# Regularity of Weakly Well-Posed Characteristic Boundary Value Problems

Authors: Alessandro Morando; Paolo Secchi
Journal: International Journal of Differential Equations
 (2010)

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## Abstract

We study the boundary value problem for a linear first-order partial differential system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a uniqueL2-solution exists, for sufficiently smooth data, and obeys an a priori
energy estimate with a finite loss of tangential/conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Provided that the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework
of weighted conormal Sobolev spaces.


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## Body

## 1. Introduction and Main Results

Forn≥2, let ℝ+n denote the n-dimensional positive half-space (1.1)R+n≔{x=(x1,x′),x1>0,x′≔(x2,…,xn)∈Rn-1}.
The boundary of ℝ+n will be systematically identified with ℝx′n-1.We are interested in the following stationary boundary value problem (BVP):(1.2)(γ+L+B)u=F,inR+n,(1.3)Mu=G,onRn-1,
where L is the first-order linear partial differential operator(1.4)L=∑j=1nAj∂j;
for each j=1,…,n, the short notation ∂j≔∂/∂xj is used.The coefficientsAj (j=1,…,n) of L are N×N matrix-valued functions in C(0)∞(ℝ+n), the space of restrictions to ℝ+n of functions of C0∞(ℝn). In (1.2), ℬ stands for a lower-order term whose form and nature will be specified later; compare to Theorem 1.1 and Section 3.2.The source termF, as well as the unknown u, is a ℝN-valued function of x; we may assume that they are both supported in the unitary positive half-ball 𝔹+≔{x=(x1,x′):x1≥0,|x|<1}.The BVP hascharacteristic boundary of constant multiplicity1≤r<N in the following sense; the coefficient A1 of the normal derivative in L displays the blockwise structure(1.5)A1(x)=(A1I,IA1I,IIA1II,IA1II,II),
where A1I,I, A1I,II, A1II,I, A1II,II are, respectively, r×r, r×(N-r), (N-r)×r, (N-r)×(N-r) submatrices, such that(1.6)A1∣x1=0I,II=0,A1∣x1=0II,I=0,A1∣x1=0II,II=0,
and A1I,I is invertible over 𝔹+. According to the representation above, we split the unknown u as u=(uI,uII); uI∈ℝr and uII∈ℝN-r are said to be, respectively, the noncharacteristic and the characteristic components of u.Concerning the boundary condition (1.3), M is assumed to be the matrix (Id0), where Id denotes the identity matrix of order d, 0 is the zero matrix of size d×(N-d), and d is a given positive integer ≤r. The datum G is a given ℝd-valued function of x′=(x2,…,xn) and is supported in the unitary (n-1)-dimensional ball B(0,1)≔{|x′|<1}.Section4 will be devoted to prove the following regularity result.Theorem 1.1.
Letk,r,s be fixed nonnegative integer numbers such that s≥r≥0, s>0, and suppose that the coefficients Aj (j=1,…,n) of the operator L in (1.4) are given in C(0)∞(ℝ+n) and A1 fulfils conditions (1.5), (1.6). One assumes that for any h>0 there exist some constants C0=C0(h)>0, γ0=γ0(h)≥1 such that for every γ≥γ0, for every operator ℬ=Op♯γ(b), whose symbol b belongs to Γ0 and satisfies |b|0,k≤h, and for all functions F∈ℋtan,γs,r(ℝ+n), and G∈Hγs(ℝn-1), the corresponding BVP (1.2)-(1.3) admits a unique solution u∈L2(ℝ+n), with u∣x1=0I∈L2(ℝn-1), and the following a priori energy estimate is satisfied:
(1.7)γ‖u‖L2(R+n)2+‖u∣x1=0I‖L2(Rn-1)2≤C0(1γ2s+1‖F‖Htan,γs,r(R+n)2+1γ2s‖G‖Hγs(Rn-1)2).
Then, for every m∈ℕ and any matrix-valued function B∈C(0)∞(ℝ+n), there exist some constants Cm>0, γm (with γm≥γm-1≥1) such that if γ≥γm and we are given arbitrary functions F∈ℋtan,γs+m,r+m(ℝ+n) and G∈Hγs+m(ℝn-1), the unique L2-solution u of (1.2)-(1.3) (with data F, G, and lower-order term ℬ≡multiplicationbyB) belongs to Htan,γm(ℝ+n), u∣x1=0I∈Hγm(ℝn-1) and the a priori estimate of order m(1.8)γ‖u‖Htan,γm(R+n)2+‖u∣x1=0I‖Hγm(Rn-1)2≤Cm(1γ2s+1‖F‖Htan,γs+m,r+m(R+n)2+1γ2s‖G‖Hγs+m(Rn-1)2)
is satisfied.The function spaces involved in the statement of Theorem1.1, as well as the norms appearing in (1.7), (1.8), will be described in Section 2. The kind of lower-order operator ℬ involved in (1.2), that is allowed in Theorem 1.1, will be introduced in Section 3.2.The BVP (1.2)-(1.3), with the aforedescribed structure, naturally arises from the study of a mixed evolution problem for a symmetric (or Friedrichs'symmetrizable) hyperbolic system, with characteristic boundary. The analysis of the regularity of the stationary problem, presented in this work, plays an important role for the study of the regularity of time-dependent hyperbolic problems, constituting the final goal of our investigation and developed in [1]. In view of the well-posedness property that problems (1.2)-(1.3) enjoy in the statement of Theorem 1.1, here we do not need to assume the hyperbolicity of the linear operator L in (1.4); the only condition required on the structure of L is that expressed by conditions (1.5) and (1.6). In the hyperbolic problems, the number d of the scalar boundary conditions prescribed in (1.3) equals the number of positive eigenvalues of A1 on {x1=0}∩𝔹+ (the so-called incoming characteristics of problem (1.2)-(1.3)), compare to [1]; this value d remains constant along the boundary, as a combined effect of the hyperbolicity and the fact that A1∣{x1=0}∩𝔹+ has constant rank.In [2], the regularity of weak solutions to the characteristic BVP (1.2)-(1.3) was studied, under the assumption that the problem is stronglyL2-well posed, namely, that a unique L2-solution exists for arbitrarily given L2-data and the solution obeys an a priori energy inequality without loss of regularity with respect to the data; this means that the L2-norms of the interior and boundary values of the solution are measured by the L2-norms of the corresponding data F,G.The statement of Theorem1.1 extends the result of [2, Theorem 15], to the case where only a weak well posedness property is assumed on the BVP (1.2)-(1.3). Here, the L2-solvability of (1.2)-(1.3) requires an additional regularity of the corresponding data F,G; the integer s represents the minimal amount of regularity, needed for data, in order to estimate the L2-norm of the solution u in the interior of the domain, and its trace on the boundary, by the energy inequality (1.7).Several problems, appearing in a variety of different physical contexts, such as fluid dynamics and magneto-hydrodynamics, exhibit a finite loss of derivatives with respect to the data, as considered by estimate (1.7) in the statement of Theorem 1.1. This is the case of some problems that do not satisfy the so-called uniform Kreiss-Lopatinskiĭ condition; see, for example, [3, 4]. For instance, when the Lopatinskiĭ determinant associated to the problem has a simple root in the hyperbolic region, estimating the L2-norm of the solution makes the loss of one tangential derivative with respect to the data; see, for example, [5, 6]. In [7], Coulombel and Guès show that, in this case, the loss of regularity of order one is optimal; they also prove that the weak well posedness, with the loss of one derivative, is independent of Lipschitzean lower-order terms, but not independent of bounded lower-order terms. This is a major difference with the strongly well-posed case, where there is no loss of derivatives and one can treat lower-order terms as source terms in the energy estimates. Also, this yields that the techniques we used in [2], for studying the regularity of strongly L2-well-posed BVPs, cannot be successfully performed in the case of weakly well-posed problems (see Section 4 for a better explanation).The paper is organized as follows. In Section2 we introduce the function spaces to be used in the following and the main related notations. In Section 3 we collect some technical tools, and the basic concerned results, that will be useful for the proof of the regularity of BVP (1.2)-(1.3), given in Section 4.A final Appendix contains the proof of the most of the technical results used in Section4.


## 2. Function Spaces

The purpose of this section is to introduce the main function spaces to be used in the following and collect their basic properties.Forj=1,2,…,n, we set (2.1)Z1≔x1∂1,Zj≔∂j,forj≥2.
Then, for every multi-index α=(α1,…,αn)∈ℕn, the conormal derivative Zα is defined by (2.2)Zα≔Z1α1⋯Znαn;
we also write ∂α=∂1α1⋯∂nαn for the usual partial derivative corresponding to α.Forγ≥1 and s∈ℝ, we set(2.3)λs,γ(ξ)≔(γ2+|ξ|2)s/2
and, in particular, λs,1≔λs.The Sobolev space of orders∈ℝ in ℝn is defined to be the set of all tempered distributions u∈𝒮′(ℝn) such that λsû∈L2(ℝn), being û the Fourier transform of u; in particular, for s∈ℕ, the Sobolev space of order s reduces to the set of all functions u∈L2(ℝn), for which ∂αu∈L2(ℝn) for all α∈ℕn with |α|≤s.Throughout the paper, for realγ≥1, Hγs(ℝn) will denote the Sobolev space of order s, equipped with the γ-depending norm ∥·∥s,γ defined by(2.4)‖u‖s,γ2≔(2π)-n∫Rnλ2s,γ(ξ)|û(ξ)|2dξ,
where (ξ=(ξ1,…,ξn) are the dual Fourier variables of x=(x1,…,xn)). The norms defined by (2.4), with different values of the parameter γ, are equivalent to each other. For γ=1 we set for brevity ∥·∥s≔∥·∥s,1 (and, accordingly, Hs(ℝn)≔H1s(ℝn)).It is clear that, fors∈ℕ, the norm in (2.4) turns out to be equivalent, uniformly with respect toγ, to the norm ∥·∥Hγs(ℝn) defined by(2.5)‖u‖Hγs(Rn)2≔∑|α|≤sγ2(s-|α|)‖∂αu‖L2(Rn)2.
Another useful remark about the parameter depending norms defined in (2.4) is provided by the following counterpart of the usual Sobolev imbedding inequality:(2.6)‖u‖s,γ≤γs-r‖u‖r,γ,
for arbitrary s≤r and γ≥1.In Section4, the ordinary Sobolev spaces, endowed with the weighted norms above, will be considered in ℝn-1 (interpreted as the boundary of the half-space ℝ+n); regardless to the different dimension, the same notations and conventions as before will be used there.Let us introduce now some classes of function spaces of Sobolev type, defined over the half-spaceℝ+n; these spaces will be used to measure the regularity of solutions to characteristic BVPs with sufficiently smooth data (cf. Theorem 1.1 and Section 4).Given an integerm≥1, the conormal Sobolev space of order m is defined as the set of functions u∈L2(ℝ+n) such that Zαu∈L2(ℝ+n), for all multi-indices α with |α|≤m. Agreeing with the notations set for the usual Sobolev spaces, for γ≥1, Htan,γm(ℝ+n) will denote the conormal space of order m equipped with the γ-depending norm(2.7)‖u‖Htan,γm(R+n)2≔∑|α|≤mγ2(m-|α|)‖Zαu‖L2(R+n)2,
and we again write Htanm(ℝ+n)≔Htan,1m(ℝ+n).For later use, we need to consider also a class ofmixed tangential/conormal spaces, where different orders of tangential and conormal smoothness are allowed. Namely, for every m,r∈ℕ, with m≥r, we let ℋtanm,r(ℝ+n) denote the space of all functions u∈L2(ℝ+n) such that Zαu∈L2(ℝ+n), whenever |α|≤m and 0≤α1≤r: here derivatives Zα are required belonging to L2 up to the order m, but conormal derivatives (namely, derivatives involving the operatorZ1) are allowed only up to the lower order r, the remaining m-r derivatives being purely tangential (i.e, involving only differentiation with respect to tangential variables x′). This space is provided with the expected γ-depending norm(2.8)‖u‖Htan,γm,r(R+n)2≔∑|α|≤m,0≤α1≤rγ2(m-|α|)‖Zαu‖L2(R+n)2.
The notation ℋtan,γm,r(ℝ+n) is used, here and below, with the same meaning as for usual and conormal Sobolev spaces (accordingly, one has ℋtanm,r(ℝ+n)=ℋtan,1m,r(ℝ+n)).For a given Banach spaceY (with norm ∥·∥Y) and 1≤p≤+∞, Lp(0,+∞;Y) will denote the space of the Y-valued measurable functions on (0,+∞) such that ∫0+∞∥u(t)∥Ypdt<+∞.It is easy to see that the following identities hold true forℋtanm,r(ℝ+n), in the border cases r=0 and r=m:(2.9)Htanm,0(R+n)=L2(0,+∞;Hm(Rn-1)),Htanm,m(R+n)=Htanm(R+n).
Actually all of the previously collected observations and properties of γ-weighted norms on usual Sobolev spaces can be readily extended to the weighted norms defined on conormal and mixed spaces.Remark 2.1.
The above-considered tangential-conormal spacesℋtanm,r(ℝ+n) can be viewed as a conormal counterpart, by the action of the ♯ mapping introduced below, of corresponding mixed spaces of Sobolev type in ℝn, studied in Hörmander’s [8].


## 3. Preliminaries and Technical Tools

In this section, we collect several technical tools that will be used in the subsequent analysis (cf. Section4).We start by recalling the definition of two operators♯ and ♮, introduced by Nishitani and Takayama in [9], with the main property of mapping isometrically square integrable (resp., essentially bounded) functions over the half-space ℝ+n onto square integrable (resp., essentially bounded) functions over the full space ℝn.The mappings♯:L2(ℝ+n)→L2(ℝn) and ♮:L∞(ℝ+n)→L∞(ℝn) are, respectively, defined by(3.1)w♯(x)≔w(ex1,x′)ex1/2,a♮(x)=a(ex1,x′),∀x=(x1,x′)∈Rn.
They are both norm preserving bijections.It is also useful to notice that the above operators can be extended to the set𝒟′(ℝ+n) of Schwartz distributions in ℝ+n. It is easily seen that both ♯ and ♮ are topological isomorphisms of the space C0∞(ℝ+n) of test functions in ℝ+n (resp., C∞(ℝ+n)) onto the space C0∞(ℝn) of test functions in ℝn (resp., C∞(ℝn)). Therefore, a standard duality argument leads to define ♯ and ♮ on 𝒟′(ℝ+n), by setting for every φ∈C0∞(ℝn)(3.2)〈u♯,φ〉≔〈u,φ♯-1〉,〈u♮,φ〉≔〈u,φ♭〉
(〈·,·〉 is used to denote the duality pairing between distributions and test functions either in the half-space ℝ+n or the full space ℝn). In the right-hand sides of (3.2), ♯-1 is just the inverse operator of ♯, while the operator ♭ is defined by(3.3)φ♭(x)=1x1φ(log⁡x1,x′),∀x1>0,x′∈Rn-1,
for functions φ∈C0∞(ℝn). The operators ♯-1 and ♭ arise by explicitly calculating the formal adjoints of ♯ and ♮, respectively.Of course, one has thatu♯,u♮∈𝒟′(ℝn); moreover the following relations can be easily verified (cf. [9]):(3.4)(ψu)♯=ψ♮u♯,(3.5)∂j(u♮)=(Zju)♮,j=1,…,n,(3.6)∂1(u♯)=(Z1u)♯+12u♯,(3.7)∂j(u♯)=(Zju)♯,j=2,…,n,
whenever u∈𝒟′(ℝ+n) and ψ∈C∞(ℝ+n) (in (3.4) u∈L2(ℝ+n) and ψ∈L∞(ℝ+n) are even allowed).From formulas (3.6), (3.7) and the L2-boundedness of ♯, it also follows that ♯:Htan,γm(ℝ+n)→Hγm(ℝn) is a topological isomorphism, for each integer m≥1 and real γ≥1.Following [9] (see also [2]), in the next subsection the lastly mentioned property of ♯ will be exploited to shift some remarkable properties of the ordinary Sobolev spaces in ℝn to the functional framework of conormal Sobolev spaces over the half-space ℝ+n.In the end, we observe that the operator♯ continuously maps the space C(0)∞(ℝ+n) into the Schwartz space 𝒮(ℝn) of rapidly decreasing functions in ℝn (note also that the same is no longer true for the image of C(0)∞(ℝ+n) under the operator ♮, which is only included into the space Cb∞(ℝn) of infinitely smooth functions in ℝn, with bounded derivatives of all orders).


### 3.1. Parameter-Depending Norms on Sobolev Spaces

We recall a classical characterization of ordinary Sobolev spaces inℝn, according to Hörmander’s [8], based upon the uniform boundedness of a suitable family of parameter-depending norms.For givens∈ℝ, γ≥1 and for each δ∈]0,1] a norm in Hs-1(ℝn) is defined by setting(3.8)‖u‖s-1,γ,δ2≔(2π)-n∫Rnλ2s,γ(ξ)λ-2,γ(δξ)|û(ξ)|2dξ.
According to Section 2, for γ=1 and any 0<δ≤1, we set ∥·∥s-1,δ≔∥·∥s-1,1,δ; the family of δ-weighted norms {∥·∥s-1,δ}0<δ≤1 was deeply studied in [8]; easy arguments (relying essentially on a γ-rescaling of functions) lead to get the same properties for the norms {∥·∥s-1,γ,δ}0<δ≤1 defined in (3.8) with an arbitrary γ≥1.Of course, one has∥·∥s-1,γ,1=∥·∥s-1,γ (cf. (2.4), with s-1 instead of s). It is also clear that, for each fixed δ∈]0,1[, the norm ∥·∥s-1,γ,δ is equivalent to ∥·∥s-1,γ in Hγs-1(ℝn), uniformly with respect toγ; notice, however, that the constants appearing in the equivalence inequalities will generally depend on δ (see (3.18)).The next characterization of Sobolev spaces readily follows by taking account of the parameterγ into the arguments used in [8, Theorem 2.4.1].Proposition 3.1.
For everys∈ℝ and γ≥1, u∈Hγs(ℝn) if and only if u∈Hγs-1(ℝn), and the set {∥u∥s-1,γ,δ}0<δ≤1 is bounded. In this case, one has
(3.9)‖u‖s-1,γ,δ↑‖u‖s,γ,asδ↓0.In order to show the regularity result stated in Theorem1.1, it is useful to provide the conormal Sobolev space Htan,γm-1(ℝ+n), m∈ℕ, γ≥1, with a family of parameter-depending norms satisfying analogous properties to those of norms defined in (3.8). Nishitani and Takayama [9] introduced such norms in the “unweighted” case γ=1, just applying the ordinary Sobolev norms ∥·∥m-1,δ in (3.8) to the pull-back of functions on ℝ+n, by the ♯ operator; then these norms were used in [2] to characterize the conormal regularity of functions.Following [9], for γ≥1, δ∈]0,1], and all u∈Htanm-1(ℝ+n), we set(3.10)‖u‖R+n,m-1,tan,γ,δ2≔‖u♯‖m-1,γ,δ2=(2π)-n∫Rnλ2m,γ(ξ)λ-2,γ(δξ)|u♯̂(ξ)|2dξ.
Because ♯ is an isomorphism of Htan,γm-1(ℝ+n) onto Hγm-1(ℝn), the family of norms {∥·∥ℝ+n,m-1,tan,γ,δ}0<δ≤1 keeps all the properties enjoyed by the family of norms defined in (3.8).In particular, the same characterization of ordinary Sobolev spaces onℝn, given by Proposition 3.1, applies also to conormal Sobolev spaces in ℝ+n (cf. [2, 9]).Proposition 3.2.
For every positive integerm and γ≥1, u∈Htan,γm(ℝ+n) if and only if u∈Htan,γm-1(ℝ+n), and the set {∥u∥ℝ+n,m-1,tan,γ,δ}0<δ≤1 is bounded. In this case, one has
(3.11)‖u‖R+n,m-1,tan,γ,δ↑‖u‖R+n,m,tan,γ,asδ↓0.As regards to the mixed spaceℋtanm,r(ℝ+n), it is worthwhile noticing that it can be endowed with the γ-weighted norm defined, by the Fourier transformation, as(3.12)‖u‖R+n,m,r,tan,γ2≔(2π)-n∫Rnλ2r,γ(ξ)λ(ξ′)2(m-r),γ|u♯̂(ξ)|2dξ;
here and below ξ′≔(ξ2,…,ξn) denotes the Fourier dual variables of the tangential space variables x′=(x2,…,xn), and, with a slight abuse of notation, we write λr,γ(ξ′) to mean in fact λr,γ(0,ξ′).Of course, the norm in (3.12) is equivalent, uniformly with respect to γ, to the norm (2.8).


