Sd1,δn,ℰ1(r,z,ϵ)-𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD0,1)≤(M~1max0≤j≤S-1Jn,j+M~1′D~n)
for all n≥0, where 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ) is defined in Lemma 5.5 and solves the problem (5.80), (5.81) and D~n is the sequence (which tends to zero as n→+∞) defined in Lemma  5.7.Proof.
From the estimates (4.54), we can write
(5.93)V0,Sd1,δn,ℰ1(τ,z,ϵ)=∑β≥0‍V0,β,Sd1,δn,ℰ1(τ,ϵ)zββ!,
where V0,β,Sd1,δn,ℰ1(τ,ϵ) are holomorphic functions such that there exist a constant u1 with 0<u1<uℰ1 (depending on uℰ1, 𝒮, and b, σ), a constant x1 such that 0<x1<ρ (depending on S, uℰ1, 𝒮, b, σ, |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), where x0<ρ), and a constant CΩ(d1,ℰ1),n>0 (depending on max0≤j≤S-1Wj,Sd1,δn,ℰ1(uℰ1) (where Wj,Sd1,δn,ℰ1 are defined in (5.23)), |λ|, max(s,k0,k1)∈𝒮|b|s,k0,k1(x0), max(s,k0,k1)∈𝒮|b~|s,k0,k1(x0), S, uℰ1, x0, 𝒮, b) with
(5.94)|V0,β,Sd1,δn,ℰ1(τ,ϵ)|≤CΩ(d1,ℰ1),nβ!(2x1)β(1+|τ|2|ϵ|2)-1exp(σ2|ϵ|rb(β)|τ|)
for all τ∈Sd1,δn∪D(0,τ0), ϵ∈ℰ1, all β≥0, and all n≥0. We deduce that
(5.95)|V0,β,Sd1,δn,ℰ1(reiξn,ϵ)|≤CΩ(d1,ℰ1),n(2x1)ββ!exp(σ2|ϵ|rb(β)r)
for all r≥0, all ϵ∈ℰ1, all β≥0, and all n≥0. In particular, r↦V0,β,Sd1,δn,ℰ1(reiξn,ϵ) belongs to Lβ,σˇ/2,ϵ. From Proposition 2.7, we deduce that r↦V0,β,Sd1,δn,ℰ1(reiξn,ϵ) belongs to 𝒟β,σˇ,ϵ′. From Proposition 2.7 and (5.95), we get a universal constant C1>0 such that
(5.96)∥V0,β,Sd1,δn,ℰ1(reiξn,ϵ)∥β,σˇ,ϵ,d≤C1∥V0,β,Sd1,δn,ℰ1(reiξn,ϵ)∥β,σˇ/2,ϵ≤C1CΩ(d1,ℰ1),n2|ϵ|σˇ-σ(2x1)ββ!
for all β≥0 and all n≥0. From (5.96), we deduce that the distribution
(5.97)𝕍0,Sd1,δn,ℰ1(r,z,ϵ)=∑β≥0‍V0,β,Sd1,δn,ℰ1(reiξn,ϵ)zββ!∈𝒟'(σˇ,ϵ,δˇ)
for all ϵ∈ℰ0∩ℰ1, all δˇ<x1/2, and all n≥0.
From (4.20), (4.21), we have that the distribution 𝕍0,Sd1,δn,ℰ1(r,z,ϵ) solves the following problem:
(5.98)(reiξn+1)∂zS𝕍0,Sd1,δn,ℰ1(r,z,ϵ)=∑(s,k0,k1)∈𝒮‍ϵk0-sbs,k0,k1(z,ϵ)(ei(s-k0)ξn∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p∂zk1𝕍0,Sd1,δn,ℰ1(r,z,ϵ)),
where 𝒪s-k01 is the set and αm,p1 are the integers from (5.80), with initial data
(5.99)(∂zj𝕍0,Sd1,δn,ℰ1)(r,0,ϵ)=v0,j,Sd1,δn,ℰ1(reiξn,ϵ),0≤j≤S-1.
In the next lemma, we give estimates for the coefficients of (5.98) and (5.80). The proof is exactly the same as the one described for Lemma  5.4.