### 3.2. A Class of Conormal Operators

The♯ operator, defined at the beginning of Section 3, can be used to allow pseudodifferential operators in ℝn  acting conormally on functions only defined over the positive half-space ℝ+n. Then the standard machinery of pseudodifferential calculus (in the parameter depending version introduced in [10, 11]) can be rearranged into a functional calculus properly behaved on conormal Sobolev spaces described in Section 2. In Section 4, this calculus will be usefully applied to study the conormal regularity of the stationary BVP (1.2)-(1.3).Let us introduce the pseudodifferential symbols, with a parameter, to be used later; here we closely follow the terminology and notations of [12].Definition 3.3.
A parameter-depending pseudodifferential symbol of orderm∈ℝ is a real-(or complex-) valued measurable function a(x,ξ,γ) on ℝn×ℝn×[1,+∞[, such that a is C∞ with respect to x and ξ, and for all multi-indices α,β∈ℕn there exists a positive constant Cα,β satisfying
(3.13)|∂ξα∂xβa(x,ξ,γ)|≤Cα,βλm-|α|,γ(ξ),
for all x,ξ∈ℝn and γ≥1.The same definition as above extends to functionsa(x,ξ,γ) taking values in the space ℝN×N (resp., ℂN×N) of N×N real (resp., complex) valued matrices, for all integers N>1 (where the module |·| is replaced in (3.13) by any equivalent norm in ℝN×N (resp., ℂN×N)). We denote by Γm the set of γ-depending symbols of order m∈ℝ (the same notation being used for both scalar-or matrix-valued symbols). Γm is equipped with the obvious norms(3.14)|a|m,k≔max⁡|α|+|β|≤ksup⁡(x,ξ)∈Rn×Rn,γ≥1λ-m+|α|,γ(ξ)|∂ξα∂xβa(x,ξ,γ)|,∀k∈N,
which turn it into a Fréchet space. For all m,m′∈ℝ, with m≤m′, the continuous imbedding Γm⊂Γm′ can be easily proven.For allm∈ℝ, the function λm,γ is of course a (scalar-valued) symbol in Γm.To perform the analysis of Section4, it is important to consider the behavior of the weight function λm,γ(·)λ-1,γ(δ·), involved in the definition of the parameter-depending norms in (3.8), (3.10), as a γ-depending symbol according to Definition 3.3.In order to simplify the forthcoming statements, henceforth the following short notations will be used:(3.15)λδm-1,γ(ξ)≔λm,γ(ξ)λ-1,γ(δξ),λ̃δ-m+1,γ(ξ)≔(λδm-1,γ(ξ))-1(=λ-m,γ(ξ)λ1,γ(δξ)),
for all real numbers m∈ℝ, γ≥1, and δ∈]0,1]. One has the obvious identities λ1m-1,γ(ξ)≡λm-1,γ(ξ), λ̃1-m+1,γ(ξ)≡λ1-m+1,γ(ξ)≡λ-m+1,γ(ξ). However, to avoid confusion in the following, it is worthwhile to remark that functions λδ-m+1,γ(ξ) and λ̃δ-m+1,γ(ξ) are no longer the same as soon as δ becomes strictly smaller than 1; indeed (3.15) gives λδ-m+1,γ(ξ)=λ-m+2,γ(ξ)λ-1,γ(δξ).A straightforward application of Leibniz's rule leads to the following result.Lemma 3.4.
For everym∈ℝ and all α∈ℕn, there exists a positive constant Cm,α such that
(3.16)|∂ξαλδm-1,γ(ξ)|≤Cm,αλδm-1-|α|,γ(ξ),∀ξ∈Rn,∀γ≥1,∀δ∈]0,1].Because of estimates (3.16), λδm-1,γ(ξ) can be regarded as a γ-depending symbol, in two different ways. On one hand, combining estimates (3.16) with the trivial inequality(3.17)λ-1,γ(δξ)≤1
immediately gives that {λδm-1,γ}0<δ≤1 is a bounded subset of Γm.On the other hand, the left inequality in(3.18)δλ1,γ(ξ)≤λ1,γ(δξ)≤λ1,γ(ξ),∀ξ∈Rn,∀δ∈]0,1],
together with (3.16), also gives(3.19)|∂ξαλδm-1,γ(ξ)|≤Cm,αδ-1λm-1-|α|,γ(ξ),∀ξ∈Rn,∀γ≥1.
According to Definition 3.3, (3.19) means that λδm-1,γ actually belongs, for each fixed δ, to Γm-1; nevertheless, the family {λδm-1,γ}0<δ≤1 is unbounded as a subset of Γm-1.For later use, we also need to study the behavior of functionsλ̃δ-m+1,γ as γ-depending symbols.Analogously to Lemma3.4, one can prove the following result.Lemma 3.5.
For allm∈ℝ and α∈ℕn, there exists C̃m,α>0 such that
(3.20)|∂ξαλ̃δ-m+1,γ(ξ)|≤C̃m,αλ̃δ-m+1-|α|,γ(ξ),∀ξ∈Rn,∀γ≥1,∀δ∈]0,1].In particular, Lemma3.5 implies that the family {λ̃δ-m+1,γ}0<δ≤1 is a bounded subset of Γ-m+1 (it suffices to combine (3.20) with the right inequality in (3.18)).Any symbola=a(x,ξ,γ)∈Γm defines a pseudodifferential operatorOpγ(a)=a(x,D,γ) on the Schwartz space 𝒮(ℝn), by the standard formula(3.21)∀u∈S(Rn),∀x∈Rn,Opγ(a)u(x)=a(x,D,γ)u(x)=:(2π)-n∫Rneix⋅ξa(x,ξ,γ)û(ξ)dξ,
where, of course, we denote x·ξ≔∑j=1nxjξj. a is called the symbol of the operator (3.21), and m is its order. It comes from the classical theory that Opγ(a) defines a linear-bounded operator (3.22)Opγ(a):S(Rn)⟶S(Rn);
moreover, the latter extends to a linear-bounded operator on the space 𝒮′(ℝn) of tempered distributions in ℝn.An exhaustive account of the symbolic calculus for pseudodifferential operators with symbols inΓm can be found in [11] (see also [12]). Here, we just recall the following result, concerning the product and the commutator of two pseudodifferential operators.Proposition 3.6.
Leta∈Γm and b∈Γl, for l,m∈ℝ. Then Opγ(a)Opγ(b) is a pseudodifferential operator with symbol in Γm+l; moreover, if one lets a#b denote the symbol of the product, one has for every integer N≥1(3.23)a#b-∑|α|<N(-i)|α|α!∂ξαa∂xαb∈Γm+l-N.
Under the same assumptions, the commutator [Opγ(a),Opγ(b)]≔Opγ(a)Opγ(b)-Opγ(b)Opγ(a) is again a pseudodifferential operator with symbol c∈Γm+l. If one further assumes that one of the two symbols a or b is scalar-valued (so that a and b commute in the pointwise product), then the symbol c of [Opγ(a),Opγ(b)] has order m+l-1.We point out that when the symbolb∈Γl of the preceding statement does not depend on the x variables (i.e., b=b(ξ,γ)), then the symbol a#b of the product Opγ(a)Opγ(b) reduces to the pointwise product of symbols a and b; in this case, the asymptotic formula (3.23) is replaced by the exact formula(3.24)(a#b)(x,ξ,γ)=a(x,ξ,γ)b(ξ,γ).
According to (3.15), (3.21), we write(3.25)λδm-1,γ(D)≔Opγ(λδm-1,γ),λ̃δ-m+1,γ(D)≔Opγ(λ̃δ-m+1,γ).
In view of (3.15) and (3.24), the operator λδm-1,γ(D) is invertible, and its two-sided inverse is given by λ̃δ-m+1,γ(D).Starting from the symbolic classesΓm, m∈ℝ, we introduce now the class of conormal operators in ℝ+n, to be used in the sequel.Leta(x,ξ,γ) be a γ-depending symbol in Γm, m∈ℝ. The conormal operator with symbola, denoted by Op♯γ(a) (or equivalently a(x,Z,γ)), is defined by setting(3.26)∀u∈C(0)∞(R+n),(Op♯γ(a)u)♯=(Opγ(a))(u♯).
In other words, the operator Op♯γ(a) is the composition of mappings(3.27)Op♯γ(a)=♯-1∘Opγ(a)∘♯.
As we already noted, u♯∈𝒮(ℝn) whenever u∈C(0)∞(ℝ+n); hence formula (3.26) makes sense and gives that Op♯γ(a)u is a C∞-function in ℝ+n. Also Op♯γ(a):C(0)∞(ℝ+n)→C∞(ℝ+n) is a linear-bounded operator that extends to a linear-bounded operator from the space of distributions u∈𝒟′(ℝ+n) satisfying u♯∈𝒮′(ℝn) into 𝒟′(ℝ+n) itself. (In principle, Op♯γ(a) could be defined by (3.26) over all functions u∈C∞(ℝ+n), such that u♯∈𝒮(ℝn). Then Op♯γ(a) defines a linear-bounded operator on the latter function space, provided that it is equipped with the topology induced, via ♯, from the Fréchet topology of 𝒮(ℝn).) Throughout the paper, we continue to denote this extension by Op♯γ(a) (or a(x,Z,γ) equivalently).As an immediate consequence of (3.27), we have that for all symbols a∈Γm, b∈Γl, with m,l∈ℝ, there holds(3.28)∀u∈C(0)∞(R+n),Op♯γ(a)Op♯γ(b)u=(Opγ(a)Opγ(b)(u♯))♯-1.
Then, it is clear that a functional calculus of conormal operators can be straightforwardly borrowed from the corresponding pseudodifferential calculus in ℝn; in particular we find that products and commutators of conormal operators are still operators of the same type, and their symbols are computed according to the rules collected in Proposition 3.6.Below, let us consider the main examples of conormal operators that will be met in Section4.As a first example, we quote the multiplication by a matrix-valued functionB∈C(0)∞(ℝ+n). It is clear that this makes an operator of order zero according to (3.26); indeed (3.4) gives for any vector-valued u∈C(0)∞(ℝ+n)(3.29)(Bu)♯(x)=B♮(x)u♯(x),
and B♮ is a C∞-function in ℝn, with bounded derivatives of any order, hence a symbol in Γ0.We remark that, when computed forB♮, the norm of order k∈ℕ, defined on symbols by (3.14), just reduces to(3.30)|B♮|0,k=max⁡|α|≤k‖∂αB♮‖L∞(Rn)=max⁡|α|≤k‖ZαB‖L∞(R+n),
where the second identity above exploits formulas (3.5) and that ♮ maps isometrically L∞(ℝ+n) onto L∞(ℝn).Now, letℒ≔γIN+∑j=1nAj(x)Zj be a first-order linear partial differential operator, with matrix-valued coefficients Aj∈C(0)∞(ℝ+n) for j=1,…,n and γ≥1. Since the leading part of ℒ only involves conormal derivatives, applying (3.4), (3.6), and (3.7) then gives (3.31)(γu+∑j=1nAjZju)♯=(γI-12A1♮)u♯+∑j=1nAj♮∂ju♯=Opγ(a)u♯,
where a=a(x,ξ,γ)≔(γIN-(1/2)A1♮(x))+i∑j=1nAj♮(x)ξj is a symbol in Γ1. Then ℒ is a conormal operator of order 1, according to (3.26).In Section4, we will be mainly interested in the family of conormal operators(3.32)λδm-1,γ(Z)≔Op♯γ(λδm-1,γ),λ̃δ-m+1,γ(Z)≔Op♯γ(λ̃δ-m+1,γ).
The operators λδm-1,γ(Z) are involved in the characterization of conormal regularity provided by Proposition 3.2 (remember that, after Lemma 3.4, λδm-1,γ∈Γm-1). Indeed, from Plancherel's formula and the fact that the operator ♯ preserves the L2-norm, the following identities(3.33)‖u‖R+n,m-1,tan,γ,δ≡‖λδm-1,γ(Z)u‖L2(R+n)
can be straightforwardly established; hence, Proposition 3.2 can be restated in terms of the boundedness, with respect to δ, of the L2-norms of functions λδm-1,γ(Z)u. This observation is the key point that leads to the analysis performed in Section 4.Another main feature of the conormal operators (3.32) is that λ̃δ-m+1,γ(Z) provides a two-sided inverse of λδm-1,γ(Z); this comes at once from the analogous property of the operators in (3.25) and formulas (3.26), (3.28).


### 3.3. Sobolev Continuity of Conormal Operators

Proposition 3.7.
Ifs,m∈ℝ then for all a∈Γm the pseudodifferential operator Opγ(a) extends as a linear-bounded operator from Hγs+m(ℝn) into Hγs(ℝn), and the operator norm of such an extension is uniformly bounded with respect to γ.We refer the reader to [11] for a detailed proof of Proposition 3.7. A thorough analysis shows that the norm of Opγ(a), as a linear-bounded operator from Hγs+m(ℝn) to Hγs(ℝn), actually depends only on a norm of type (3.14) of the symbol a, besides the Sobolev order s and the symbolic order m (cf. [11] for detailed calculations). This observation entails, in particular, that the operator norm is uniformly bounded with respect to γ and other additional parameters from which the symbol of the operator might possibly depend, as a bounded mapping.From the Sobolev continuity of pseudodifferential operators quoted above, and using that the operator♯ maps isomorphically conormal Sobolev spaces in ℝ+n onto ordinary Sobolev spaces in ℝn, we easily derive the following result.Proposition 3.8.
Ifm∈ℤ and a∈Γm, then the conormal operator Op♯γ(a) extends to a linear-bounded operator from Htan,γs+m(ℝ+n) to Htan,γs(ℝ+n), for every integer s≥0, such that s+m≥0; moreover the operator norm of such an extension is uniformly bounded with respect to γ.Remark 3.9.
We point out that the statement above only deals with integer orders of symbols and conormal Sobolev spaces. The reason is that, in Section2, conormal Sobolev spaces were only defined for positive integer orders. In principle, this lack could be removed by extending the definition of conormal spaces Htans(ℝ+n) to any real order s: this could be trivially done, just defining Htans(ℝ+n) to be the pull-back, by the operator ♯, of functions in Hs(ℝn). However, this extension to fractional exponents seems to be useless for the subsequent developments.As regards to the action of conormal operators on the mixed spacesℋtan,γs,r(ℝ+n), similar arguments to those used in the proof of Proposition 3.8 lead to the following.Proposition 3.10.
Leta=a(x,ξ,γ) be a symbol in Γm, for m∈ℤ. Then for all integers r,s∈ℕ, such that s≥r, s>0, and r+m≥0, Op♯γ(a)=a(x,Z,γ) extends by continuity to a linear-bounded operator
(3.34)Op♯γ(a):Htan,γs+m,r+m(R+n)⟶Htan,γs,r(R+n).
Moreover, the operator norm of such an extension is uniformly bounded with respect to γ.


## 3.1. Parameter-Depending Norms on Sobolev Spaces

We recall a classical characterization of ordinary Sobolev spaces inℝn, according to Hörmander’s [8], based upon the uniform boundedness of a suitable family of parameter-depending norms.For givens∈ℝ, γ≥1 and for each δ∈]0,1] a norm in Hs-1(ℝn) is defined by setting(3.8)‖u‖s-1,γ,δ2≔(2π)-n∫Rnλ2s,γ(ξ)λ-2,γ(δξ)|û(ξ)|2dξ.
According to Section 2, for γ=1 and any 0<δ≤1, we set ∥·∥s-1,δ≔∥·∥s-1,1,δ; the family of δ-weighted norms {∥·∥s-1,δ}0<δ≤1 was deeply studied in [8]; easy arguments (relying essentially on a γ-rescaling of functions) lead to get the same properties for the norms {∥·∥s-1,γ,δ}0<δ≤1 defined in (3.8) with an arbitrary γ≥1.Of course, one has∥·∥s-1,γ,1=∥·∥s-1,γ (cf. (2.4), with s-1 instead of s). It is also clear that, for each fixed δ∈]0,1[, the norm ∥·∥s-1,γ,δ is equivalent to ∥·∥s-1,γ in Hγs-1(ℝn), uniformly with respect toγ; notice, however, that the constants appearing in the equivalence inequalities will generally depend on δ (see (3.18)).The next characterization of Sobolev spaces readily follows by taking account of the parameterγ into the arguments used in [8, Theorem 2.4.1].Proposition 3.1.
For everys∈ℝ and γ≥1, u∈Hγs(ℝn) if and only if u∈Hγs-1(ℝn), and the set {∥u∥s-1,γ,δ}0<δ≤1 is bounded. In this case, one has
(3.9)‖u‖s-1,γ,δ↑‖u‖s,γ,asδ↓0.In order to show the regularity result stated in Theorem1.1, it is useful to provide the conormal Sobolev space Htan,γm-1(ℝ+n), m∈ℕ, γ≥1, with a family of parameter-depending norms satisfying analogous properties to those of norms defined in (3.8). Nishitani and Takayama [9] introduced such norms in the “unweighted” case γ=1, just applying the ordinary Sobolev norms ∥·∥m-1,δ in (3.8) to the pull-back of functions on ℝ+n, by the ♯ operator; then these norms were used in [2] to characterize the conormal regularity of functions.Following [9], for γ≥1, δ∈]0,1], and all u∈Htanm-1(ℝ+n), we set(3.10)‖u‖R+n,m-1,tan,γ,δ2≔‖u♯‖m-1,γ,δ2=(2π)-n∫Rnλ2m,γ(ξ)λ-2,γ(δξ)|u♯̂(ξ)|2dξ.
Because ♯ is an isomorphism of Htan,γm-1(ℝ+n) onto Hγm-1(ℝn), the family of norms {∥·∥ℝ+n,m-1,tan,γ,δ}0<δ≤1 keeps all the properties enjoyed by the family of norms defined in (3.8).In particular, the same characterization of ordinary Sobolev spaces onℝn, given by Proposition 3.1, applies also to conormal Sobolev spaces in ℝ+n (cf. [2, 9]).Proposition 3.2.
For every positive integerm and γ≥1, u∈Htan,γm(ℝ+n) if and only if u∈Htan,γm-1(ℝ+n), and the set {∥u∥ℝ+n,m-1,tan,γ,δ}0<δ≤1 is bounded. In this case, one has
(3.11)‖u‖R+n,m-1,tan,γ,δ↑‖u‖R+n,m,tan,γ,asδ↓0.As regards to the mixed spaceℋtanm,r(ℝ+n), it is worthwhile noticing that it can be endowed with the γ-weighted norm defined, by the Fourier transformation, as(3.12)‖u‖R+n,m,r,tan,γ2≔(2π)-n∫Rnλ2r,γ(ξ)λ(ξ′)2(m-r),γ|u♯̂(ξ)|2dξ;
here and below ξ′≔(ξ2,…,ξn) denotes the Fourier dual variables of the tangential space variables x′=(x2,…,xn), and, with a slight abuse of notation, we write λr,γ(ξ′) to mean in fact λr,γ(0,ξ′).Of course, the norm in (3.12) is equivalent, uniformly with respect to γ, to the norm (2.8).