Lemma  5.7.Let(5.100)bs,k0,k1(z,ϵ)=∑β≥0‍bs,k0,k1,β(ϵ)zββ!be the convergent Taylor expansion ofbs,k0,k1with respect toznear 0. Then, there exist positive constantsA~,B~,ρ~,ρ~′,μ~,μ~′and a sequenceD~n>0such thatlimn→+∞D~n=0  with(5.101)|∂rq(bs,k0,k1,β(ϵ)ei(s-k0)arg(λ)reiarg(λ)+1)|≤A~B~-ββ!q!(ρ~(r+μ~))q+1,|∂rq(bs,k0,k1,β(ϵ)ei(s-k0)ξnreiξn+1)|≤A~B~-ββ!q!(ρ~(r+μ~))q+1,|∂rq(bs,k0,k1,β(ϵ)ei(s-k0)arg(λ)reiarg(λ)+1)-∂rq(bs,k0,k1,β(ϵ)ei(s-k0)ξnreiξn+1)|≤D~nB~-ββ!q!(ρ~′(r+μ~′))q+1for allq≥0, allβ≥0, alln≥0, allr≥0and allϵ∈ℰ0∩ℰ1.
Now, we consider the distribution(5.102)𝕍0,Sd1,δn,ℰ1Δ(r,z,ϵ):=𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)-𝕍0,Sd1,δn,ℰ1(r,z,ϵ)
for all r≥0, all z∈D(0,δD,Z0)∩D(0,δˇ), with 0<δˇ<x1/2 and δD,Z0 defined in Lemma 5.5, for all ϵ∈ℰ0∩ℰ1. One writes the Taylor expansions as follows:
(5.103)𝕍0,Sd1,δn,ℰ1Δ(r,z,ϵ)=∑β≥0‍𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ)zββ!,𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)=∑β≥0‍𝕍β,arg(λ),Sd0,ℰ0(r,ϵ)zββ!,
for z∈D(0,δD,Z0)∩D(0,δˇ); then the coefficients 𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ) satisfy the next recursion:
(5.104)𝕍0,β+S,Sd1,δn,ℰ1Δ(r,ϵ)=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!ϵk0-sei(s-k0)ξnreiξn+1×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍0,β2+k1,Sd1,δn,ℰ1Δ(r,ϵ)β2!)+∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍β!bs,k0,k1,β1(ϵ)β1!ϵk0-s(ei(s-k0)arg(λ)reiarg(λ)+1-ei(s-k0)ξnreiξn+1)×(∑(m,p)∈𝒪s-k01‍αm,p1rm∂r-p𝕍β2+k1,arg(λ),Sd0,ℰ0(r,ϵ)β2!)
for all h≥0, all β≥0. We put 𝕍0,β,n,ℰ1Δ(ϵ)=||𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ)||β,σˇ,ϵ,d. Using Corollary 2.10, Propositions 2.11 and 2.12 and Lemma  5.7, we get a constant C23.21>0 (depending on 𝒮,σˇ,S,ρ~,μ~) and C23.22>0 (depending on 𝒮,σˇ,S,ρ~′,μ~′) such that the next inequalities:
(5.105)𝕍0,β+S,n,ℰ1Δ(ϵ)≤∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.21β!A~B~-β1(β+S+1)b(s-k0)𝕍0,β2+k1,n,ℰ1Δ(ϵ)β2!+∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.22β!D~nB~-β1(β+S+1)b(s-k0)×∥𝕍β2+k1,arg(λ),Sd0,ℰ0(r,ϵ)∥β2+k1,σˇ,ϵ,dβ2!
hold for all β≥0, where A~,B~>0 and the sequence D~n, n≥0 are defined in Lemma  5.7.
We consider the following sequence of Cauchy problems:(5.106)∂xS𝕎n,ℰ1Δ(x)=∑(s,k0,k1)∈𝒮‍C23.21(x∂x+S+1)b(s-k0)(A~1-x/B~∂xk1𝕎n,ℰ1Δ(x))+𝔻~n(x),
where
(5.107)𝔻~n(x)=∑(s,k0,k1)∈𝒮‍C23.22(x∂x+S+1)b(s-k0)(D~n1-x/B~∂xk1𝕎arg(λ),ℰ0(x))
with
(5.108)𝕎arg(λ),ℰ0(x)=∑β≥0‍supϵ∈ℰ0∩ℰ1∥𝕍β,arg(λ),Sd0,ℰ0(x)∥β,σˇ,ϵ,dxββ!
for given initial data
(5.109)(∂xj𝕎n,ℰ1Δ)(0)=𝕎j,n,ℰ1Δ=supϵ∈ℰ0∩ℰ1∥𝕍j,arg(λ),Sd0,ℰ0(r,ϵ)-v0,j,Sd1,δn,ℰ1(reiξn,ϵ)∥j,σˇ,ϵ,d,0≤j≤S-1,
which are finite positive numbers due to the assumption (5.27) and Remark 2.4. Moreover, the initial data satisfy the estimates
(5.110)|𝕎j,n,ℰ1Δ|≤Jn,j
for all 0≤j≤S-1 and all n≥0.
On the other hand, we have that𝕎arg(λ),ℰ0(x) is convergent for all |x|≤Xℳd0/4 (where Xℳd0 is chosen in (5.58)). Indeed, we know, from (5.90), that
(5.111)𝕍h,λ,ℳd0,ℰ0(r,z,ϵ)=∑β≥0‍𝕍h,β,λ,ℳd0,ℰ0(r,ϵ)zββ!
is convergent for all |z|<δD,Z0, all r>0, and all h≥0. From (5.86) and (5.89), we know that
(5.112)∥𝕍h,β,λ,ℳd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d≤C3A(|ϵ|B(σˇ,b,ϵ)/σˇ)hh!∥𝕍h,β,ℳd0,ℰ0(r,ϵ)∥β,σ˘,ϵ,d
for all h≥0 and all β≥0. From (5.58) and (5.112), we deduce that
(5.113)∥𝕍β,arg(λ),Sd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d=∥∑h≥0‍𝕍h,β,λ,ℳd0,ℰ0(r,ϵ)∥β,σˇ,ϵ,d≤C3ACℳd0β!(2Xℳd0)β∑h≥0‍(2|ϵ|B(σˇ,b,ϵ)σˇUℳd0)h
and this last sum is convergent provided that ϵ0 is small enough. We deduce that 𝕎arg(λ),ℰ0(x) belongs to G(U,Xℳd0/4), for any U>0. Let C~ℳd0:=||𝕎arg(λ),ℰ0(x)||(U,Xℳd0/4).
From Lemmas2.20 and 2.21, we get constants D~ℳd0>0 (depending on 𝒮,σˇ,S,ρ~′,μ~′,B~,U,Xℳd0), 0<U~1,ℳd0<U, and 0<X~1,ℳd0<Xℳd0/4 such that
(5.114)∥𝔻~n(x)∥(U~1,ℳd0,X~1,ℳd0)≤D~nD~ℳd0C~ℳd0
for all n≥0.
From the assumption (4.42) and the fact that b>1, we deduce that
(5.115)S>b(s-k0)+k1
for all (s,k0,k1)∈𝒮. Hence, the assumption (2.108) is satisfied in Proposition 2.22 for the problem (5.106), (5.109). Moreover, the initial data 𝕎j,nΔ can be seen as constant functions (therefore analytic) with respect to a variable u on the closed disc D-(0,U) for any given U>0 and the coefficients of (5.106) are analytic with respect to x on D-(0,B~/2) and constant (therefore analytic) with respect to u on D-(0,U). We deduce that all the hypotheses of Proposition 2.22 for the problem (5.106), (5.109) are fulfilled. A direct computation shows that the problem (5.106), (5.109) has a unique formal solution 𝕎n,ℰ1Δ(x)=∑β≥0‍wβ,n,ℰ1Δxβ/β!, with wβ,n,ℰ1Δ∈ℂ. From Proposition 2.22, we deduce that 𝕎n,ℰ1Δ(x)∈G(U~1,X~1) where 0<U~1<U~1,ℳd0 (depending on 𝒮) and 0<X~1<min(B~/2,X~1,ℳd0) (depending on 𝒮,S,σˇ,A~,B~,ρ~,μ~). Moreover, from (2.111) and (5.114), there exist constants M~1>0 (depending on 𝒮,S,σˇ,A~,B~,ρ~,μ~) and M~2>0 (depending on 𝒮,B~,S) such that
(5.116)∥𝕎n,ℰ1Δ(x)∥(U~1,X~1)≤M~1max0≤j≤S-1Jn,j+D~nM~2D~ℳd0C~ℳd0
for all n≥0.