## 3.2. A Class of Conormal Operators

The♯ operator, defined at the beginning of Section 3, can be used to allow pseudodifferential operators in ℝn  acting conormally on functions only defined over the positive half-space ℝ+n. Then the standard machinery of pseudodifferential calculus (in the parameter depending version introduced in [10, 11]) can be rearranged into a functional calculus properly behaved on conormal Sobolev spaces described in Section 2. In Section 4, this calculus will be usefully applied to study the conormal regularity of the stationary BVP (1.2)-(1.3).Let us introduce the pseudodifferential symbols, with a parameter, to be used later; here we closely follow the terminology and notations of [12].Definition 3.3.
A parameter-depending pseudodifferential symbol of orderm∈ℝ is a real-(or complex-) valued measurable function a(x,ξ,γ) on ℝn×ℝn×[1,+∞[, such that a is C∞ with respect to x and ξ, and for all multi-indices α,β∈ℕn there exists a positive constant Cα,β satisfying
(3.13)|∂ξα∂xβa(x,ξ,γ)|≤Cα,βλm-|α|,γ(ξ),
for all x,ξ∈ℝn and γ≥1.The same definition as above extends to functionsa(x,ξ,γ) taking values in the space ℝN×N (resp., ℂN×N) of N×N real (resp., complex) valued matrices, for all integers N>1 (where the module |·| is replaced in (3.13) by any equivalent norm in ℝN×N (resp., ℂN×N)). We denote by Γm the set of γ-depending symbols of order m∈ℝ (the same notation being used for both scalar-or matrix-valued symbols). Γm is equipped with the obvious norms(3.14)|a|m,k≔max⁡|α|+|β|≤ksup⁡(x,ξ)∈Rn×Rn,γ≥1λ-m+|α|,γ(ξ)|∂ξα∂xβa(x,ξ,γ)|,∀k∈N,
which turn it into a Fréchet space. For all m,m′∈ℝ, with m≤m′, the continuous imbedding Γm⊂Γm′ can be easily proven.For allm∈ℝ, the function λm,γ is of course a (scalar-valued) symbol in Γm.To perform the analysis of Section4, it is important to consider the behavior of the weight function λm,γ(·)λ-1,γ(δ·), involved in the definition of the parameter-depending norms in (3.8), (3.10), as a γ-depending symbol according to Definition 3.3.In order to simplify the forthcoming statements, henceforth the following short notations will be used:(3.15)λδm-1,γ(ξ)≔λm,γ(ξ)λ-1,γ(δξ),λ̃δ-m+1,γ(ξ)≔(λδm-1,γ(ξ))-1(=λ-m,γ(ξ)λ1,γ(δξ)),
for all real numbers m∈ℝ, γ≥1, and δ∈]0,1]. One has the obvious identities λ1m-1,γ(ξ)≡λm-1,γ(ξ), λ̃1-m+1,γ(ξ)≡λ1-m+1,γ(ξ)≡λ-m+1,γ(ξ). However, to avoid confusion in the following, it is worthwhile to remark that functions λδ-m+1,γ(ξ) and λ̃δ-m+1,γ(ξ) are no longer the same as soon as δ becomes strictly smaller than 1; indeed (3.15) gives λδ-m+1,γ(ξ)=λ-m+2,γ(ξ)λ-1,γ(δξ).A straightforward application of Leibniz's rule leads to the following result.Lemma 3.4.
For everym∈ℝ and all α∈ℕn, there exists a positive constant Cm,α such that
(3.16)|∂ξαλδm-1,γ(ξ)|≤Cm,αλδm-1-|α|,γ(ξ),∀ξ∈Rn,∀γ≥1,∀δ∈]0,1].Because of estimates (3.16), λδm-1,γ(ξ) can be regarded as a γ-depending symbol, in two different ways. On one hand, combining estimates (3.16) with the trivial inequality(3.17)λ-1,γ(δξ)≤1
immediately gives that {λδm-1,γ}0<δ≤1 is a bounded subset of Γm.On the other hand, the left inequality in(3.18)δλ1,γ(ξ)≤λ1,γ(δξ)≤λ1,γ(ξ),∀ξ∈Rn,∀δ∈]0,1],
together with (3.16), also gives(3.19)|∂ξαλδm-1,γ(ξ)|≤Cm,αδ-1λm-1-|α|,γ(ξ),∀ξ∈Rn,∀γ≥1.
According to Definition 3.3, (3.19) means that λδm-1,γ actually belongs, for each fixed δ, to Γm-1; nevertheless, the family {λδm-1,γ}0<δ≤1 is unbounded as a subset of Γm-1.For later use, we also need to study the behavior of functionsλ̃δ-m+1,γ as γ-depending symbols.Analogously to Lemma3.4, one can prove the following result.Lemma 3.5.
For allm∈ℝ and α∈ℕn, there exists C̃m,α>0 such that
(3.20)|∂ξαλ̃δ-m+1,γ(ξ)|≤C̃m,αλ̃δ-m+1-|α|,γ(ξ),∀ξ∈Rn,∀γ≥1,∀δ∈]0,1].In particular, Lemma3.5 implies that the family {λ̃δ-m+1,γ}0<δ≤1 is a bounded subset of Γ-m+1 (it suffices to combine (3.20) with the right inequality in (3.18)).Any symbola=a(x,ξ,γ)∈Γm defines a pseudodifferential operatorOpγ(a)=a(x,D,γ) on the Schwartz space 𝒮(ℝn), by the standard formula(3.21)∀u∈S(Rn),∀x∈Rn,Opγ(a)u(x)=a(x,D,γ)u(x)=:(2π)-n∫Rneix⋅ξa(x,ξ,γ)û(ξ)dξ,
where, of course, we denote x·ξ≔∑j=1nxjξj. a is called the symbol of the operator (3.21), and m is its order. It comes from the classical theory that Opγ(a) defines a linear-bounded operator (3.22)Opγ(a):S(Rn)⟶S(Rn);
moreover, the latter extends to a linear-bounded operator on the space 𝒮′(ℝn) of tempered distributions in ℝn.An exhaustive account of the symbolic calculus for pseudodifferential operators with symbols inΓm can be found in [11] (see also [12]). Here, we just recall the following result, concerning the product and the commutator of two pseudodifferential operators.Proposition 3.6.
Leta∈Γm and b∈Γl, for l,m∈ℝ. Then Opγ(a)Opγ(b) is a pseudodifferential operator with symbol in Γm+l; moreover, if one lets a#b denote the symbol of the product, one has for every integer N≥1(3.23)a#b-∑|α|<N(-i)|α|α!∂ξαa∂xαb∈Γm+l-N.
Under the same assumptions, the commutator [Opγ(a),Opγ(b)]≔Opγ(a)Opγ(b)-Opγ(b)Opγ(a) is again a pseudodifferential operator with symbol c∈Γm+l. If one further assumes that one of the two symbols a or b is scalar-valued (so that a and b commute in the pointwise product), then the symbol c of [Opγ(a),Opγ(b)] has order m+l-1.We point out that when the symbolb∈Γl of the preceding statement does not depend on the x variables (i.e., b=b(ξ,γ)), then the symbol a#b of the product Opγ(a)Opγ(b) reduces to the pointwise product of symbols a and b; in this case, the asymptotic formula (3.23) is replaced by the exact formula(3.24)(a#b)(x,ξ,γ)=a(x,ξ,γ)b(ξ,γ).
According to (3.15), (3.21), we write(3.25)λδm-1,γ(D)≔Opγ(λδm-1,γ),λ̃δ-m+1,γ(D)≔Opγ(λ̃δ-m+1,γ).
In view of (3.15) and (3.24), the operator λδm-1,γ(D) is invertible, and its two-sided inverse is given by λ̃δ-m+1,γ(D).Starting from the symbolic classesΓm, m∈ℝ, we introduce now the class of conormal operators in ℝ+n, to be used in the sequel.Leta(x,ξ,γ) be a γ-depending symbol in Γm, m∈ℝ. The conormal operator with symbola, denoted by Op♯γ(a) (or equivalently a(x,Z,γ)), is defined by setting(3.26)∀u∈C(0)∞(R+n),(Op♯γ(a)u)♯=(Opγ(a))(u♯).
In other words, the operator Op♯γ(a) is the composition of mappings(3.27)Op♯γ(a)=♯-1∘Opγ(a)∘♯.
As we already noted, u♯∈𝒮(ℝn) whenever u∈C(0)∞(ℝ+n); hence formula (3.26) makes sense and gives that Op♯γ(a)u is a C∞-function in ℝ+n. Also Op♯γ(a):C(0)∞(ℝ+n)→C∞(ℝ+n) is a linear-bounded operator that extends to a linear-bounded operator from the space of distributions u∈𝒟′(ℝ+n) satisfying u♯∈𝒮′(ℝn) into 𝒟′(ℝ+n) itself. (In principle, Op♯γ(a) could be defined by (3.26) over all functions u∈C∞(ℝ+n), such that u♯∈𝒮(ℝn). Then Op♯γ(a) defines a linear-bounded operator on the latter function space, provided that it is equipped with the topology induced, via ♯, from the Fréchet topology of 𝒮(ℝn).) Throughout the paper, we continue to denote this extension by Op♯γ(a) (or a(x,Z,γ) equivalently).As an immediate consequence of (3.27), we have that for all symbols a∈Γm, b∈Γl, with m,l∈ℝ, there holds(3.28)∀u∈C(0)∞(R+n),Op♯γ(a)Op♯γ(b)u=(Opγ(a)Opγ(b)(u♯))♯-1.
Then, it is clear that a functional calculus of conormal operators can be straightforwardly borrowed from the corresponding pseudodifferential calculus in ℝn; in particular we find that products and commutators of conormal operators are still operators of the same type, and their symbols are computed according to the rules collected in Proposition 3.6.Below, let us consider the main examples of conormal operators that will be met in Section4.As a first example, we quote the multiplication by a matrix-valued functionB∈C(0)∞(ℝ+n). It is clear that this makes an operator of order zero according to (3.26); indeed (3.4) gives for any vector-valued u∈C(0)∞(ℝ+n)(3.29)(Bu)♯(x)=B♮(x)u♯(x),
and B♮ is a C∞-function in ℝn, with bounded derivatives of any order, hence a symbol in Γ0.We remark that, when computed forB♮, the norm of order k∈ℕ, defined on symbols by (3.14), just reduces to(3.30)|B♮|0,k=max⁡|α|≤k‖∂αB♮‖L∞(Rn)=max⁡|α|≤k‖ZαB‖L∞(R+n),
where the second identity above exploits formulas (3.5) and that ♮ maps isometrically L∞(ℝ+n) onto L∞(ℝn).Now, letℒ≔γIN+∑j=1nAj(x)Zj be a first-order linear partial differential operator, with matrix-valued coefficients Aj∈C(0)∞(ℝ+n) for j=1,…,n and γ≥1. Since the leading part of ℒ only involves conormal derivatives, applying (3.4), (3.6), and (3.7) then gives (3.31)(γu+∑j=1nAjZju)♯=(γI-12A1♮)u♯+∑j=1nAj♮∂ju♯=Opγ(a)u♯,
where a=a(x,ξ,γ)≔(γIN-(1/2)A1♮(x))+i∑j=1nAj♮(x)ξj is a symbol in Γ1. Then ℒ is a conormal operator of order 1, according to (3.26).In Section4, we will be mainly interested in the family of conormal operators(3.32)λδm-1,γ(Z)≔Op♯γ(λδm-1,γ),λ̃δ-m+1,γ(Z)≔Op♯γ(λ̃δ-m+1,γ).
The operators λδm-1,γ(Z) are involved in the characterization of conormal regularity provided by Proposition 3.2 (remember that, after Lemma 3.4, λδm-1,γ∈Γm-1). Indeed, from Plancherel's formula and the fact that the operator ♯ preserves the L2-norm, the following identities(3.33)‖u‖R+n,m-1,tan,γ,δ≡‖λδm-1,γ(Z)u‖L2(R+n)
can be straightforwardly established; hence, Proposition 3.2 can be restated in terms of the boundedness, with respect to δ, of the L2-norms of functions λδm-1,γ(Z)u. This observation is the key point that leads to the analysis performed in Section 4.Another main feature of the conormal operators (3.32) is that λ̃δ-m+1,γ(Z) provides a two-sided inverse of λδm-1,γ(Z); this comes at once from the analogous property of the operators in (3.25) and formulas (3.26), (3.28).


## 3.3. Sobolev Continuity of Conormal Operators

Proposition 3.7.
Ifs,m∈ℝ then for all a∈Γm the pseudodifferential operator Opγ(a) extends as a linear-bounded operator from Hγs+m(ℝn) into Hγs(ℝn), and the operator norm of such an extension is uniformly bounded with respect to γ.We refer the reader to [11] for a detailed proof of Proposition 3.7. A thorough analysis shows that the norm of Opγ(a), as a linear-bounded operator from Hγs+m(ℝn) to Hγs(ℝn), actually depends only on a norm of type (3.14) of the symbol a, besides the Sobolev order s and the symbolic order m (cf. [11] for detailed calculations). This observation entails, in particular, that the operator norm is uniformly bounded with respect to γ and other additional parameters from which the symbol of the operator might possibly depend, as a bounded mapping.From the Sobolev continuity of pseudodifferential operators quoted above, and using that the operator♯ maps isomorphically conormal Sobolev spaces in ℝ+n onto ordinary Sobolev spaces in ℝn, we easily derive the following result.Proposition 3.8.
Ifm∈ℤ and a∈Γm, then the conormal operator Op♯γ(a) extends to a linear-bounded operator from Htan,γs+m(ℝ+n) to Htan,γs(ℝ+n), for every integer s≥0, such that s+m≥0; moreover the operator norm of such an extension is uniformly bounded with respect to γ.Remark 3.9.
We point out that the statement above only deals with integer orders of symbols and conormal Sobolev spaces. The reason is that, in Section2, conormal Sobolev spaces were only defined for positive integer orders. In principle, this lack could be removed by extending the definition of conormal spaces Htans(ℝ+n) to any real order s: this could be trivially done, just defining Htans(ℝ+n) to be the pull-back, by the operator ♯, of functions in Hs(ℝn). However, this extension to fractional exponents seems to be useless for the subsequent developments.As regards to the action of conormal operators on the mixed spacesℋtan,γs,r(ℝ+n), similar arguments to those used in the proof of Proposition 3.8 lead to the following.Proposition 3.10.
Leta=a(x,ξ,γ) be a symbol in Γm, for m∈ℤ. Then for all integers r,s∈ℕ, such that s≥r, s>0, and r+m≥0, Op♯γ(a)=a(x,Z,γ) extends by continuity to a linear-bounded operator
(3.34)Op♯γ(a):Htan,γs+m,r+m(R+n)⟶Htan,γs,r(R+n).
Moreover, the operator norm of such an extension is uniformly bounded with respect to γ.


## 4. Proof of Theorem1.1

This section is entirely devoted to the proof of Theorem1.1.


### 4.1. The Strategy of the Proof: Comparison with the Strongly Well-Posed Case

As it was already pointed out in the Introduction, in order to solve the BVP (1.2)-(1.3) in L2, Theorem 1.1 requires an additional tangential/conormal regularity of the corresponding data. The precise increase of regularity needed for the data is prescribed by the energy inequality (1.7): to estimate the L2-norm of the solution, in the interior and on the boundary of the domain, we lose r conormal derivatives and s-r tangential derivatives with respect to the interior source term F, and s (tangential) derivatives with respect to the boundary datum G.In [2], the conormal regularity of weak solutions to the BVP (1.2)-(1.3) was studied, under the assumption that no loss of derivatives occurs in the estimate of the solution by the data. To prove the result of [2, Theorem 15], the solution u to (1.2)-(1.3) was regularized by a family of tangential mollifiers Jɛ, 0<ɛ<1, defined by Nishitani and Takayama in [9] as a suitable combination of the operator ♯ and the standard Friedrichs'mollifiers. The essential point of the analysis performed in [2] was to notice that the mollified solution Jɛu solves the same problem (1.2)-(1.3), as the original solution u. The data of the problem for Jɛu come from the regularization, by Jɛ, of the data involved in the original problem for u; furthermore, an additional term [Jɛ,L]u, where [Jɛ,L] is the commutator between the differential operator L and the tangential mollifier Jɛ, appears into the equation satisfied by Jɛu. Because the energy estimate associated to a strong L2-well-posed problem does not lose derivatives, actually this term can be incorporated into the source term of the equation satisfied by Jɛu.In the case of Theorem1.1, where the a priori estimate (1.7) exhibits a finite loss of regularity with respect to the data, this technique seems to be unsuccessful, since [Jɛ,L]u cannot be absorbed into the right-hand side without losing derivatives on the solution u; on the other hand, it seems that the same term cannot be merely reduced to a lower-order term involving the smoothed solution Jɛu, as well (this should require a sharp control of the error term u-Jɛu).These observations lead to develop another technique, where the tangential mollifierJɛ is replaced by the family of operators (3.32), involved in the characterization of regularity given by Proposition 3.2. Instead of studying the problem satisfied by the smoothed solution Jɛu, here we consider the problem satisfied by λδm-1,γ(Z)u. As before, a new term [λδm-1,γ(Z),L]u appears which takes account of the commutator between the differential operator L and the conormal operator λδm-1,γ(Z). Since we assume the weak well-posedness of the BVP (1.2)-(1.3) to be preserved under lower order terms, the approach consists of treating the commutator [λδm-1,γ(Z),L]u as a lower-order term within the interior equation for λδm-1,γ(Z)u (see (4.10)) (differently from the strongly L2-well-posed case studied in [2], the stability of problem (1.2)-(1.3) under lower-order perturbations is no longer a trivial consequence of the well-posedness itself. In Theorem 1.1, this stability is required as an additional hypothesis about the BVP); this is made possible by taking advantage from the invertibility of the operator λδm-1,γ(Z).We argue by induction on the integer orderm≥1. Let us take arbitrary data F∈ℋtan,γs+m,r+m(ℝ+n), G∈Hγs+m(ℝn-1), and fix an arbitrary matrix-valued function B∈C(0)∞(ℝ+n) (as the lower order term in the interior equation (1.2)).Because of the inductive hypothesis, we already know that the uniqueL2-solution u to (1.2)-(1.3) actually belongs to Htan,γm-1(ℝ+n) and u∣x1=0I belongs to Hγm-1(ℝn-1), provided that γ is taken to be larger than a certain constant γm-1≥1; moreover the solution u obeys the estimate (1.8) of order m-1(4.1)γ‖u‖Htan,γm-1(R+n)2+‖u∣x1=0I‖Hγm-1(Rn-1)2≤Cm-1(1γ2s+1‖F‖Htan,γs+m-1,r+m-1(R+n)2+1γ2s‖G‖Hγs+m-1(Rn-1)2),
where the positive constant Cm-1, as well as γm-1, only depends on the smoothness order m and the L∞-norm of a finite number (depending on m itself) of conormal derivatives of B (cf. (3.30)), besides the coefficients Aj  (1≤j≤n) of L and the integer numbers r and s.In order to increase the conormal regularity of the solutionu by order one, we are going to act on u by the conormal operator λδm-1,γ(Z); then we consider the analogue of the original problem (1.2)-(1.3) satisfied by λδm-1,γ(Z)u.