Now, the coefficientswβ,n,ℰ1Δ satisfy the following equalities:
(5.117)wβ+S,n,ℰ1Δ=∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.21β!A~B~-β1(β+S+1)b(s-k0)wβ2+k1,n,ℰ1Δβ2!+∑(s,k0,k1)∈𝒮‍∑β1+β2=β‍C23.22β!D~nB~-β1(β+S+1)b(s-k0)×supϵ∈ℰ0∩ℰ1∥𝕍β2+k1,arg(λ),Sd0,ℰ0(r,ϵ)∥β2+k1,σ˘,ϵ,dβ2!
for all β≥0 and all n≥0, with
(5.118)wj,n,ℰ1Δ=𝕎j,n,ℰ1Δ,0≤j≤S-1.
Gathering the inequalities (5.105) and the equalities (5.117), with the initial data (5.118), one gets that
(5.119)supϵ∈ℰ0∩ℰ1|𝕍0,β,n,ℰ1Δ(ϵ)|≤wβ,n,ℰ1Δ
for all β,n≥0.
From (5.119) and the estimates (5.116), we deduce that
(5.120)supϵ∈ℰ0∩ℰ1∥𝕍0,β,Sd1,δn,ℰ1Δ(r,ϵ)∥β,σˇ,ϵ,d≤(M~1max0≤j≤S-1Jn,j+D~nM~2D~ℳd0C~ℳd0)β!(1X~1)β
for all β,n≥0. From (5.120), we get that
(5.121)supϵ∈ℰ0∩ℰ1∥𝕍0,Sd1,δn,ℰ1(r,z,ϵ)-𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD0,1)≤2(M~1max0≤j≤S-1Jn,j+D~nM~2D~ℳd0C~ℳd0)
for all n≥0 and for all 0<δD0,1<X~1/2. This implies the estimates (5.92).In the following lemma, we express the functionX0,1(t,z,ϵ) as Laplace transform of a staircase distribution.Lemma 5.7.
Letσˇ>σ~>σrb(S-1) as in Lemma 5.3. Then, one can write the function X0,1(t,z,ϵ), which by construction of Proposition 4.12, solves the singularly perturbed Cauchy problem
(5.122)ϵt2∂t∂zSX0,1(t,z,ϵ)+(ϵt+1)∂zSX0,1(t,z,ϵ)=∑(s,k0,k1)∈𝒮‍bs,k0,k1(z,ϵ)ts(∂tk0∂zk1X0,1)(t,z,ϵ)
for given initial data
(5.123)(∂zjX0,1)(t,0,ϵ)=φ0,1,j(ϵt,ϵ),0≤j≤S-1
in the form of a Laplace transform in direction arg(λ)(5.124)X0,1(t,z,ϵ)=ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δD0,1)×(ℰ0∩ℰ1), where 𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∈𝒟'(σˇ,ϵ,δD0,1) solves the Cauchy problem (5.80), (5.81).Proof.
From Proposition4.12 and the assumption (5.23), we get that the function X0,1(t,z,ϵ) can be expressed as a Laplace transform in the direction ξn,
(5.125)X0,1(t,z,ϵ)=1ϵt∫Lξn‍V0,Sd1,δn,ℰ1(τ,z,ϵ)exp(-τϵt)dτ=eiξnϵt∫0+∞‍V0,Sd1,δn,ℰ1(reiξn,z,ϵ)exp(-reiξnϵt)dr
for all (t,z,ϵ)∈(𝒯∩D(0,ι′′))×D(0,δℰ1)×(ℰ0∩ℰ1), all n≥0. Now, let t∈𝒯∩D(0,ι′′), ϵ∈ℰ0∩ℰ1. For all n≥0, we can rewite X0,1(t,z,ϵ) as a Laplace transform in the direction arg(λ) as follows:
(5.126)X0,1(t,z,ϵ)=ℒarg(λ)(𝕍0,Sd1,δn,ℰ1(r,z,ϵ))(ϵtei(arg(λ)-ξn))
for all z∈D(0,δℰ1). Using the expression (5.126), we deduce that from the estimates (3.5), there exists a constant C(t,ϵ)>0 such that
(5.127)|X0,1(t,z,ϵ)-ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)|≤C(t,ϵ)∥𝕍0,Sd1,δn,ℰ1(r,z,ϵ)-𝕍arg(λ),Sd0,ℰ0(r,z,ϵ)∥(σˇ,ϵ,d,δD0,1)+|ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵtei(arg(λ)-ξn))-ℒarg(λ)(𝕍arg(λ),Sd0,ℰ0(r,z,ϵ))(ϵt)|
for all n≥0 and all z∈D(0,δD0,1). By letting n tend to +∞ and using the estimates (5.92), we get the formula (5.124).Now, we are in the position to state the main result of our work.Theorem 5.8.
Let the assumptions (4.42), (4.44), (4.67), (4.69), (4.70), (5.1), (5.4), (5.8), (5.20), (5.23), and (5.27) hold. Then, if one denote by op>(𝒢κ0) (resp. op(𝒢κ1)) the opening of the sector 𝒢κ0 (resp. 𝒢κ1), one has that for all t∈𝒯∩D(0,ι′′), z∈D(0,δD0,1), the function s↦g0(s,t,z) (constructed in Proposition 4.15) can be analytically continued along any path Γ in the punctured sector
(5.128)S˙κ0,κ1,t,λ={s∈ℂ*κ0-op(𝒢κ0)2<arg(s)<κ1+op(𝒢κ1)2}∖⋃k=1∞{λkt},
as a function denoted by g0Γ,t,z(s). Moreover, for all k≥1, and any path Γ0,k⊂S˙κ0,κ1,t,λ from 0 to a neighborhood of λk/t, there exists a constant Ck>0 such that
(5.129)|g0Γ0,k,t,z(s)|≤Ck|log(s-λkt)|
as s tends to λk/t in a sector centered at λk/t.Proof.
The proof is based on the following version of a result on analytic continuation of Borel transforms obtained by Fruchard and Schäfke in [3]. This result extends a former statement obtained by the same authors in [28].
Theorem (FS).Letr>0and letg:D(0,r)→ℂbe a holomorphic function that can be analytically continued as a functiong+(resp.,g-) with exponential growth of order 1 on an unbounded sectorSκ+,δ+(resp.Sκ-,δ-) centered at 0, with bisecting directionκ+(resp.κ-) and openingδ+(resp.δ-). LetC>rbe a real number and letm≥1be an integer. Let{ak∈ℂ*,1≤k≤m}⊂D(0,C)be a set of aligned points and letα>0witharg(ak)=α∈(κ-,κ+), for all1≤k≤m. For all integers1≤k≤m, letSkbe an unbounded open sector centered atak, with bisecting direction which is parallel toκ-, and openingμ>0such that theSk∩D(0,C)do not intersect for all1≤k≤m.
Now, for all
1
≤
k
≤
m
, let
g
k
be a holomorphic and bounded function on a small neighborhood of 0 and with exponential growth of order 1 on the sector
S
k
-
a
k
=
{
s
∈
ℂ
/
s
+
a
k
∈
S
k
}
with bisecting direction
κ
-
. We consider the Laplace transforms
(5.130)
f
+
(
ϵ
)
=
∫
L
κ
+
‍
g
+
(
s
)
e
-
s
/
ϵ
d
s
,
f
-
(
ϵ
)
=
∫
L
κ
-
‍
g
-
(
s
)
e
-
s
/
ϵ
d
s
,
f
k
-
(
ϵ
)
=
∫
L
κ
-
‍
g
k
(
s
)
e
-
s
/
ϵ
d
s
for all
k
≥
1
, where
L
κ
+
is the half-line starting from 0 in the direction
κ
+
and
L
κ
-
is the half-line starting from 0 in the direction
κ
-
. The function
f
+
(resp.
f
-
) defines a holomorphic and bounded function on an open sector
ℰ
+
(resp.
ℰ
-
) with finite radius, with bisecting direction
κ
+
(resp.