### 4.2. A Modified Version of the Conormal Operatorλδm-1,γ(Z)

Due to some technical reasons that will be clarified in Section4.3, we need to slightly modify the conormal operator λδm-1,γ(Z) to be applied to the solution u of the original BVP (1.2)-(1.3), as was described in the preceding section.The first step is to decompose the weight functionλδm-1,γ as the sum of two contributions. To do so, we proceed as follows. Firstly, we take an arbitrary positive, even function χ∈C∞(ℝn) with the following properties:(4.2)0≤χ(x)≤1,∀x∈Rn,χ(x)≡1,for|x|≤12,χ(x)≡0,for|x|>1.
Then, we set(4.3)λχ,δm-1,γ(ξ)≔χ(D)(λδm-1,γ)(ξ)=(F-1χ*λδm-1,γ)(ξ),rm,δ(ξ,γ)≔λδm-1,γ(ξ)-λχ,δm-1,γ(ξ)=(I-χ(D))(λδm-1,γ)(ξ).
The following result (the proof of which is postponed to Appendix A) shows that the function λχ,δm-1,γ essentially behaves like λδm-1,γ.Lemma 4.1.
Let the functionχ∈C∞(ℝn) satisfy the assumptions in (4.2). Then λχ,δm-1,γ is a symbol in Γm-1; moreover for every multi-index α∈ℕn, there exists a positive constant Cm,α, independent of γ and δ, such that
(4.4)|∂ξαλχ,δm-1,γ(ξ)|≤Cm,αλδm-1-|α|,γ(ξ),∀ξ∈Rn.An immediate consequence of Lemma4.1 and (4.3) is that rm,δ is also a γ-depending symbol in Γm-1.Let us define with the obvious meaning of the notations(4.5)λχ,δm-1,γ(D)≔Opγ(λχ,δm-1,γ),rm,δ(D,γ)≔Opγ(rm,δ),λχ,δm-1,γ(Z)≔Op♯γ(λχ,δm-1,γ),rm,δ(Z,γ)≔Op♯γ(rm,δ).
The second important result is concerned with the conormal operator rm,δ(Z,γ)=Op♯γ(rm,δ) and tells that it essentially behaves as a regularizing operator on conormal Sobolev spaces.Lemma 4.2.
For everyk∈ℕ, the conormal operator rm,δ(Z,γ) extends as a linear-bounded operator, still denoted by rm,δ(Z,γ), from L2(ℝ+n) to Htan,γk(ℝ+n). Moreover there exists a positive constant Cm,k, depending only on k and m, such that for all γ≥1 and δ∈]0,1](4.6)‖rm,δ(Z,γ)u‖Htan,γk(R+n)≤Cm,kγk‖u‖L2(R+n),∀u∈L2(R+n).Remark 4.3.
In the framework of the general theory of pseudodifferential operators, the procedure adopted to define the symbolλχ,δm-1,γ is standard and is used to modify an arbitrary symbol in such a way to make properly supported the corresponding pseudodifferential operator (see [13] for the definition of a properly supported operator and an extensive description of the method). As a general issue, one can prove that the resulting properly supported operator differs from the original one by a regularizing remainder. Essentially, an easy adaptation of the same arguments to the framework of conormal spaces in ℝ+n can be employed to prove the regularizing action of the conormal operator rm,δ(Z,γ) stated by Lemma 4.2.According to (4.3), we decompose(4.7)λδm-1,γ(Z)=λχ,δm-1,γ(Z)+rm,δ(Z,γ).
As a consequence of Lemmas 4.1 and 4.2, the role of the family of conormal operators {λδm-1,γ(Z)}0<δ≤1 in the characterization of the conormal regularity provided by Proposition 3.2 (cf. also (3.33)) can also be played by the family of “modified” operators {λχ,δm-1,γ(Z)}0<δ≤1, namely, we have the following.Corollary 4.4.
For every positive integerm and γ≥1, u∈Htan,γm(ℝ+n) if and only if u∈Htan,γm-1(ℝ+n) and the set {∥λχ,δm-1,γ(Z)u∥L2(ℝ+n)}0<δ≤1 is bounded.In order to suitably handle the commutator between the differential operatorL and the conormal operator λχ,δm-1,γ(Z), that comes from writing down the problem satisfied by λχ,δm-1,γ(Z)u (see Sections 4.3.1 and 4.3.2), it is useful to analyze the behavior of the pseudodifferential operators λχ,δm-1,γ(D), when interacting with another pseudodifferential operator by composition and commutation. The following lemma analyzes these situations; for later use, it is convenient to replace in our reasoning the function λχ,δm-1,γ by a general γ-depending symbol aδ preserving the same kind of decay properties as in (4.4).Lemma 4.5.
Let{aδ}0<δ≤1 be a family of symbols aδ=aδ(x,ξ,γ)∈Γr-1, r∈ℝ, such that for all multi-indices α,β∈ℕn there exists a positive constant Cr,α,β, independent of γ and δ, for which
(4.8)|∂ξα∂xβaδ(x,ξ,γ)|≤Cr,α,βλδr-1-|α|,γ(ξ),∀x,ξ∈Rn.
Let b=b(x,ξ,γ) be another symbol in Γl, for l∈ℝ.
Then, for everyδ∈]0,1], the product Opγ(aδ)Opγ(b) is a pseudodifferential operator with symbol aδ#b in Γl+r-1. Moreover, for all multi-indices α,β∈ℕn there exists a constant Cr,l,α,β, independent of γ and δ, such that
(4.9)|∂ξα∂xβ(aδ#b)(x,ξ,γ)|≤Cr,l,α,βλδl+r-1-|α|,γ(ξ),∀x,ξ∈Rn.
Under the same hypotheses, Opγ(aδ)Opγ(b)λ̃δ-m+1,γ(D) is a pseudodifferential operator with symbol (aδ#b)λ̃δ-m+1,γ in Γl+r-m; moreover, {(aδ#b)λ̃δ-m+1,γ}0<δ≤1 is a bounded subset of Γl+r-m. Eventually, if the symbol aδ is scalar-valued, [Opγ(aδ),Opγ(b)]λ̃δ-m+1,γ(D) is a pseudodifferential operator with symbol cδ∈Γl+r-m-1, and {cδ}0<δ≤1 is a bounded subset of Γl+r-m-1.The proof of Lemma4.5 is postponed to Appendix A.Remark 4.6.
ThatOpγ(aδ)Opγ(b), Opγ(aδ)Opγ(b)λ̃δ-m+1,γ(D) and [Opγ(aδ),Opγ(b)]λ̃δ-m+1,γ(D) have symbols belonging, respectively, to Γl+r-1, Γl+r-m and Γl+r-m-1 (for scalar-valued aδ) follows at once from the standard rules of symbolic calculus summarized in Proposition 3.6.  The nontrivial part of the statement above (although deduced from the asymptotic formula (3.23) with a minor effort) is the one asserting that the symbol of Opγ(aδ)Opγ(b) enjoys estimates (4.9); indeed, these estimates give the precise dependence on δ of the decay at infinity of this symbol. Then the remaining assertions in Lemma 4.5 easily follow from (4.9) itself.Remark 4.7.
In view of Proposition3.8, the results on symbols collected in Lemma 4.5 can be used to study the conormal Sobolev continuity of the related conormal operators.
To be definite, for every nonnegative integer numbers, such that s+l+r-m is also nonnegative, Proposition 3.8 and Lemma 4.5 imply that the conormal operator Op♯γ(aδ)Op♯γ(b)λ̃δ-m+1,γ(Z) extends as a linear-bounded mapping from Htan,γs+l+r-m(ℝ+n) into Htan,γs(ℝ+n); moreover, its operator norm is uniformly bounded with respect to γ and δ.
If in additions+l+r-m≥1 and aδ are scalar-valued, then [Op♯γ(aδ),Op♯γ(b)]λ̃δ-m+1,γ(Z) extends as a linear-bounded operator from Htan,γs+l+r-m-1(ℝ+n) into Htan,γs(ℝ+n), and again its operator norm is uniformly bounded with respect to γ and δ.
These mapping properties will be usefully applied in Sections4.3 and 4.5.


### 4.3. The Interior Equation

We follow the strategy already explained in Section4.1, where now the role of the operator λδm-1,γ(Z) is replaced by λχ,δm-1,γ(Z). Since λχ,δm-1,γ∈Γm-1 (because of Lemma 4.1) and, for γ≥γm-1, u∈Htan,γm-1(ℝ+n) (from the inductive hypothesis), after Proposition 3.8 we know that λχ,δm-1,γ(Z)u∈L2(ℝ+n).Applyingλχ,δm-1,γ(Z) to (1.2) (where ℬ is just the multiplication by B), we find that λχ,δm-1,γ(Z)u must solve(4.10)(γ+L+B)(λχ,δm-1,γ(Z)u)+[λχ,δm-1,γ(Z),L+B]u=λχ,δm-1,γ(Z)F,inR+n.
We are going now to show that the commutator term [λχ,δm-1,γ(Z),L+B]u in the above equation can be actually considered as a lower-order term with respect to λχ,δm-1,γ(Z)u.To this end, we may decompose this term as the sum of two contributions corresponding, respectively, to thetangential and normal components of L.First, in view of (1.5), (1.6), we may write the coefficient A1 of the normal derivative ∂1 in the expression (1.4) of L as (4.11)A1=A11+A12,A11≔(A1I,I000),A1∣x1=02=0,
hence (4.12)A12∂1=H1Z1,
where H1(x)=x1-1A12(x)∈C(0)∞(ℝ+n). Accordingly, we split L as (4.13)L=A11∂1+Ltan,Ltan≔H1Z1+∑j=2nAjZj.
Consequently, we have(4.14)[λχ,δm-1,γ(Z),L+B]u=[λχ,δm-1,γ(Z),A11∂1]u+[λχ,δm-1,γ(Z),Ltan+B]u.
Note that Ltan+B is just a conormal operator of order 1, according to the terminology introduced in Section 3.2.


#### 4.3.1. The Tangential Commutator

Firstly, we study thetangential commutator[λχ,δm-1,γ(Z),Ltan+B]u. Using the identity λ̃δ-m+1,γ(Z)λδm-1,γ(Z)=I and (4.7), the latter can be written in terms of λχ,δγ,m-1(Z)u, modulo some smoothing reminder. Indeed we compute(4.15)[λχ,δm-1,γ(Z),Ltan+B]u=[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u+rm,δ(Z,γ)u)=[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u)+sm,δ(x,Z,γ)u,
where we have set for short(4.16)sm,δ(x,Z,γ)≔[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)rm,δ(Z,γ).


#### 4.3.2. The Normal Commutator

First of all, we notice that, due to the structure of the matrixA11, the commutator [λχ,δγ,m-1(Z),A11∂1] acts nontrivially only on the noncharacteristic component of the vector function u; namely, we have: (4.17)[λχ,δm-1,γ(Z),A11∂1]u=([λχ,δm-1,γ(Z),A1I,I∂1]uI0).
Therefore, we focus on the study of the first nontrivial component of the commutator term. Note that the commutator [λχ,δm-1,γ(Z),A1I,I∂1] cannot be merely treated by the tools of the conormal calculus developed in Section 3.2, because of the presence of the effective normal derivative ∂1 (recall that A1I,I is invertible). This section is devoted to the study of the normal commutator [λχ,δm-1,γ(Z),A1I,I∂1]uI. The following result is of fundamental importance.Proposition 4.8.
For allδ∈]0,1], γ≥1 and m∈ℕ, there exists a symbol qm,δ(x,ξ,γ)∈Γm-2 such that
(4.18)[λχ,δm-1,γ(Z),A1I,I∂1]w=qm,δ(x,Z,γ)(∂1w),∀w∈C(0)∞(R+n).
Moreover, the symbol qm,δ(x,ξ,γ) obeys the following estimates. For all α,β∈ℕn, there exists a positive constant Cm,α,β, independent of γ and δ, such that
(4.19)|∂ξα∂xβqm,δ(x,ξ,γ)|≤Cm,α,βλδm-2-|α|,γ(ξ),∀x,ξ∈Rn.Proof.
Thatqm,δ(x,ξ,γ), satisfying estimates (4.19), is a symbol in Γm-2 actually follows arguing from (4.19) and inequalities (3.18) as was already done for λδm-1,γ(ξ) and λ̃δ-m+1,γ(ξ) (see Section 3.2).
For givenw∈C(0)∞(ℝ+n), let us explicitly compute ([λχ,δm-1,γ(Z),A1I,I∂1]w)♯; using the identity (∂1w)♯=e-x1(Z1w)♯ and that λχ,δm-1,γ(Z) and Z1 commute, we find for every x∈ℝn(4.20)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1(Z1λχ,δm-1,γ(Z)w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1(λχ,δm-1,γ(Z)Z1w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1λχ,δm-1,γ(D)(Z1w)♯(x).
Observing that λχ,δm-1,γ(D) acts on the space 𝒮(ℝn) as the convolution by the inverse Fourier transform of λχ,δm-1,γ, the preceding expression can be equivalently restated as follows:
(4.21)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=F-1(λχ,δm-1,γ)*A1I,I,♮e-(⋅)1(Z1w)♯(x)-A1I,I,♮(x)e-x1F-1(λχ,δm-1,γ)*(Z1w)♯=〈F-1(λχ,δm-1,γ),A1I,I,♮(x-⋅)e-(x1-(⋅)1)(Z1w)♯(x-⋅)〉-A1I,I,♮(x)e-x1〈F-1(λχ,δm-1,γ),(Z1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)A1I,I,♮(x-⋅)e-(x1-(⋅)1)(Z1w)♯(x-⋅)〉-〈ηδm-1,γ,χ(⋅)A1I,I,♮(x)e-x1(Z1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)A1I,I,♮(x-⋅)(∂1w)♯(x-⋅)〉-〈ηδm-1,γ,χ(⋅)A1I,I,♮(x)e-(⋅)1(∂1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)[A1I,I,♮(x-⋅)-A1I,I,♮(x)e-(⋅)1](∂1w)♯(x-⋅)〉,
where ηδm-1,γ≔ℱ-1(λδm-1,γ) and the identity ℱ-1(λχ,δm-1,γ)=χηδm-1,γ (following at once from (4.3)) has been used. Just for brevity, let us further set
(4.22)K(x,y)≔[A1I,I,♮(x-y)-A1I,I,♮(x)e-y1]χ(y).
Thus the identity above reads as
(4.23)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=〈ηδm-1,γ,K(x,⋅)(∂1w)♯(x-⋅)〉,
where the “kernel” 𝒦(x,y) is a bounded function in C∞(ℝn×ℝn), with bounded derivatives of all orders. This regularity of 𝒦 is due to the presence of the function χ in formula (4.22); actually the vanishing of χ at infinity prevents the blow-up of the exponential factor e-y1, as y1→-∞. We point out that this is just the step of our analysis of the normal commutator, where this function χ is needed.
After (4.22), we also have that 𝒦(x,0)=0; then, by a Taylor expansion with respect to y, we can represent the kernel 𝒦(x,y) as follows:
(4.24)K(x,y)=∑k=1nbk(x,y)yk,
where bk(x,y) are given bounded functions in C∞(ℝn×ℝn), with bounded derivatives; it comes from (4.22) and (4.2) that bk can be defined in such a way that for some ɛ>1 and all x∈ℝn there holds
(4.25)supp⁡bk(x,⋅)⊆{|y|≤ɛ}.
Inserting (4.24) in (4.23) and using standard properties of the Fourier transform we get
(4.26)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=〈ηδm-1,γ,∑k=1nbk(x,⋅)(⋅)k(∂1w)♯(x-⋅)〉=∑k=1n〈(⋅)kF-1(λδm-1,γ),bk(x,⋅)(∂1w)♯(x-⋅)〉=-∑k=1n〈F-1(Dkλδm-1,γ),bk(x,⋅)(∂1w)♯(x-⋅)〉=-∑k=1n〈Dkλδm-1,γ,F-1(bk(x,⋅)(∂1w)♯(x-⋅))〉=-∑k=1n∫RnDkλδm-1,γ(ξ)F-1(bk(x,⋅)(∂1w)♯(x-⋅))(ξ)dξ=-∑k=1n(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)(∂1w)♯(x-y)dy)dξ,
where we have set Dk≔-i∂ξk (for each k=1,…,n);   note that for w∈C(0)∞(ℝ+n) and any x∈ℝn, the function bk(x,·)(∂1w)♯(x-·) belongs to 𝒮(ℝn); hence the last expression in (4.26) makes sense. Henceforth, we replace (∂1w)♯ by any function v∈𝒮(ℝn). Our next goal is writing the integral operator
(4.27)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ
as a pseudodifferential operator.
Firstly, we make use of the inversion formula for the Fourier transformation and Fubini's theorem to recast (4.27) as follows:
(4.28)∫Rneiξ⋅ybk(x,y)v(x-y)dy=(2π)-n∫Rneiξ⋅ybk(x,y)(∫Rnei(x-y)⋅ηv̂(η)dη)dy=(2π)-n∫Rneix⋅η(∫Rne-iy⋅(η-ξ)bk(x,y)dy)v̂(η)dη=(2π)-n∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη;
for every index k, b̂k(x,ζ) denotes the partial Fourier transform of bk(x,y) with respect to y. Then, inserting (4.28) into (4.27), we obtain
(4.29)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ=(2π)-2n∫RnDkλδm-1,γ(ξ)(∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη)dξ.
Recall that, for each x∈ℝn, the function y↦bk(x,y) belongs to C0∞(ℝn) (and its compact support does not depend on x, see (4.25)); thus, for each x∈ℝn, b̂k(x,ζ) is rapidly decreasing in ζ. Because of the estimates for derivatives of λδm-1,γ and since v̂(η) is also rapidly decreasing, Fubini's theorem can be used to change the order of the integrations within (4.29). So we get
(4.30)(2π)-2n∫RnDkλδm-1,γ(ξ)(∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη)dξ=(2π)-2n∫Rneix⋅η(∫Rnb̂k(x,η-ξ)Dkλδm-1,γ(ξ)dξ)v̂(η)dη=(2π)-n∫Rneix⋅ηqk,m,δ(x,η,γ)v̂(η)dη,
where we have set
(4.31)qk,m,δ(x,ξ,γ)≔(2π)-n∫Rnb̂k(x,η)Dkλδm-1,γ(ξ-η)dη.
Notice that formula (4.31) defines qk,m,δ as the convolution of the functions b̂k(x,·) and Dkλδm-1,γ; hence qk,m,δ is a well-defined C∞-function in ℝn×ℝn.
The proof of Proposition4.8 will be accomplished, once the following lemma is proved.
Lemma 4.9.
For everym∈ℕ, k=1,…,n and all α,β∈Nn, there exists a positive constant Ck,m,α,β, independent of γ and δ, such that
(4.32)|∂ξα∂xβqk,m,δ(x,ξ,γ)|≤Ck,m,α,βλδm-2-|α|,γ(ξ),∀x,ξ∈Rn.
It comes from Lemma4.9 and the left inequality in (3.18) that, for each index k, the function qk,m,δ is a symbol in Γm-2; notice however that the set {qk,m,δ}0<δ≤1 is bounded in Γm-1 but not in Γm-2. The proof of Lemma 4.9 is postponed to Appendix A.
Now, we continue the proof of Proposition4.8.
End of the Proof of Proposition4.8
The last row of (4.30) provides the desired representation of (4.27) as a pseudodifferential operator; actually it gives the identity
(4.33)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ=Opγ(qk,m,δ)v(x),
for every v∈𝒮(ℝn).
Inserting the above formula (withv=(∂1w)♯) into (4.26) finally gives
(4.34)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=Opγ(qm,δ)(∂1w)♯(x),
where qm,δ=qm,δ(x,ξ,γ) is the symbol in Γm-2 defined by
(4.35)qm,δ(x,ξ,γ)≔-∑k=1nqk,m,δ(x,ξ,γ).
Of course, formula (4.18) is equivalent to (4.34), in view of (3.26). Estimates (4.19) are satisfied by qm,δ by summation over k of the similar estimates satisfied by qm,δ,k (cf. Lemma 4.9).
This ends the proof of Proposition4.8.Now, we are going to show how the representation in (4.18) can be exploited to treat the normal commutator as a lower-order term in (4.10) satified by λχ,δm-1,γ(Z)u.Firstly, we notice that formula (4.18) has been deduced for smooth functions w, while u is just an L2-function (actually it belongs to Htanm-1(ℝ+n), by the inductive hypothesis). In order to use (4.18) for u, we need to approximate the latter by smooth functions. This can be done by the help of [14, Proposition 1, Theorem 1]; from there, we know that there exists a sequence {uν}ν in C(0)∞(ℝ+n) such that(4.36)uν⟶u,inL2(R+n),(L+B)uν⟶(L+B)u,inL2(R+n),uν∣x1=0I⟶u∣x1=0I,inH-1/2(Rn-1),asν⟶+∞.
For each index ν, the regular function uν∈C(0)∞(ℝ+n) solves the same BVP as the function u, with new data Fν, Gν defined by(4.37)Fν≔(γ+L+B)uν,Gν≔Muν∣x1=0.
It immediately follows from (4.36) that the regular data Fν,Gν approximate the original data F,G by(4.38)Fν⟶F,inL2(R+n),Gν⟶G,inH-1/2(Rn-1),asν⟶+∞.
The same analysis performed to the BVP (1.2)-(1.3) can be applied to the BVP solved by uν, for each ν; in particular, we find that λχ,δm-1,γ(Z)uν satisfies the analogue to (4.10), where F is replaced by Fν. Because of the regularity of uν, formula (4.18) can be used to represent the normal commutator term [λχ,δm-1,γ(Z),A1I,I∂1]uνI. Directly from system (γ+L+B)uν=Fν, ∂1uνI can be represented in terms of tangential derivatives of uν only, as follows:(4.39)∂1uνI=(A1I,I)-1FνI+LIuν,
where ℒI=ℒI(x,Z,γ) denotes the tangential partial differential operator(4.40)LIuν≔-(A1I,I)-1[γuνI+H1Z1uνII+(∑j=2nAjZjuν+Buν)I],
and we have set H1≔x1-1A1I,II (recall that H1∈C(0)∞(ℝ+n) since A1∣x1=0I,II=0). Inserting (4.39) into (4.18), written for w=uνI, leads to(4.41)[λχ,δm-1,γ(Z),A1I,I∂1]uνI=qm,δ(x,Z,γ)((A1I,I)-1FνI+LIuν).
On the other hand, plugging λ̃δ-m+1,γ(Z)λχ,δm-1,γ(Z)+λ̃δ-m+1,γ(Z)rm,δ(Z,γ)=I into (4.41) gives(4.42)[λχ,δm-1,γ(Z),A1I,I∂1]uνI=qm,δ(x,Z,γ)((A1I,I)-1FνI)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)uν)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)uν.
Now, we let ν→+∞.From (1.2) (written for u and uν), and using that uν♯→u♯ in L2(ℝn), one finds (4.43)(A1∂1uν)♯⟶(A1∂1u)♯,inS′(Rn),
hence (because ♯-1 is a linear continuous operator from 𝒮′(ℝn)⊂𝒟′(ℝn) to 𝒟′(ℝ+n))(4.44)λχ,δm-1,γ(Z)(A1I,I∂1uνI)⟶λχ,δm-1,γ(Z)(A1I,I∂1uI),inD′(R+n).
On the other hand, uν♯→u♯ in L2(ℝn) implies that(4.45)λχ,δm-1,γ(Z)(uν)⟶λχ,δm-1,γ(Z)(u),inD′(R+n),
hence(4.46)A1I,I∂1(λχ,δm-1,γ(Z)(uνI))⟶A1I,I∂1(λχ,δm-1,γ(Z)(uI)),inD′(R+n).
Adding (4.44), (4.46) then gives(4.47)[λχ,δm-1,γ(Z),A1I,I∂1]uνI⟶[λχ,δm-1,γ(Z),A1I,I∂1]uI,inD′(R+n).
As to the right-hand side of (4.42), all the involved operators, acting on Fν and uν, are conormal. Hence the L2-convergences uν→u and Fν→F and the fact that conormal operators continuously extend to the space of distributions u∈𝒟′(ℝ+n), for which u♯∈𝒮′(ℝn), give the convergences(4.48)qm,δ(x,Z,γ)((A1I,I)-1FνI)⟶qm,δ(x,Z,γ)((A1I,I)-1FI),qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)uν)⟶qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u),qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)uν⟶qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)u,inD′(R+n).
Therefore, the uniqueness of the limit in 𝒟′(ℝ+n) together with (4.47), (4.48) implies that (4.42) holds true for the L2-solution u of (1.2)-(1.3), that is,(4.49)[λχ,δm-1,γ(Z),A1I,I∂1]uI=qm,δ(x,Z,γ)((A1I,I)-1FI)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)u.
Let us come back to the commutator term [λχ,δm-1,γ(Z),L+B]u appearing in the interior equation (4.10).Substituting (4.15), (4.49) into (4.14) gives for this term the following representation:(4.50)[λχ,δm-1,γ(Z),L+B]u=ρm,δ(x,Z,γ)(λχ,δm-1,γ(Z)u)+τm,δ(x,Z,γ)u+ηm,δ(x,Z,γ)F,
where we have set for short(4.51)ρm,δ(x,Z,γ)≔[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)+(qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)0),τm,δ(x,Z,γ)≔sm,δ(x,Z,γ)+(qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)0),ηm,δ(x,Z,γ)≔(qm,δ(x,Z,γ)(A1I,I)-1000).
Consequently, the interior equation (4.10) can be restated as(4.52)(γ+L+B+ρm,δ(x,Z,γ))(λχ,δm-1,γ(Z)u)+τm,δ(x,Z,γ)u+ηm,δ(x,Z,γ)F=λχ,δm-1,γ(Z)F,inR+n.
Since Ltan+B and ℒI are conormal operators with symbol in Γ1, Lemma 4.5 and Proposition 4.8 imply that ρm,δ(x,Z,γ) is a conormal operator with symbol in Γ0, for each 0<δ≤1; moreover, it amounts that the family of symbols {ρm,δ}0<δ≤1 is a bounded subset of Γ0. Therefore, ρm,δ(x,Z,γ) can be regarded, jointly with B, as a lower-order term in (4.52), as considered in the statement of Theorem 1.1.Concerning the termsτm,δ(x,Z,γ)u, ηm,δ(x,Z,γ)F, they can be both moved into the right-hand side of (4.52), to be treated as a part of the interior source term, as will be detailed in Section 4.5.