κ
-
) and opening
π
+
δ
+
(resp.
π
+
δ
-
). The sectors
ℰ
+
,
ℰ
-
are chosen in such a way that
ℰ
+
∩
ℰ
-
is contained in a sector with direction
α
and with opening less than
π
. Assume that the following Stokes relation
(5.131)
f
+
(
ϵ
)
=
f
-
(
ϵ
)
+
∑
k
=
1
m
‍
e
-
a
k
/
ϵ
k
!
f
k
-
(
ϵ
)
+
𝒪
(
e
-
C
e
i
α
/
ϵ
)
holds for all
ϵ
∈
ℰ
+
∩
ℰ
-
, where
𝒪
(
e
-
C
e
i
α
/
ϵ
)
is a holomorphic function
R
(
ϵ
)
on
ℰ
+
∩
ℰ
-
such that there exists a constant
H
>
0
with
(5.132)
|
R
(
ϵ
)
|
≤
H
|
e
-
C
e
i
α
/
ϵ
|
=
H
e
-
(
C
/
|
ϵ
|
)
cos
(
α
-
arg
(
ϵ
)
)
for all
ϵ
∈
ℰ
+
∩
ℰ
-
.
Then, the function
g
:
D
(
0
,
r
)
→
ℂ
can be analytically continued along any path
Γ
in the punctured sector
(5.133)
S
˙
κ
-
,
κ
+
,
C
=
{
s
∈
ℂ
*
|
s
|
<
C
,
κ
-
-
δ
-
2
<
arg
(
s
)
<
κ
+
+
δ
+
2
}
∖
⋃
k
=
1
m
{
a
k
}
.
Moreover, for all
1
≤
k
≤
m
, and any path
Γ
0
,
k
⊂
S
˙
κ
-
,
κ
+
,
C
from 0 to a neighborhood of
a
k
, if we denote by
g
Γ
0
,
k
(
s
)
the analytic continuation of
g
along
Γ
0
,
k
, then there exists a constant
C
k
>
0
such that
(5.134)
|
g
Γ
0
,
k
(
s
)
|
≤
C
k
|
log
(
s
-
a
k
)
|
as
s
tends to
a
k
in a sector centered at
a
k.
Proof. For the sake of completeness, we give a sketch of proof of this theorem. In the first step, let us consider the following sums of Cauchy integrals
(5.135)h(t)=12iπ∑k=1m‍1k!∫Lak,κ-,C‍gk(τ-ak)τ-tdτ,
where Lak,κ-,C is the segment starting from ak in the direction κ- with length C. The multivalued function h(t) can be analytically continued along any path Γ in ℂ∖{a1,…,am} by deforming the path of integration Lak,κ-,C in the sector Sk and keeping the endpoints of the segment Lak,κ-,C fixed for all 1≤k≤m. Moreover, let 1≤k≤m and t∈Lak,κ-∖{ak}, where Lak,κ- denotes the half-line starting from ak in the direction κ-. We denote by hΓak,t,ρ(t) the analytic continuation of h(t) along a loop Γak,t,ρ around ak constructed as follows: the loop follows a segment starting from t in the direction ak then turns around ak along a circle Γak,ρ of small radius ρ>0 positively oriented and then goes back to t following the same segment. We have that
(5.136)h(t)-hΓak,t,ρ(t)=gk(t-ak)k!.
Indeed, by the Cauchy theorem, one can write h(t)-hΓak,t,ρ(t) as a Cauchy integral
(5.137)Ik=12iπk!∫𝒞ak,C‍gk(τ-ak)τ-tdτ,
where 𝒞ak,C is a positively oriented closed curve enclosing t starting from ak and containing the point ak+Ceiκ-. By the residue theorem, one gets that Ik=gk(t-ak)/k!. From the relation (5.136), we also deduce the existence of a holomorphic function b(t) near ak such that
(5.138)h(t)=-gk(t-ak)2iπk!log(t-ak)+b(t)
for all t near ak, for a well-chosen determination of the logarithm log(x).