### 4.4. The Boundary Condition

Now we are going to seek for an appropriate boundary condition to be coupled with the interior equation (4.10), in order to state a BVP solved by λχ,δm-1,γ(Z)u.To this end, it is worthwhile to make an additional hypothesis about the smooth functionχ involved in the definition of λχ,δm-1,γ(Z). We assume that χ has the form(4.53)∀x=(x1,x′)∈Rn,χ(x)=χ1(x1)χ̃(x′),
where χ1∈C∞(ℝ) and χ̃∈C∞(ℝn-1) are given positive even functions, to be chosen in such a way that conditions (4.2) are made satisfied.As it was done for the analysis of the normal commutator (cf. Proposition4.8), we start our reasoning by dealing with smooth functions. In this case, following closely the arguments employed to prove Proposition 4.8 and Lemma 4.9, we are able to get the following.Proposition 4.10.
Assume thatχ obeys the assumptions (4.2), (4.53). Then, for all δ∈]0,1], γ≥1, and m∈ℕ, the function bm,δ′(ξ′,γ) defined by
(4.54)bm,δ′(ξ′,γ)≔(2π)-n∫Rnλδm-1,γ(η1,η′+ξ′)(e(⋅)1/2χ1)∧1(η1)χ̃̂(η′)dη,∀ξ′∈Rn-1,
(where ξ′≔(ξ2,…,ξn) are the Fourier dual variables of the tangential variables x′=(x2,…,xn)) is a γ-depending symbol in ℝn-1 belonging to Γm-1. Moreover, for all functions w∈C(0)∞(ℝ+n) there holds
(4.55)∀x′∈Rn-1,(λχ,δm-1,γ(Z)w)∣x1=0(x′)=bm,δ′(D′,γ)(w∣x1=0)(x′),
where one has set D′=(D2,…,Dn), Dj=-i∂j (for j=2,…,n) and bm,δ′(D′,γ) stands for the ordinary pseudodifferential operator in ℝn-1 with symbol bm,δ′.
Eventually, the following estimates are satisfied by the symbolbm,δ′(ξ′,γ): for all α′=(α2,…,αn)∈ℕn-1 there exists a positive constant Cm,α′, independent of γ and δ, such that
(4.56)|∂ξ′α′b′m,δ(ξ′,γ)|≤Cm,α′λδm-1-|α′|,γ(ξ′),∀ξ′∈Rn-1.Proof.
Thatbm,δ′ belongs to Γm-1 immediately follows from estimates (4.56), using the (n-1)-dimensional counterpart of (3.18).
Letw∈C(0)∞(ℝ+n); to find a symbol bm,δ′ satisfying (4.55), from (4.3) we firstly compute
(4.57)(λχ,δm-1,γ(Z)w)♯(x)=λχ,δm-1,γ(D)(w♯)(x)=(F-1(λχ,δm-1,γ)*w♯)(x)=〈F-1(λχ,δm-1,γ),w♯(x-⋅)〉=〈F-1(λδm-1,γ),χ(⋅)e(x1-(⋅)1)/2w(ex1-(⋅)1,x′-(⋅)′)〉,∀(x1,x′)∈Rn,
hence
(4.58)λχ,δm-1,γ(Z)w(x)=〈F-1(λδm-1,γ),χ(⋅)e-(⋅)1/2w(x1e-(⋅)1,x′-(⋅)′)〉=〈λδm-1,γ,F-1(χ(⋅)e-(⋅)1/2w(x1e-(⋅)1,x′-(⋅)′))〉=(2π)-n∫λδm-1,γ(ξ)(∫eiξ⋅yχ(y)e-y1/2w(x1e-y1,x′-y′)dy)dξ,∀x1>0,∀x′∈Rn-1.
The regularity of w legitimates all the above calculations. Setting x1=0 in the last expression above, we deduce the corresponding expression for the trace on the boundary of λχ,δm-1,γ(Z)w(4.59)(λχ,δm-1,γ(Z)w)∣x1=0(x′)=(2π)-n∫λδm-1,γ(ξ)(∫eiξ⋅yχ(y)e-y1/2(w∣x1=0)(x′-y′)dy)dξ.
Now we substitute (4.53) into the y-integral appearing in the last expression above; then Fubini's theorem gives
(4.60)∫eiξ⋅yχ1(y1)χ̃(y′)e-y1/2(w∣x1=0)(x′-y′)dy=∫eiξ′⋅y′(∫eiξ1y1e-y1/2χ1(y1)dy1)χ̃(y′)(w∣x1=0)(x′-y′)dy′=(e(⋅)1/2χ1)∧1(ξ1)∫eiξ′⋅y′χ̃(y′)(w∣x1=0)(x′-y′)dy′,
where ∧1 is used to denote the one-dimensional Fourier transformation with respect to y1. Writing, by the inversion formula, (w∣x1=0)(x′-y′)=(2π)-n+1∫ei(x′-y′)·η′w∣x1=0̂(η′)dη′ and using once more Fubini's theorem, we further obtain
(4.61)∫eiξ′⋅y′χ̃(y′)(w∣x1=0)(x′-y′)dy′=(2π)-n+1∫eiξ′⋅y′χ̃(y′)(∫ei(x′-y′)⋅η′w∣x1=0̂(η′)dη′)dy′=∫eix′⋅η′((2π)-n+1∫ei(ξ′-η′)⋅y′χ̃(y′)dy′)w∣x1=0̂(η′)dη′=(2π)-n+1∫eix′⋅η′χ̃̂(ξ′-η′)w∣x1=0̂(η′)dη′;∧ is used here to denote the (n-1)-dimensional Fourier transformation with respect to x′. Inserting (4.60), (4.61) into (4.59) then leads to
(4.62)(λχ,δm-1,γ(Z)w)∣x1=0(x′)=(2π)-n∫λδm-1,γ(ξ)(e(⋅)1/2χ1)∧1(ξ1)((2π)-n+1∫eix′⋅η′χ̃̂(ξ′-η′)w∣x1=0̂(η′)dη′)dξ.
Because (e(·)1/2χ1)∧1∈𝒮(ℝ), χ̃̂∈𝒮(ℝn-1), and w∣x1=0̂∈𝒮(ℝn-1), the double integral
(4.63)∬eix′⋅η′λδm-1,γ(ξ)(e(⋅)1/2χ1)∧1(ξ1)χ̃̂(ξ′-η′)w∣x1=0̂(η′)dη′dξ
converges absolutely; hence Fubini's theorem allows to exchange the order of the integrations in (4.62) and find
(4.64)(λχ,δm-1,γ(Z)w)∣x1=0(x′)=(2π)-n+1∫eix′⋅η′bm,δ′(η′,γ)w∣x1=0̂(η′)dη′,
where bm,δ′(η′,γ) is defined by (4.54). This shows the identity (4.55).
The proof of estimates (4.56) is similar to that of estimates (4.32) in Lemma 4.9 (see Appendix A); so we will omit it.Let us now illustrate how formula (4.55) can be used to derive the desired boundary condition satisfied by λχ,δm-1,γ(Z)u.Again, letu be the L2-solution to the original BVP (1.2)-(1.3) and {uν}ν=1+∞ the corresponding sequence in C(0)∞(ℝ+n), approximating u in the sense of (4.36).The last convergence in (4.36) and the Sobolev continuity of standard pseudodifferential operators gives in particular that(4.65)bm,δ′(D′,γ)(uν∣x1=0I)⟶bm,δ′(D′,γ)(u∣x1=0I),inH-m+1/2(Rn-1).
On the other hand, (4.36) and (4.52) (written for u and uν) can be used to prove that (λχ,δm-1,γ(Z)uI)∣x1=0 and (λχ,δm-1,γ(Z)uνI)∣x1=0, for each ν, are traces well defined in H-1/2(ℝn-1) and the convergence(4.66)(λχ,δm-1,γ(Z)uνI)∣x1=0⟶(λχ,δm-1,γ(Z)uI)∣x1=0
holds true, at least in 𝒟′(ℝn-1). The proof of this assertion is postponed to Appendix A (see Lemma A.2).Sinceuν are smooth functions, from Proposition 4.10, it comes that for each ν:(4.67)(λχ,δm-1,γ(Z)uνI)∣x1=0=bm,δ′(D′,γ)(uν∣x1=0I).
Then, letting ν→+∞, (4.65) and (4.66) yield(4.68)(λχ,δm-1,γ(Z)uI)∣x1=0=bm,δ′(D′,γ)(u∣x1=0I).
Recalling that M=(Id,0), from the boundary condition (1.3) and (4.68), we immediately find(4.69)(M(λχ,δm-1,γ(Z)u))∣x1=0=M(λχ,δm-1,γ(Z)uI)∣x1=0=Mbm,δ′(D′,γ)(u∣x1=0I)=bm,δ′(D′,γ)(Mu∣x1=0)=bm,δ′(D′,γ)G.


### 4.5. Derivation of the Conormal Regularity at the Orderm

We are now in the position to get the desired conormal regularity of the solutionu to (1.2)-(1.3), under the assumptions that F∈ℋtan,γs+m,r+m(ℝ+n), G∈Hγs+m(ℝn-1). From now on, assume that γ≥γm-1; so far, from the inductive hypothesis we know that, for such a γ, u belongs to Htan,γm-1(ℝ+n), u∣x1=0I∈Hγm-1(ℝn-1) and the estimate (4.1) is satisfied (see the end of Section 4.1). Because of the calculations made in Sections 4.3 and 4.4, it follows that λχ,δm-1,γ(Z)u is an L2 solution of the problem(4.70)(γ+L+B+ρm,δ(x,Z,γ))(λχ,δγ,m-1(Z)u)=(λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F-τm,δ(x,Z,γ)u,inR+n,(4.71)M(λχ,δm-1,γ(Z)u)=bm,δ′(D′,γ)G,onRn-1.
The previous one reads as the original BVP (1.2)-(1.3) solved by u, where the role of the lower-order term ℬ is played here by the conormal operator B+ρm,δ(x,Z,γ). As already discussed in the end of Section 4.3, in view of Proposition 3.8 and Lemma 4.5, the symbol of B+ρm,δ(x,Z,γ) belongs to Γ0, and {ρm,δ}0<δ≤1 is a bounded subset of Γ0.As regards to the termsτm,δ(x,Z,γ)u and ηm,δ(x,Z,γ)F appearing into the right-hand side of (4.70), they can be regarded as a part of the source term in the interior equation (4.70) (this is the reason why they have been moved in the right-hand side of (4.70)).Let us firstly focus onτm,δ(x,Z,γ)u. After Lemma 4.5 and Proposition 3.8 (see also Proposition 4.8, Remark 4.7, and formulas (4.16), (4.51)), we know that for any k∈ℕ the operators [λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z) and qm,δ(x,Z,γ)ℒIλ̃δ-m+1,γ(Z) extend as linear-bounded mappings from Htan,γk(ℝ+n) into itself, and their operator norms are uniformly bounded with respect to γ and δ. Combining with the result of Lemma 4.2, it follows that a positive constant Cs>0, independent of γ and δ, can be found in such a way that(4.72)‖τm,δ(x,Z,γ)u‖Htan,γs,r(R+n)≤‖τm,δ(x,Z,γ)u‖Htan,γs(R+n)≤Csγs‖u‖L2(R+n).
Concerning now the term (λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F, after Lemma 4.5 and Proposition 4.8 we already know that the symbol ηm,δ(x,ξ,γ) has order m-2 and obeys the same decay estimates as in (4.19). From (3.17) it then follows that {ηm,δ}0<δ≤1 is a bounded subset of Γm-1; because {λχ,δm-1,γ}0<δ≤1 is also a bounded subset of Γm (as a consequence of (4.4) and (3.17) again), after Proposition 3.10 we conclude that there exists a positive constant Cm,s,r such that(4.73)‖(λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F‖Htan,γs,r(R+n)≤Cm,s,r‖F‖Htan,γs+m,r+m(R+n).
As regards to the boundary datum bm,δ′(D′,γ)G in (4.71), the family of symbols {bm,δ′}0<δ≤1 in ℝn-1 defines a bounded subset of Γm; this follows from estimates (4.56) and the inequality (3.17) (in dimension n-1). Therefore, the Sobolev continuity of ordinary pseudodifferential operators in ℝn-1 implies the existence of a positive constant Cm,s, independent of γ and δ, such that:(4.74)‖bm,δ′(D′,γ)G‖Hγs(Rn-1)≤Cm,s‖G‖Hγs+m(Rn-1).
From the assumptions made about the BVP (1.2)-(1.3) in Theorem 1.1, we find some constants γ̃m≥1,  C̃m>0 such that for all γ≥γm≔max{γm-1,γ̃m} and every δ∈]0,1], λχ,δm-1,γ(Z)u is the only L2-solution of (4.70)-(4.71) and obeys the estimate(4.75)γ‖λχ,δm-1,γ(Z)u‖L2(R+n)2+‖(λχ,δm-1,γ(Z)uI)∣x1=0‖L2(Rn-1)2≤C̃m(1γ2s+1‖(λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F-τm,δ(x,Z,γ)u‖Htan,γs,r(R+n)2+1γ2s‖bm,δ′(D′,γ)G‖Hγs(Rn-1)2).
We remark that, according to the statement of Theorem 1.1, the constants γ̃m, C̃m are only dependent on a δ,γ-uniform upper bound of the kth-order norm (3.14), computed on the symbol of B+ρm,δ(x,Z,γ), besides the coefficients Aj, 1≤j≤n, and the integer numbers r,s.Then, using (4.72)–(4.74) and (1.7) to estimate the right-hand side of (4.75), we get(4.76)γ‖λχ,δm-1,γ(Z)u‖L2(R+n)2+‖(λχ,δm-1,γ(Z)uI)∣x1=0‖L2(Rn-1)2≤C̃m,s,r(1γ2s+1‖F‖Htan,γs+m,r+m(R+n)2+1γ2s‖G‖Hγs+m(Rn-1)2),
where C̃m,s,r is a suitable positive constant independent of γ≥γm and δ. Since the L2-norms ∥λχ,δm-1,γ(Z)u∥L2(ℝ+n) are bounded by (4.76), uniformly with respect to δ∈]0,1], Corollary 4.4 gives u∈Htan,γm(ℝ+n).As to the Sobolev regularity of the trace on the boundary of the noncharacteristic componentuI of the solution, the estimate (4.76) gives a bound of ∥(λχ,δm-1,γ(Z)uI)∣x1=0∥L2(ℝn-1)=∥bm,δ′(D′,γ)(u∣x1=0I)∥L2(ℝn-1) uniform with respect to δ∈]0,1] (cf. (4.68)). Then u∣x1=0I∈Hγm(ℝn-1) can be derived from the next result, the proof of which will be given in Appendix A.Lemma 4.11.
Form∈ℕ and δ∈]0,1], let bm,δ′(ξ′,γ) be defined by (4.54). Then there exists a symbol βm,δ(ξ′,γ)∈Γm-2 such that
(4.77)bm,δ′(ξ′,γ)=λδm-1,γ(ξ′)+βm,δ(ξ′,γ),∀ξ′∈Rn-1.
In addition, the symbol βm,δ satisfies the following estimates: for every α′∈ℕn-1, there exists a positive constant Cm,α′, independent of γ and δ, such that
(4.78)|∂ξ′α′βm,δ(ξ′,γ)|≤Cm,α′λδm-2-|α′|,γ(ξ′),∀ξ′∈Rn-1.Arguing as was done to derive Corollary4.4 from Lemma 4.2, from Lemma 4.11 we deduce the following.Corollary 4.12.
For every positive integerm and γ≥1, v∈Hγm(ℝn-1) if and only if v∈Hγm-1(ℝn-1) and the set {∥bm,δ′(D′,γ)v∥L2(ℝn-1)}0<δ≤1 is bounded.After the result of Corollary4.12, we conclude that u∣x1=0I∈Hγm(ℝn-1).It remains to prove that the solution of (1.2)-(1.3) satisfies the a priori estimate (1.8) of order m. From estimate (4.76) and the use of the identities (4.7), (4.68), (4.77), we also deduce
(4.79)γ‖λδm-1,γ(Z)u‖L2(R+n)2+‖λδm-1,γ(D′)(u∣x1=0I)‖L2(Rn-1)2≤Cm,s,r′(1γ2s+1‖F‖Htan,γs+m,r+m(R+n)2+1γ2s‖G‖Hγs+m(Rn-1)2)+2γ‖rm,δ(Z,γ)u‖L2(R+n)2+2‖βm,δ(D′,γ)(u∣x1=0I)‖L2(Rn-1)2,
where the positive constant Cm,s,r′ is again independent of γ≥γm and δ. On the other hand, using Lemma 4.2 and that {βm,δ}0<δ≤1 is a bounded subset of Γm-1 (that follows at once from Lemma 4.11 and inequality (3.17)), one can estimate(4.80)γ‖rm,δ(Z,γ)u‖L2(R+n)2+‖βm,δ(D′,γ)(u∣x1=0I)‖L2(Rn-1)2≤Cm(γ‖u‖L2(R+n)2+‖u∣x1=0I‖Hγm-1(Rn-1)2)
with positive constant Cm independent, once again, of γ and δ.In the end, combining (4.79), (4.80) and using the a priori estimate (4.1) of order m-1 on u, which holds true by the inductive assumption, we conclude that there exists a constant Cm,s,r′′>0 such that(4.81)γ‖λδm-1,γ(Z)u‖L2(R+n)2+‖λδm-1,γ(D′)(u∣x1=0I)‖L2(Rn-1)2≤Cm,s,r′′(1γ2s+1‖F‖Htan,γs+m,r+m(R+n)2+1γ2s‖G‖Hγs+m(Rn-1)2)
for all γ≥γm and δ∈]0,1].The energy estimate (1.8) of order m hence follows by letting δ→0 into the left-hand side of (4.81) (for an arbitrarily fixed γ≥γm) and exploiting the results of Propositions 3.1 and 3.2.


## 4.1. The Strategy of the Proof: Comparison with the Strongly Well-Posed Case

As it was already pointed out in the Introduction, in order to solve the BVP (1.2)-(1.3) in L2, Theorem 1.1 requires an additional tangential/conormal regularity of the corresponding data. The precise increase of regularity needed for the data is prescribed by the energy inequality (1.7): to estimate the L2-norm of the solution, in the interior and on the boundary of the domain, we lose r conormal derivatives and s-r tangential derivatives with respect to the interior source term F, and s (tangential) derivatives with respect to the boundary datum G.In [2], the conormal regularity of weak solutions to the BVP (1.2)-(1.3) was studied, under the assumption that no loss of derivatives occurs in the estimate of the solution by the data. To prove the result of [2, Theorem 15], the solution u to (1.2)-(1.3) was regularized by a family of tangential mollifiers Jɛ, 0<ɛ<1, defined by Nishitani and Takayama in [9] as a suitable combination of the operator ♯ and the standard Friedrichs'mollifiers. The essential point of the analysis performed in [2] was to notice that the mollified solution Jɛu solves the same problem (1.2)-(1.3), as the original solution u. The data of the problem for Jɛu come from the regularization, by Jɛ, of the data involved in the original problem for u; furthermore, an additional term [Jɛ,L]u, where [Jɛ,L] is the commutator between the differential operator L and the tangential mollifier Jɛ, appears into the equation satisfied by Jɛu. Because the energy estimate associated to a strong L2-well-posed problem does not lose derivatives, actually this term can be incorporated into the source term of the equation satisfied by Jɛu.In the case of Theorem1.1, where the a priori estimate (1.7) exhibits a finite loss of regularity with respect to the data, this technique seems to be unsuccessful, since [Jɛ,L]u cannot be absorbed into the right-hand side without losing derivatives on the solution u; on the other hand, it seems that the same term cannot be merely reduced to a lower-order term involving the smoothed solution Jɛu, as well (this should require a sharp control of the error term u-Jɛu).These observations lead to develop another technique, where the tangential mollifierJɛ is replaced by the family of operators (3.32), involved in the characterization of regularity given by Proposition 3.2. Instead of studying the problem satisfied by the smoothed solution Jɛu, here we consider the problem satisfied by λδm-1,γ(Z)u. As before, a new term [λδm-1,γ(Z),L]u appears which takes account of the commutator between the differential operator L and the conormal operator λδm-1,γ(Z). Since we assume the weak well-posedness of the BVP (1.2)-(1.3) to be preserved under lower order terms, the approach consists of treating the commutator [λδm-1,γ(Z),L]u as a lower-order term within the interior equation for λδm-1,γ(Z)u (see (4.10)) (differently from the strongly L2-well-posed case studied in [2], the stability of problem (1.2)-(1.3) under lower-order perturbations is no longer a trivial consequence of the well-posedness itself. In Theorem 1.1, this stability is required as an additional hypothesis about the BVP); this is made possible by taking advantage from the invertibility of the operator λδm-1,γ(Z).We argue by induction on the integer orderm≥1. Let us take arbitrary data F∈ℋtan,γs+m,r+m(ℝ+n), G∈Hγs+m(ℝn-1), and fix an arbitrary matrix-valued function B∈C(0)∞(ℝ+n) (as the lower order term in the interior equation (1.2)).Because of the inductive hypothesis, we already know that the uniqueL2-solution u to (1.2)-(1.3) actually belongs to Htan,γm-1(ℝ+n) and u∣x1=0I belongs to Hγm-1(ℝn-1), provided that γ is taken to be larger than a certain constant γm-1≥1; moreover the solution u obeys the estimate (1.8) of order m-1(4.1)γ‖u‖Htan,γm-1(R+n)2+‖u∣x1=0I‖Hγm-1(Rn-1)2≤Cm-1(1γ2s+1‖F‖Htan,γs+m-1,r+m-1(R+n)2+1γ2s‖G‖Hγs+m-1(Rn-1)2),
where the positive constant Cm-1, as well as γm-1, only depends on the smoothness order m and the L∞-norm of a finite number (depending on m itself) of conormal derivatives of B (cf. (3.30)), besides the coefficients Aj  (1≤j≤n) of L and the integer numbers r and s.In order to increase the conormal regularity of the solutionu by order one, we are going to act on u by the conormal operator λδm-1,γ(Z); then we consider the analogue of the original problem (1.2)-(1.3) satisfied by λδm-1,γ(Z)u.


## 4.2. A Modified Version of the Conormal Operatorλδm-1,γ(Z)

Due to some technical reasons that will be clarified in Section4.3, we need to slightly modify the conormal operator λδm-1,γ(Z) to be applied to the solution u of the original BVP (1.2)-(1.3), as was described in the preceding section.The first step is to decompose the weight functionλδm-1,γ as the sum of two contributions. To do so, we proceed as follows. Firstly, we take an arbitrary positive, even function χ∈C∞(ℝn) with the following properties:(4.2)0≤χ(x)≤1,∀x∈Rn,χ(x)≡1,for|x|≤12,χ(x)≡0,for|x|>1.
Then, we set(4.3)λχ,δm-1,γ(ξ)≔χ(D)(λδm-1,γ)(ξ)=(F-1χ*λδm-1,γ)(ξ),rm,δ(ξ,γ)≔λδm-1,γ(ξ)-λχ,δm-1,γ(ξ)=(I-χ(D))(λδm-1,γ)(ξ).
The following result (the proof of which is postponed to Appendix A) shows that the function λχ,δm-1,γ essentially behaves like λδm-1,γ.Lemma 4.1.
Let the functionχ∈C∞(ℝn) satisfy the assumptions in (4.2). Then λχ,δm-1,γ is a symbol in Γm-1; moreover for every multi-index α∈ℕn, there exists a positive constant Cm,α, independent of γ and δ, such that
(4.4)|∂ξαλχ,δm-1,γ(ξ)|≤Cm,αλδm-1-|α|,γ(ξ),∀ξ∈Rn.An immediate consequence of Lemma4.1 and (4.3) is that rm,δ is also a γ-depending symbol in Γm-1.Let us define with the obvious meaning of the notations(4.5)λχ,δm-1,γ(D)≔Opγ(λχ,δm-1,γ),rm,δ(D,γ)≔Opγ(rm,δ),λχ,δm-1,γ(Z)≔Op♯γ(λχ,δm-1,γ),rm,δ(Z,γ)≔Op♯γ(rm,δ).
The second important result is concerned with the conormal operator rm,δ(Z,γ)=Op♯γ(rm,δ) and tells that it essentially behaves as a regularizing operator on conormal Sobolev spaces.Lemma 4.2.
For everyk∈ℕ, the conormal operator rm,δ(Z,γ) extends as a linear-bounded operator, still denoted by rm,δ(Z,γ), from L2(ℝ+n) to Htan,γk(ℝ+n). Moreover there exists a positive constant Cm,k, depending only on k and m, such that for all γ≥1 and δ∈]0,1](4.6)‖rm,δ(Z,γ)u‖Htan,γk(R+n)≤Cm,kγk‖u‖L2(R+n),∀u∈L2(R+n).Remark 4.3.
In the framework of the general theory of pseudodifferential operators, the procedure adopted to define the symbolλχ,δm-1,γ is standard and is used to modify an arbitrary symbol in such a way to make properly supported the corresponding pseudodifferential operator (see [13] for the definition of a properly supported operator and an extensive description of the method). As a general issue, one can prove that the resulting properly supported operator differs from the original one by a regularizing remainder. Essentially, an easy adaptation of the same arguments to the framework of conormal spaces in ℝ+n can be employed to prove the regularizing action of the conormal operator rm,δ(Z,γ) stated by Lemma 4.2.According to (4.3), we decompose(4.7)λδm-1,γ(Z)=λχ,δm-1,γ(Z)+rm,δ(Z,γ).
As a consequence of Lemmas 4.1 and 4.2, the role of the family of conormal operators {λδm-1,γ(Z)}0<δ≤1 in the characterization of the conormal regularity provided by Proposition 3.2 (cf. also (3.33)) can also be played by the family of “modified” operators {λχ,δm-1,γ(Z)}0<δ≤1, namely, we have the following.Corollary 4.4.
For every positive integerm and γ≥1, u∈Htan,γm(ℝ+n) if and only if u∈Htan,γm-1(ℝ+n) and the set {∥λχ,δm-1,γ(Z)u∥L2(ℝ+n)}0<δ≤1 is bounded.In order to suitably handle the commutator between the differential operatorL and the conormal operator λχ,δm-1,γ(Z), that comes from writing down the problem satisfied by λχ,δm-1,γ(Z)u (see Sections 4.3.1 and 4.3.2), it is useful to analyze the behavior of the pseudodifferential operators λχ,δm-1,γ(D), when interacting with another pseudodifferential operator by composition and commutation. The following lemma analyzes these situations; for later use, it is convenient to replace in our reasoning the function λχ,δm-1,γ by a general γ-depending symbol aδ preserving the same kind of decay properties as in (4.4).Lemma 4.5.
Let{aδ}0<δ≤1 be a family of symbols aδ=aδ(x,ξ,γ)∈Γr-1, r∈ℝ, such that for all multi-indices α,β∈ℕn there exists a positive constant Cr,α,β, independent of γ and δ, for which
(4.8)|∂ξα∂xβaδ(x,ξ,γ)|≤Cr,α,βλδr-1-|α|,γ(ξ),∀x,ξ∈Rn.
Let b=b(x,ξ,γ) be another symbol in Γl, for l∈ℝ.
Then, for everyδ∈]0,1], the product Opγ(aδ)Opγ(b) is a pseudodifferential operator with symbol aδ#b in Γl+r-1. Moreover, for all multi-indices α,β∈ℕn there exists a constant Cr,l,α,β, independent of γ and δ, such that
(4.9)|∂ξα∂xβ(aδ#b)(x,ξ,γ)|≤Cr,l,α,βλδl+r-1-|α|,γ(ξ),∀x,ξ∈Rn.
Under the same hypotheses, Opγ(aδ)Opγ(b)λ̃δ-m+1,γ(D) is a pseudodifferential operator with symbol (aδ#b)λ̃δ-m+1,γ in Γl+r-m; moreover, {(aδ#b)λ̃δ-m+1,γ}0<δ≤1 is a bounded subset of Γl+r-m. Eventually, if the symbol aδ is scalar-valued, [Opγ(aδ),Opγ(b)]λ̃δ-m+1,γ(D) is a pseudodifferential operator with symbol cδ∈Γl+r-m-1, and {cδ}0<δ≤1 is a bounded subset of Γl+r-m-1.The proof of Lemma4.5 is postponed to Appendix A.Remark 4.6.
ThatOpγ(aδ)Opγ(b), Opγ(aδ)Opγ(b)λ̃δ-m+1,γ(D) and [Opγ(aδ),Opγ(b)]λ̃δ-m+1,γ(D) have symbols belonging, respectively, to Γl+r-1, Γl+r-m and Γl+r-m-1 (for scalar-valued aδ) follows at once from the standard rules of symbolic calculus summarized in Proposition 3.6.  The nontrivial part of the statement above (although deduced from the asymptotic formula (3.23) with a minor effort) is the one asserting that the symbol of Opγ(aδ)Opγ(b) enjoys estimates (4.9); indeed, these estimates give the precise dependence on δ of the decay at infinity of this symbol. Then the remaining assertions in Lemma 4.5 easily follow from (4.9) itself.Remark 4.7.
In view of Proposition3.8, the results on symbols collected in Lemma 4.5 can be used to study the conormal Sobolev continuity of the related conormal operators.
To be definite, for every nonnegative integer numbers, such that s+l+r-m is also nonnegative, Proposition 3.8 and Lemma 4.5 imply that the conormal operator Op♯γ(aδ)Op♯γ(b)λ̃δ-m+1,γ(Z) extends as a linear-bounded mapping from Htan,γs+l+r-m(ℝ+n) into Htan,γs(ℝ+n); moreover, its operator norm is uniformly bounded with respect to γ and δ.
If in additions+l+r-m≥1 and aδ are scalar-valued, then [Op♯γ(aδ),Op♯γ(b)]λ̃δ-m+1,γ(Z) extends as a linear-bounded operator from Htan,γs+l+r-m-1(ℝ+n) into Htan,γs(ℝ+n), and again its operator norm is uniformly bounded with respect to γ and δ.
These mapping properties will be usefully applied in Sections4.3 and 4.5.