In the second step, let us define the truncated Laplace transforms and Laplace transforms(5.139)HC′+(ϵ)=∫Lκ+,C′‍h(s)e-s/ϵds,HC′-(ϵ)=∫Lκ-,C′‍h(s)e-s/ϵds,H+(ϵ)=∫Lκ+‍h(s)e-s/ϵds,H-(ϵ)=∫Lκ-‍h(s)e-s/ϵds,
where Lκ+,C′ is the segment starting from 0 to C′eiκ+ and Lκ-,C′ is the segment starting from 0 to C′eiκ-, for any fixed C′>C. By the Cauchy formula, one can write the difference HC′+(ϵ)-HC′-(ϵ) as the sum
(5.140)HC′+(ϵ)-HC′-(ϵ)=-∑k=1m‍∫Γak,ρ‍h(s)e-s/ϵds+∫Lak,ρ,C′,κ-‍(h(s)-hΓak,s,ρ(s))e-s/ϵds+𝒪(e-Ceiα/ϵ),
where Lak,ρ,C′,κ- is the segment starting from ak+ρeiκ- to ak+C′eiκ- for any ρ>0 small enough. Due to the decomposition (5.138), h(s) is integrable at ak. By letting ρ tending to 0 and C′ tending to infinity, using the relation (5.136) in (5.140), ones gets that
(5.141)H+(ϵ)-H-(ϵ)=∑k=1m‍1k!∫Lak,κ-‍gk(s-ak)e-s/ϵds+𝒪(e-Ceiα/ϵ)=∑k=1m‍e-ak/ϵk!∫Lκ-‍gk(s)e-s/ϵds+𝒪(e-Ceiα/ϵ),
where Lak,κ- is the half-line starting from ak in the direction κ-.
Now, one considers the differencesD+(ϵ)=f+(ϵ)-H+(ϵ) and D-(ϵ)=f-(ϵ)-H-(ϵ). From the Stokes relations (5.131) and (5.141), one deduces that
(5.142)D+(ϵ)-D-(ϵ)=𝒪(e-Ceiα/ϵ)
for all ϵ∈ℰ+∩ℰ-. Using a similar Borel transform integral representation as in the proof of Theorem 1 in [28], one can show that the difference g(s)-h(s), which is by construction analytic near the origin in ℂ, can be analytically continued to a function G(s), which is holomorphic on the sector Sκ-,κ+,C={s∈ℂ*/|s|<C,κ-<arg(s)<κ+}. Since h can be analytically continued along any path in ℂ∖{a1,…,an}, one gets that the function g can be analytically continued along any path in S˙κ-,κ+,C and from the decomposition (5.138) one deduces the estimates (5.134).
Now, we return to the proof of Theorem5.8. From the formula (4.76) and Proposition 5.2, the following equality
(5.143)∫Lκ1‍g0,1(s,t,z)e-s/ϵds=∫Lκ0‍g0,0(s,t,z)e-s/ϵds+∑h≥1‍exp(-hλ/ϵt)h!∫Lκ0‍gh,0(s,t,z)e-s/ϵds
holds for all ϵ∈ℰ0∩ℰ1, and all t∈𝒯∩D(0,ι′′), all z∈D(0,δD0,1). Let t∈𝒯∩D(0,ι′′) and z∈D(0,δD0,1) fixed. Let m≥1 be an integer. From the estimates (4.78), we get that
(5.144)∑h≥m+1‍|exp(-hλ/ϵt)h!∫Lκ0‍gh,0(s,t,z)e-s/ϵds|≤2C~0∑h≥m+1‍|exp(-hλϵt)|(2u1)h≤2C~0(2u1)m+1|exp(-(m+1)λϵt)|11-2|exp(-λ/ϵt)|/u1
for all ϵ∈ℰ0∩ℰ1. From (5.143) and (5.144), we deduce that the following Stokes relation
(5.145)∫Lκ1‍g0,1(s,t,z)e-s/ϵds=∫Lκ0‍g0,0(s,t,z)e-s/ϵds+∑h=1m‍exp(-hλ/ϵt)h!∫Lκ0‍gh,0(s,t,z)e-s/ϵ+𝒪(e-(m+1)λ/(ϵt)).