## 4.3. The Interior Equation

We follow the strategy already explained in Section4.1, where now the role of the operator λδm-1,γ(Z) is replaced by λχ,δm-1,γ(Z). Since λχ,δm-1,γ∈Γm-1 (because of Lemma 4.1) and, for γ≥γm-1, u∈Htan,γm-1(ℝ+n) (from the inductive hypothesis), after Proposition 3.8 we know that λχ,δm-1,γ(Z)u∈L2(ℝ+n).Applyingλχ,δm-1,γ(Z) to (1.2) (where ℬ is just the multiplication by B), we find that λχ,δm-1,γ(Z)u must solve(4.10)(γ+L+B)(λχ,δm-1,γ(Z)u)+[λχ,δm-1,γ(Z),L+B]u=λχ,δm-1,γ(Z)F,inR+n.
We are going now to show that the commutator term [λχ,δm-1,γ(Z),L+B]u in the above equation can be actually considered as a lower-order term with respect to λχ,δm-1,γ(Z)u.To this end, we may decompose this term as the sum of two contributions corresponding, respectively, to thetangential and normal components of L.First, in view of (1.5), (1.6), we may write the coefficient A1 of the normal derivative ∂1 in the expression (1.4) of L as (4.11)A1=A11+A12,A11≔(A1I,I000),A1∣x1=02=0,
hence (4.12)A12∂1=H1Z1,
where H1(x)=x1-1A12(x)∈C(0)∞(ℝ+n). Accordingly, we split L as (4.13)L=A11∂1+Ltan,Ltan≔H1Z1+∑j=2nAjZj.
Consequently, we have(4.14)[λχ,δm-1,γ(Z),L+B]u=[λχ,δm-1,γ(Z),A11∂1]u+[λχ,δm-1,γ(Z),Ltan+B]u.
Note that Ltan+B is just a conormal operator of order 1, according to the terminology introduced in Section 3.2.


### 4.3.1. The Tangential Commutator

Firstly, we study thetangential commutator[λχ,δm-1,γ(Z),Ltan+B]u. Using the identity λ̃δ-m+1,γ(Z)λδm-1,γ(Z)=I and (4.7), the latter can be written in terms of λχ,δγ,m-1(Z)u, modulo some smoothing reminder. Indeed we compute(4.15)[λχ,δm-1,γ(Z),Ltan+B]u=[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u+rm,δ(Z,γ)u)=[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u)+sm,δ(x,Z,γ)u,
where we have set for short(4.16)sm,δ(x,Z,γ)≔[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)rm,δ(Z,γ).


### 4.3.2. The Normal Commutator

First of all, we notice that, due to the structure of the matrixA11, the commutator [λχ,δγ,m-1(Z),A11∂1] acts nontrivially only on the noncharacteristic component of the vector function u; namely, we have: (4.17)[λχ,δm-1,γ(Z),A11∂1]u=([λχ,δm-1,γ(Z),A1I,I∂1]uI0).
Therefore, we focus on the study of the first nontrivial component of the commutator term. Note that the commutator [λχ,δm-1,γ(Z),A1I,I∂1] cannot be merely treated by the tools of the conormal calculus developed in Section 3.2, because of the presence of the effective normal derivative ∂1 (recall that A1I,I is invertible). This section is devoted to the study of the normal commutator [λχ,δm-1,γ(Z),A1I,I∂1]uI. The following result is of fundamental importance.Proposition 4.8.
For allδ∈]0,1], γ≥1 and m∈ℕ, there exists a symbol qm,δ(x,ξ,γ)∈Γm-2 such that
(4.18)[λχ,δm-1,γ(Z),A1I,I∂1]w=qm,δ(x,Z,γ)(∂1w),∀w∈C(0)∞(R+n).
Moreover, the symbol qm,δ(x,ξ,γ) obeys the following estimates. For all α,β∈ℕn, there exists a positive constant Cm,α,β, independent of γ and δ, such that
(4.19)|∂ξα∂xβqm,δ(x,ξ,γ)|≤Cm,α,βλδm-2-|α|,γ(ξ),∀x,ξ∈Rn.Proof.
Thatqm,δ(x,ξ,γ), satisfying estimates (4.19), is a symbol in Γm-2 actually follows arguing from (4.19) and inequalities (3.18) as was already done for λδm-1,γ(ξ) and λ̃δ-m+1,γ(ξ) (see Section 3.2).
For givenw∈C(0)∞(ℝ+n), let us explicitly compute ([λχ,δm-1,γ(Z),A1I,I∂1]w)♯; using the identity (∂1w)♯=e-x1(Z1w)♯ and that λχ,δm-1,γ(Z) and Z1 commute, we find for every x∈ℝn(4.20)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1(Z1λχ,δm-1,γ(Z)w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1(λχ,δm-1,γ(Z)Z1w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1λχ,δm-1,γ(D)(Z1w)♯(x).
Observing that λχ,δm-1,γ(D) acts on the space 𝒮(ℝn) as the convolution by the inverse Fourier transform of λχ,δm-1,γ, the preceding expression can be equivalently restated as follows:
(4.21)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=F-1(λχ,δm-1,γ)*A1I,I,♮e-(⋅)1(Z1w)♯(x)-A1I,I,♮(x)e-x1F-1(λχ,δm-1,γ)*(Z1w)♯=〈F-1(λχ,δm-1,γ),A1I,I,♮(x-⋅)e-(x1-(⋅)1)(Z1w)♯(x-⋅)〉-A1I,I,♮(x)e-x1〈F-1(λχ,δm-1,γ),(Z1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)A1I,I,♮(x-⋅)e-(x1-(⋅)1)(Z1w)♯(x-⋅)〉-〈ηδm-1,γ,χ(⋅)A1I,I,♮(x)e-x1(Z1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)A1I,I,♮(x-⋅)(∂1w)♯(x-⋅)〉-〈ηδm-1,γ,χ(⋅)A1I,I,♮(x)e-(⋅)1(∂1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)[A1I,I,♮(x-⋅)-A1I,I,♮(x)e-(⋅)1](∂1w)♯(x-⋅)〉,
where ηδm-1,γ≔ℱ-1(λδm-1,γ) and the identity ℱ-1(λχ,δm-1,γ)=χηδm-1,γ (following at once from (4.3)) has been used. Just for brevity, let us further set
(4.22)K(x,y)≔[A1I,I,♮(x-y)-A1I,I,♮(x)e-y1]χ(y).
Thus the identity above reads as
(4.23)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=〈ηδm-1,γ,K(x,⋅)(∂1w)♯(x-⋅)〉,
where the “kernel” 𝒦(x,y) is a bounded function in C∞(ℝn×ℝn), with bounded derivatives of all orders. This regularity of 𝒦 is due to the presence of the function χ in formula (4.22); actually the vanishing of χ at infinity prevents the blow-up of the exponential factor e-y1, as y1→-∞. We point out that this is just the step of our analysis of the normal commutator, where this function χ is needed.
After (4.22), we also have that 𝒦(x,0)=0; then, by a Taylor expansion with respect to y, we can represent the kernel 𝒦(x,y) as follows:
(4.24)K(x,y)=∑k=1nbk(x,y)yk,
where bk(x,y) are given bounded functions in C∞(ℝn×ℝn), with bounded derivatives; it comes from (4.22) and (4.2) that bk can be defined in such a way that for some ɛ>1 and all x∈ℝn there holds
(4.25)supp⁡bk(x,⋅)⊆{|y|≤ɛ}.
Inserting (4.24) in (4.23) and using standard properties of the Fourier transform we get
(4.26)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=〈ηδm-1,γ,∑k=1nbk(x,⋅)(⋅)k(∂1w)♯(x-⋅)〉=∑k=1n〈(⋅)kF-1(λδm-1,γ),bk(x,⋅)(∂1w)♯(x-⋅)〉=-∑k=1n〈F-1(Dkλδm-1,γ),bk(x,⋅)(∂1w)♯(x-⋅)〉=-∑k=1n〈Dkλδm-1,γ,F-1(bk(x,⋅)(∂1w)♯(x-⋅))〉=-∑k=1n∫RnDkλδm-1,γ(ξ)F-1(bk(x,⋅)(∂1w)♯(x-⋅))(ξ)dξ=-∑k=1n(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)(∂1w)♯(x-y)dy)dξ,
where we have set Dk≔-i∂ξk (for each k=1,…,n);   note that for w∈C(0)∞(ℝ+n) and any x∈ℝn, the function bk(x,·)(∂1w)♯(x-·) belongs to 𝒮(ℝn); hence the last expression in (4.26) makes sense. Henceforth, we replace (∂1w)♯ by any function v∈𝒮(ℝn). Our next goal is writing the integral operator
(4.27)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ
as a pseudodifferential operator.
Firstly, we make use of the inversion formula for the Fourier transformation and Fubini's theorem to recast (4.27) as follows:
(4.28)∫Rneiξ⋅ybk(x,y)v(x-y)dy=(2π)-n∫Rneiξ⋅ybk(x,y)(∫Rnei(x-y)⋅ηv̂(η)dη)dy=(2π)-n∫Rneix⋅η(∫Rne-iy⋅(η-ξ)bk(x,y)dy)v̂(η)dη=(2π)-n∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη;
for every index k, b̂k(x,ζ) denotes the partial Fourier transform of bk(x,y) with respect to y. Then, inserting (4.28) into (4.27), we obtain
(4.29)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ=(2π)-2n∫RnDkλδm-1,γ(ξ)(∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη)dξ.
Recall that, for each x∈ℝn, the function y↦bk(x,y) belongs to C0∞(ℝn) (and its compact support does not depend on x, see (4.25)); thus, for each x∈ℝn, b̂k(x,ζ) is rapidly decreasing in ζ. Because of the estimates for derivatives of λδm-1,γ and since v̂(η) is also rapidly decreasing, Fubini's theorem can be used to change the order of the integrations within (4.29). So we get
(4.30)(2π)-2n∫RnDkλδm-1,γ(ξ)(∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη)dξ=(2π)-2n∫Rneix⋅η(∫Rnb̂k(x,η-ξ)Dkλδm-1,γ(ξ)dξ)v̂(η)dη=(2π)-n∫Rneix⋅ηqk,m,δ(x,η,γ)v̂(η)dη,
where we have set
(4.31)qk,m,δ(x,ξ,γ)≔(2π)-n∫Rnb̂k(x,η)Dkλδm-1,γ(ξ-η)dη.
Notice that formula (4.31) defines qk,m,δ as the convolution of the functions b̂k(x,·) and Dkλδm-1,γ; hence qk,m,δ is a well-defined C∞-function in ℝn×ℝn.
The proof of Proposition4.8 will be accomplished, once the following lemma is proved.
Lemma 4.9.
For everym∈ℕ, k=1,…,n and all α,β∈Nn, there exists a positive constant Ck,m,α,β, independent of γ and δ, such that
(4.32)|∂ξα∂xβqk,m,δ(x,ξ,γ)|≤Ck,m,α,βλδm-2-|α|,γ(ξ),∀x,ξ∈Rn.
It comes from Lemma4.9 and the left inequality in (3.18) that, for each index k, the function qk,m,δ is a symbol in Γm-2; notice however that the set {qk,m,δ}0<δ≤1 is bounded in Γm-1 but not in Γm-2. The proof of Lemma 4.9 is postponed to Appendix A.
Now, we continue the proof of Proposition4.8.
End of the Proof of Proposition4.8
The last row of (4.30) provides the desired representation of (4.27) as a pseudodifferential operator; actually it gives the identity
(4.33)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ=Opγ(qk,m,δ)v(x),
for every v∈𝒮(ℝn).
Inserting the above formula (withv=(∂1w)♯) into (4.26) finally gives
(4.34)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=Opγ(qm,δ)(∂1w)♯(x),
where qm,δ=qm,δ(x,ξ,γ) is the symbol in Γm-2 defined by
(4.35)qm,δ(x,ξ,γ)≔-∑k=1nqk,m,δ(x,ξ,γ).
Of course, formula (4.18) is equivalent to (4.34), in view of (3.26). Estimates (4.19) are satisfied by qm,δ by summation over k of the similar estimates satisfied by qm,δ,k (cf. Lemma 4.9).
This ends the proof of Proposition4.8.Now, we are going to show how the representation in (4.18) can be exploited to treat the normal commutator as a lower-order term in (4.10) satified by λχ,δm-1,γ(Z)u.Firstly, we notice that formula (4.18) has been deduced for smooth functions w, while u is just an L2-function (actually it belongs to Htanm-1(ℝ+n), by the inductive hypothesis). In order to use (4.18) for u, we need to approximate the latter by smooth functions. This can be done by the help of [14, Proposition 1, Theorem 1]; from there, we know that there exists a sequence {uν}ν in C(0)∞(ℝ+n) such that(4.36)uν⟶u,inL2(R+n),(L+B)uν⟶(L+B)u,inL2(R+n),uν∣x1=0I⟶u∣x1=0I,inH-1/2(Rn-1),asν⟶+∞.
For each index ν, the regular function uν∈C(0)∞(ℝ+n) solves the same BVP as the function u, with new data Fν, Gν defined by(4.37)Fν≔(γ+L+B)uν,Gν≔Muν∣x1=0.
It immediately follows from (4.36) that the regular data Fν,Gν approximate the original data F,G by(4.38)Fν⟶F,inL2(R+n),Gν⟶G,inH-1/2(Rn-1),asν⟶+∞.
The same analysis performed to the BVP (1.2)-(1.3) can be applied to the BVP solved by uν, for each ν; in particular, we find that λχ,δm-1,γ(Z)uν satisfies the analogue to (4.10), where F is replaced by Fν. Because of the regularity of uν, formula (4.18) can be used to represent the normal commutator term [λχ,δm-1,γ(Z),A1I,I∂1]uνI. Directly from system (γ+L+B)uν=Fν, ∂1uνI can be represented in terms of tangential derivatives of uν only, as follows:(4.39)∂1uνI=(A1I,I)-1FνI+LIuν,
where ℒI=ℒI(x,Z,γ) denotes the tangential partial differential operator(4.40)LIuν≔-(A1I,I)-1[γuνI+H1Z1uνII+(∑j=2nAjZjuν+Buν)I],
and we have set H1≔x1-1A1I,II (recall that H1∈C(0)∞(ℝ+n) since A1∣x1=0I,II=0). Inserting (4.39) into (4.18), written for w=uνI, leads to(4.41)[λχ,δm-1,γ(Z),A1I,I∂1]uνI=qm,δ(x,Z,γ)((A1I,I)-1FνI+LIuν).
On the other hand, plugging λ̃δ-m+1,γ(Z)λχ,δm-1,γ(Z)+λ̃δ-m+1,γ(Z)rm,δ(Z,γ)=I into (4.41) gives(4.42)[λχ,δm-1,γ(Z),A1I,I∂1]uνI=qm,δ(x,Z,γ)((A1I,I)-1FνI)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)uν)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)uν.
Now, we let ν→+∞.From (1.2) (written for u and uν), and using that uν♯→u♯ in L2(ℝn), one finds (4.43)(A1∂1uν)♯⟶(A1∂1u)♯,inS′(Rn),
hence (because ♯-1 is a linear continuous operator from 𝒮′(ℝn)⊂𝒟′(ℝn) to 𝒟′(ℝ+n))(4.44)λχ,δm-1,γ(Z)(A1I,I∂1uνI)⟶λχ,δm-1,γ(Z)(A1I,I∂1uI),inD′(R+n).
On the other hand, uν♯→u♯ in L2(ℝn) implies that(4.45)λχ,δm-1,γ(Z)(uν)⟶λχ,δm-1,γ(Z)(u),inD′(R+n),
hence(4.46)A1I,I∂1(λχ,δm-1,γ(Z)(uνI))⟶A1I,I∂1(λχ,δm-1,γ(Z)(uI)),inD′(R+n).
Adding (4.44), (4.46) then gives(4.47)[λχ,δm-1,γ(Z),A1I,I∂1]uνI⟶[λχ,δm-1,γ(Z),A1I,I∂1]uI,inD′(R+n).
As to the right-hand side of (4.42), all the involved operators, acting on Fν and uν, are conormal. Hence the L2-convergences uν→u and Fν→F and the fact that conormal operators continuously extend to the space of distributions u∈𝒟′(ℝ+n), for which u♯∈𝒮′(ℝn), give the convergences(4.48)qm,δ(x,Z,γ)((A1I,I)-1FνI)⟶qm,δ(x,Z,γ)((A1I,I)-1FI),qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)uν)⟶qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u),qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)uν⟶qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)u,inD′(R+n).
Therefore, the uniqueness of the limit in 𝒟′(ℝ+n) together with (4.47), (4.48) implies that (4.42) holds true for the L2-solution u of (1.2)-(1.3), that is,(4.49)[λχ,δm-1,γ(Z),A1I,I∂1]uI=qm,δ(x,Z,γ)((A1I,I)-1FI)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)u.
Let us come back to the commutator term [λχ,δm-1,γ(Z),L+B]u appearing in the interior equation (4.10).Substituting (4.15), (4.49) into (4.14) gives for this term the following representation:(4.50)[λχ,δm-1,γ(Z),L+B]u=ρm,δ(x,Z,γ)(λχ,δm-1,γ(Z)u)+τm,δ(x,Z,γ)u+ηm,δ(x,Z,γ)F,
where we have set for short(4.51)ρm,δ(x,Z,γ)≔[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)+(qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)0),τm,δ(x,Z,γ)≔sm,δ(x,Z,γ)+(qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)0),ηm,δ(x,Z,γ)≔(qm,δ(x,Z,γ)(A1I,I)-1000).
Consequently, the interior equation (4.10) can be restated as(4.52)(γ+L+B+ρm,δ(x,Z,γ))(λχ,δm-1,γ(Z)u)+τm,δ(x,Z,γ)u+ηm,δ(x,Z,γ)F=λχ,δm-1,γ(Z)F,inR+n.
Since Ltan+B and ℒI are conormal operators with symbol in Γ1, Lemma 4.5 and Proposition 4.8 imply that ρm,δ(x,Z,γ) is a conormal operator with symbol in Γ0, for each 0<δ≤1; moreover, it amounts that the family of symbols {ρm,δ}0<δ≤1 is a bounded subset of Γ0. Therefore, ρm,δ(x,Z,γ) can be regarded, jointly with B, as a lower-order term in (4.52), as considered in the statement of Theorem 1.1.Concerning the termsτm,δ(x,Z,γ)u, ηm,δ(x,Z,γ)F, they can be both moved into the right-hand side of (4.52), to be treated as a part of the interior source term, as will be detailed in Section 4.5.


## 4.3.1. The Tangential Commutator

Firstly, we study thetangential commutator[λχ,δm-1,γ(Z),Ltan+B]u. Using the identity λ̃δ-m+1,γ(Z)λδm-1,γ(Z)=I and (4.7), the latter can be written in terms of λχ,δγ,m-1(Z)u, modulo some smoothing reminder. Indeed we compute(4.15)[λχ,δm-1,γ(Z),Ltan+B]u=[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u+rm,δ(Z,γ)u)=[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u)+sm,δ(x,Z,γ)u,
where we have set for short(4.16)sm,δ(x,Z,γ)≔[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)rm,δ(Z,γ).