Holds, where 𝒪(e-(m+1)λ/(ϵt)) is a holomorphic function R(ϵ) on ℰ0∩ℰ1 such that there exists a constant H>0 with
(5.146)|R(ϵ)|≤H|e-(m+1)λ/(ϵt)|
for all ϵ∈ℰ0∩ℰ1. We can apply Theorem (FS) with ak=kλ/t, for 1≤k≤m, C=|λ|(m+1)/|t| to get that the function s↦g0(s,t,z) (constructed in Proposition 4.15) can be analytically continued along any path in the punctured sector
(5.147)S˙κ0,κ1,t,λ,m={s∈ℂ*|s|<|λ|(m+1)|t|,κ0-op(𝒢κ0)2<arg(s)<κ1+op(𝒢κ1)2}∖⋃k=1m{λkt}
as a function denoted by g0Γ,t,z(s). Moreover, for all 1≤k≤m, and any path Γ0,k⊂S˙κ0,κ1,t,λ,m from 0 to a neighborhood of λk/t, there exists a constant Ck>0 such that |g0Γ0,k,t,z(s)|≤Ck|log(s-λk/t)| as s tends to λk/t in a sector centered at λk/t. Since this result is true for all m≥1, Theorem 5.8 follows.In the next result, Onee show that under the additional hypothesis that the coefficients of (4.90) are polynomials in the parameter ϵ, the function g0(s,t,z) solves a singular linear partial differential equation in ℂ3.Corollary 5.9.
Let the assumptions of Theorem5.8 hold. We assume moreover that, for all tuple (s,k0,k1) chosen in the set 𝒮, the coefficients bs,k0,k1(z,ϵ) belong to ℂ{z}[ϵ] with the following expansion in ϵ:
(5.148)bs,k0,k1(z,ϵ)=∑m=k0ds,k0,k1‍bs,k0,k1m(z)ϵm
for some ds,k0,k1≥k0. Then, for all K∈ℕ with K≥1 and K≥max{ds,k0,k1∈ℕ/(s,k0,k1)∈𝒮}, the function g0(u,t,z) (constructed in Proposition 4.15) satisfies the following singular linear partial differential equation
(5.149)t2∂t∂uK-1∂zSg0(u,t,z)+∂uK∂zSg0(u,t,z)=-t∂uK-1∂zSg0(u,t,z)+∑(s,k0,k1)∈𝒮‍∑m=k0ds,k0,k1‍bs,k0,k1m(z)ts(∂uK-m∂tk0∂zk1g0)(u,t,z)
for all (u,t,z)∈D(0,s0)×(𝒯∩D(0,ι′′))×D(0,δD0,1). From Theorem 5.8, for all (t,z)∈(𝒯∩D(0,ι′′))×D(0,δD0,1), this solution g0(u,t,z) can be analytically continued with respect to u along any path in the punctured sector S˙κ0,κ1,t,λ with logarithmic estimates (5.129) near the singular points λk/t for all k≥1.Proof.
From Proposition4.15, we have that the function
(5.150)X0,0(t,z,ϵ)=ϵ-1∫Lκ0‍g0,0(s,t,z)e-s/ϵds
solves (5.122) on (𝒯∩D(0,ι′′))×D(0,δD0,1)×ℰ0. From the formulas in Proposition 4.2, we deduce that the function g0,0(u,t,z) solves the singular integrodifferential equation
(5.151)t2∂t∂u-1∂zSg0,0(u,t,z)+∂zSg0,0(u,t,z)=-t∂u-1∂zSg0,0(u,t,z)+∑(s,k0,k1)∈𝒮‍∑m=k0ds,k0,k1‍bs,k0,k1m(z)ts(∂u-m∂tk0∂zk1g0,0)(u,t,z)
for all (u,t,z)∈(𝒢κ0∪D(0,s0))×(𝒯∩D(0,ι′′))×D(0,δD0,1). Since g0(u,t,z) is holomorphic on D(0,s0)×(𝒯∩D(0,ι′′))×D(0,δD0,1) and has g0,0(u,t,z) as analytic continuation on (𝒢κ0∪D(0,s0))×(𝒯∩D(0,ι′′))×D(0,δD0,1), we get that g0(u,t,z) also solves (5.151). By differentiating K times of each hand side of the equation with respect to u, one gets that g0(u,t,z) solves the partial differential equation (5.149).



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*Source: 930385-2012-12-31.xml*