## 4.3.2. The Normal Commutator

First of all, we notice that, due to the structure of the matrixA11, the commutator [λχ,δγ,m-1(Z),A11∂1] acts nontrivially only on the noncharacteristic component of the vector function u; namely, we have: (4.17)[λχ,δm-1,γ(Z),A11∂1]u=([λχ,δm-1,γ(Z),A1I,I∂1]uI0).
Therefore, we focus on the study of the first nontrivial component of the commutator term. Note that the commutator [λχ,δm-1,γ(Z),A1I,I∂1] cannot be merely treated by the tools of the conormal calculus developed in Section 3.2, because of the presence of the effective normal derivative ∂1 (recall that A1I,I is invertible). This section is devoted to the study of the normal commutator [λχ,δm-1,γ(Z),A1I,I∂1]uI. The following result is of fundamental importance.Proposition 4.8.
For allδ∈]0,1], γ≥1 and m∈ℕ, there exists a symbol qm,δ(x,ξ,γ)∈Γm-2 such that
(4.18)[λχ,δm-1,γ(Z),A1I,I∂1]w=qm,δ(x,Z,γ)(∂1w),∀w∈C(0)∞(R+n).
Moreover, the symbol qm,δ(x,ξ,γ) obeys the following estimates. For all α,β∈ℕn, there exists a positive constant Cm,α,β, independent of γ and δ, such that
(4.19)|∂ξα∂xβqm,δ(x,ξ,γ)|≤Cm,α,βλδm-2-|α|,γ(ξ),∀x,ξ∈Rn.Proof.
Thatqm,δ(x,ξ,γ), satisfying estimates (4.19), is a symbol in Γm-2 actually follows arguing from (4.19) and inequalities (3.18) as was already done for λδm-1,γ(ξ) and λ̃δ-m+1,γ(ξ) (see Section 3.2).
For givenw∈C(0)∞(ℝ+n), let us explicitly compute ([λχ,δm-1,γ(Z),A1I,I∂1]w)♯; using the identity (∂1w)♯=e-x1(Z1w)♯ and that λχ,δm-1,γ(Z) and Z1 commute, we find for every x∈ℝn(4.20)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1(Z1λχ,δm-1,γ(Z)w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1(λχ,δm-1,γ(Z)Z1w)♯(x)=λχ,δm-1,γ(D)(A1I,I,♮e-(⋅)1(Z1w)♯)(x)-A1I,I,♮(x)e-x1λχ,δm-1,γ(D)(Z1w)♯(x).
Observing that λχ,δm-1,γ(D) acts on the space 𝒮(ℝn) as the convolution by the inverse Fourier transform of λχ,δm-1,γ, the preceding expression can be equivalently restated as follows:
(4.21)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=F-1(λχ,δm-1,γ)*A1I,I,♮e-(⋅)1(Z1w)♯(x)-A1I,I,♮(x)e-x1F-1(λχ,δm-1,γ)*(Z1w)♯=〈F-1(λχ,δm-1,γ),A1I,I,♮(x-⋅)e-(x1-(⋅)1)(Z1w)♯(x-⋅)〉-A1I,I,♮(x)e-x1〈F-1(λχ,δm-1,γ),(Z1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)A1I,I,♮(x-⋅)e-(x1-(⋅)1)(Z1w)♯(x-⋅)〉-〈ηδm-1,γ,χ(⋅)A1I,I,♮(x)e-x1(Z1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)A1I,I,♮(x-⋅)(∂1w)♯(x-⋅)〉-〈ηδm-1,γ,χ(⋅)A1I,I,♮(x)e-(⋅)1(∂1w)♯(x-⋅)〉=〈ηδm-1,γ,χ(⋅)[A1I,I,♮(x-⋅)-A1I,I,♮(x)e-(⋅)1](∂1w)♯(x-⋅)〉,
where ηδm-1,γ≔ℱ-1(λδm-1,γ) and the identity ℱ-1(λχ,δm-1,γ)=χηδm-1,γ (following at once from (4.3)) has been used. Just for brevity, let us further set
(4.22)K(x,y)≔[A1I,I,♮(x-y)-A1I,I,♮(x)e-y1]χ(y).
Thus the identity above reads as
(4.23)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=〈ηδm-1,γ,K(x,⋅)(∂1w)♯(x-⋅)〉,
where the “kernel” 𝒦(x,y) is a bounded function in C∞(ℝn×ℝn), with bounded derivatives of all orders. This regularity of 𝒦 is due to the presence of the function χ in formula (4.22); actually the vanishing of χ at infinity prevents the blow-up of the exponential factor e-y1, as y1→-∞. We point out that this is just the step of our analysis of the normal commutator, where this function χ is needed.
After (4.22), we also have that 𝒦(x,0)=0; then, by a Taylor expansion with respect to y, we can represent the kernel 𝒦(x,y) as follows:
(4.24)K(x,y)=∑k=1nbk(x,y)yk,
where bk(x,y) are given bounded functions in C∞(ℝn×ℝn), with bounded derivatives; it comes from (4.22) and (4.2) that bk can be defined in such a way that for some ɛ>1 and all x∈ℝn there holds
(4.25)supp⁡bk(x,⋅)⊆{|y|≤ɛ}.
Inserting (4.24) in (4.23) and using standard properties of the Fourier transform we get
(4.26)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=〈ηδm-1,γ,∑k=1nbk(x,⋅)(⋅)k(∂1w)♯(x-⋅)〉=∑k=1n〈(⋅)kF-1(λδm-1,γ),bk(x,⋅)(∂1w)♯(x-⋅)〉=-∑k=1n〈F-1(Dkλδm-1,γ),bk(x,⋅)(∂1w)♯(x-⋅)〉=-∑k=1n〈Dkλδm-1,γ,F-1(bk(x,⋅)(∂1w)♯(x-⋅))〉=-∑k=1n∫RnDkλδm-1,γ(ξ)F-1(bk(x,⋅)(∂1w)♯(x-⋅))(ξ)dξ=-∑k=1n(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)(∂1w)♯(x-y)dy)dξ,
where we have set Dk≔-i∂ξk (for each k=1,…,n);   note that for w∈C(0)∞(ℝ+n) and any x∈ℝn, the function bk(x,·)(∂1w)♯(x-·) belongs to 𝒮(ℝn); hence the last expression in (4.26) makes sense. Henceforth, we replace (∂1w)♯ by any function v∈𝒮(ℝn). Our next goal is writing the integral operator
(4.27)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ
as a pseudodifferential operator.
Firstly, we make use of the inversion formula for the Fourier transformation and Fubini's theorem to recast (4.27) as follows:
(4.28)∫Rneiξ⋅ybk(x,y)v(x-y)dy=(2π)-n∫Rneiξ⋅ybk(x,y)(∫Rnei(x-y)⋅ηv̂(η)dη)dy=(2π)-n∫Rneix⋅η(∫Rne-iy⋅(η-ξ)bk(x,y)dy)v̂(η)dη=(2π)-n∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη;
for every index k, b̂k(x,ζ) denotes the partial Fourier transform of bk(x,y) with respect to y. Then, inserting (4.28) into (4.27), we obtain
(4.29)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ=(2π)-2n∫RnDkλδm-1,γ(ξ)(∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη)dξ.
Recall that, for each x∈ℝn, the function y↦bk(x,y) belongs to C0∞(ℝn) (and its compact support does not depend on x, see (4.25)); thus, for each x∈ℝn, b̂k(x,ζ) is rapidly decreasing in ζ. Because of the estimates for derivatives of λδm-1,γ and since v̂(η) is also rapidly decreasing, Fubini's theorem can be used to change the order of the integrations within (4.29). So we get
(4.30)(2π)-2n∫RnDkλδm-1,γ(ξ)(∫Rneix⋅ηb̂k(x,η-ξ)v̂(η)dη)dξ=(2π)-2n∫Rneix⋅η(∫Rnb̂k(x,η-ξ)Dkλδm-1,γ(ξ)dξ)v̂(η)dη=(2π)-n∫Rneix⋅ηqk,m,δ(x,η,γ)v̂(η)dη,
where we have set
(4.31)qk,m,δ(x,ξ,γ)≔(2π)-n∫Rnb̂k(x,η)Dkλδm-1,γ(ξ-η)dη.
Notice that formula (4.31) defines qk,m,δ as the convolution of the functions b̂k(x,·) and Dkλδm-1,γ; hence qk,m,δ is a well-defined C∞-function in ℝn×ℝn.
The proof of Proposition4.8 will be accomplished, once the following lemma is proved.
Lemma 4.9.
For everym∈ℕ, k=1,…,n and all α,β∈Nn, there exists a positive constant Ck,m,α,β, independent of γ and δ, such that
(4.32)|∂ξα∂xβqk,m,δ(x,ξ,γ)|≤Ck,m,α,βλδm-2-|α|,γ(ξ),∀x,ξ∈Rn.
It comes from Lemma4.9 and the left inequality in (3.18) that, for each index k, the function qk,m,δ is a symbol in Γm-2; notice however that the set {qk,m,δ}0<δ≤1 is bounded in Γm-1 but not in Γm-2. The proof of Lemma 4.9 is postponed to Appendix A.
Now, we continue the proof of Proposition4.8.
End of the Proof of Proposition4.8
The last row of (4.30) provides the desired representation of (4.27) as a pseudodifferential operator; actually it gives the identity
(4.33)(2π)-n∫RnDkλδm-1,γ(ξ)(∫Rneiξ⋅ybk(x,y)v(x-y)dy)dξ=Opγ(qk,m,δ)v(x),
for every v∈𝒮(ℝn).
Inserting the above formula (withv=(∂1w)♯) into (4.26) finally gives
(4.34)([λχ,δm-1,γ(Z),A1I,I∂1]w)♯(x)=Opγ(qm,δ)(∂1w)♯(x),
where qm,δ=qm,δ(x,ξ,γ) is the symbol in Γm-2 defined by
(4.35)qm,δ(x,ξ,γ)≔-∑k=1nqk,m,δ(x,ξ,γ).
Of course, formula (4.18) is equivalent to (4.34), in view of (3.26). Estimates (4.19) are satisfied by qm,δ by summation over k of the similar estimates satisfied by qm,δ,k (cf. Lemma 4.9).
This ends the proof of Proposition4.8.Now, we are going to show how the representation in (4.18) can be exploited to treat the normal commutator as a lower-order term in (4.10) satified by λχ,δm-1,γ(Z)u.Firstly, we notice that formula (4.18) has been deduced for smooth functions w, while u is just an L2-function (actually it belongs to Htanm-1(ℝ+n), by the inductive hypothesis). In order to use (4.18) for u, we need to approximate the latter by smooth functions. This can be done by the help of [14, Proposition 1, Theorem 1]; from there, we know that there exists a sequence {uν}ν in C(0)∞(ℝ+n) such that(4.36)uν⟶u,inL2(R+n),(L+B)uν⟶(L+B)u,inL2(R+n),uν∣x1=0I⟶u∣x1=0I,inH-1/2(Rn-1),asν⟶+∞.
For each index ν, the regular function uν∈C(0)∞(ℝ+n) solves the same BVP as the function u, with new data Fν, Gν defined by(4.37)Fν≔(γ+L+B)uν,Gν≔Muν∣x1=0.
It immediately follows from (4.36) that the regular data Fν,Gν approximate the original data F,G by(4.38)Fν⟶F,inL2(R+n),Gν⟶G,inH-1/2(Rn-1),asν⟶+∞.
The same analysis performed to the BVP (1.2)-(1.3) can be applied to the BVP solved by uν, for each ν; in particular, we find that λχ,δm-1,γ(Z)uν satisfies the analogue to (4.10), where F is replaced by Fν. Because of the regularity of uν, formula (4.18) can be used to represent the normal commutator term [λχ,δm-1,γ(Z),A1I,I∂1]uνI. Directly from system (γ+L+B)uν=Fν, ∂1uνI can be represented in terms of tangential derivatives of uν only, as follows:(4.39)∂1uνI=(A1I,I)-1FνI+LIuν,
where ℒI=ℒI(x,Z,γ) denotes the tangential partial differential operator(4.40)LIuν≔-(A1I,I)-1[γuνI+H1Z1uνII+(∑j=2nAjZjuν+Buν)I],
and we have set H1≔x1-1A1I,II (recall that H1∈C(0)∞(ℝ+n) since A1∣x1=0I,II=0). Inserting (4.39) into (4.18), written for w=uνI, leads to(4.41)[λχ,δm-1,γ(Z),A1I,I∂1]uνI=qm,δ(x,Z,γ)((A1I,I)-1FνI+LIuν).
On the other hand, plugging λ̃δ-m+1,γ(Z)λχ,δm-1,γ(Z)+λ̃δ-m+1,γ(Z)rm,δ(Z,γ)=I into (4.41) gives(4.42)[λχ,δm-1,γ(Z),A1I,I∂1]uνI=qm,δ(x,Z,γ)((A1I,I)-1FνI)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)uν)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)uν.
Now, we let ν→+∞.From (1.2) (written for u and uν), and using that uν♯→u♯ in L2(ℝn), one finds (4.43)(A1∂1uν)♯⟶(A1∂1u)♯,inS′(Rn),
hence (because ♯-1 is a linear continuous operator from 𝒮′(ℝn)⊂𝒟′(ℝn) to 𝒟′(ℝ+n))(4.44)λχ,δm-1,γ(Z)(A1I,I∂1uνI)⟶λχ,δm-1,γ(Z)(A1I,I∂1uI),inD′(R+n).
On the other hand, uν♯→u♯ in L2(ℝn) implies that(4.45)λχ,δm-1,γ(Z)(uν)⟶λχ,δm-1,γ(Z)(u),inD′(R+n),
hence(4.46)A1I,I∂1(λχ,δm-1,γ(Z)(uνI))⟶A1I,I∂1(λχ,δm-1,γ(Z)(uI)),inD′(R+n).
Adding (4.44), (4.46) then gives(4.47)[λχ,δm-1,γ(Z),A1I,I∂1]uνI⟶[λχ,δm-1,γ(Z),A1I,I∂1]uI,inD′(R+n).
As to the right-hand side of (4.42), all the involved operators, acting on Fν and uν, are conormal. Hence the L2-convergences uν→u and Fν→F and the fact that conormal operators continuously extend to the space of distributions u∈𝒟′(ℝ+n), for which u♯∈𝒮′(ℝn), give the convergences(4.48)qm,δ(x,Z,γ)((A1I,I)-1FνI)⟶qm,δ(x,Z,γ)((A1I,I)-1FI),qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)uν)⟶qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u),qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)uν⟶qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)u,inD′(R+n).
Therefore, the uniqueness of the limit in 𝒟′(ℝ+n) together with (4.47), (4.48) implies that (4.42) holds true for the L2-solution u of (1.2)-(1.3), that is,(4.49)[λχ,δm-1,γ(Z),A1I,I∂1]uI=qm,δ(x,Z,γ)((A1I,I)-1FI)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)(λχ,δm-1,γ(Z)u)+qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)u.
Let us come back to the commutator term [λχ,δm-1,γ(Z),L+B]u appearing in the interior equation (4.10).Substituting (4.15), (4.49) into (4.14) gives for this term the following representation:(4.50)[λχ,δm-1,γ(Z),L+B]u=ρm,δ(x,Z,γ)(λχ,δm-1,γ(Z)u)+τm,δ(x,Z,γ)u+ηm,δ(x,Z,γ)F,
where we have set for short(4.51)ρm,δ(x,Z,γ)≔[λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z)+(qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)0),τm,δ(x,Z,γ)≔sm,δ(x,Z,γ)+(qm,δ(x,Z,γ)LIλ̃δ-m+1,γ(Z)rm,δ(Z,γ)0),ηm,δ(x,Z,γ)≔(qm,δ(x,Z,γ)(A1I,I)-1000).
Consequently, the interior equation (4.10) can be restated as(4.52)(γ+L+B+ρm,δ(x,Z,γ))(λχ,δm-1,γ(Z)u)+τm,δ(x,Z,γ)u+ηm,δ(x,Z,γ)F=λχ,δm-1,γ(Z)F,inR+n.
Since Ltan+B and ℒI are conormal operators with symbol in Γ1, Lemma 4.5 and Proposition 4.8 imply that ρm,δ(x,Z,γ) is a conormal operator with symbol in Γ0, for each 0<δ≤1; moreover, it amounts that the family of symbols {ρm,δ}0<δ≤1 is a bounded subset of Γ0. Therefore, ρm,δ(x,Z,γ) can be regarded, jointly with B, as a lower-order term in (4.52), as considered in the statement of Theorem 1.1.Concerning the termsτm,δ(x,Z,γ)u, ηm,δ(x,Z,γ)F, they can be both moved into the right-hand side of (4.52), to be treated as a part of the interior source term, as will be detailed in Section 4.5.


## 4.4. The Boundary Condition

Now we are going to seek for an appropriate boundary condition to be coupled with the interior equation (4.10), in order to state a BVP solved by λχ,δm-1,γ(Z)u.To this end, it is worthwhile to make an additional hypothesis about the smooth functionχ involved in the definition of λχ,δm-1,γ(Z). We assume that χ has the form(4.53)∀x=(x1,x′)∈Rn,χ(x)=χ1(x1)χ̃(x′),
where χ1∈C∞(ℝ) and χ̃∈C∞(ℝn-1) are given positive even functions, to be chosen in such a way that conditions (4.2) are made satisfied.As it was done for the analysis of the normal commutator (cf. Proposition4.8), we start our reasoning by dealing with smooth functions. In this case, following closely the arguments employed to prove Proposition 4.8 and Lemma 4.9, we are able to get the following.Proposition 4.10.
Assume thatχ obeys the assumptions (4.2), (4.53). Then, for all δ∈]0,1], γ≥1, and m∈ℕ, the function bm,δ′(ξ′,γ) defined by
(4.54)bm,δ′(ξ′,γ)≔(2π)-n∫Rnλδm-1,γ(η1,η′+ξ′)(e(⋅)1/2χ1)∧1(η1)χ̃̂(η′)dη,∀ξ′∈Rn-1,
(where ξ′≔(ξ2,…,ξn) are the Fourier dual variables of the tangential variables x′=(x2,…,xn)) is a γ-depending symbol in ℝn-1 belonging to Γm-1. Moreover, for all functions w∈C(0)∞(ℝ+n) there holds
(4.55)∀x′∈Rn-1,(λχ,δm-1,γ(Z)w)∣x1=0(x′)=bm,δ′(D′,γ)(w∣x1=0)(x′),
where one has set D′=(D2,…,Dn), Dj=-i∂j (for j=2,…,n) and bm,δ′(D′,γ) stands for the ordinary pseudodifferential operator in ℝn-1 with symbol bm,δ′.
Eventually, the following estimates are satisfied by the symbolbm,δ′(ξ′,γ): for all α′=(α2,…,αn)∈ℕn-1 there exists a positive constant Cm,α′, independent of γ and δ, such that
(4.56)|∂ξ′α′b′m,δ(ξ′,γ)|≤Cm,α′λδm-1-|α′|,γ(ξ′),∀ξ′∈Rn-1.Proof.
Thatbm,δ′ belongs to Γm-1 immediately follows from estimates (4.56), using the (n-1)-dimensional counterpart of (3.18).
Letw∈C(0)∞(ℝ+n); to find a symbol bm,δ′ satisfying (4.55), from (4.3) we firstly compute
(4.57)(λχ,δm-1,γ(Z)w)♯(x)=λχ,δm-1,γ(D)(w♯)(x)=(F-1(λχ,δm-1,γ)*w♯)(x)=〈F-1(λχ,δm-1,γ),w♯(x-⋅)〉=〈F-1(λδm-1,γ),χ(⋅)e(x1-(⋅)1)/2w(ex1-(⋅)1,x′-(⋅)′)〉,∀(x1,x′)∈Rn,
hence
(4.58)λχ,δm-1,γ(Z)w(x)=〈F-1(λδm-1,γ),χ(⋅)e-(⋅)1/2w(x1e-(⋅)1,x′-(⋅)′)〉=〈λδm-1,γ,F-1(χ(⋅)e-(⋅)1/2w(x1e-(⋅)1,x′-(⋅)′))〉=(2π)-n∫λδm-1,γ(ξ)(∫eiξ⋅yχ(y)e-y1/2w(x1e-y1,x′-y′)dy)dξ,∀x1>0,∀x′∈Rn-1.
The regularity of w legitimates all the above calculations. Setting x1=0 in the last expression above, we deduce the corresponding expression for the trace on the boundary of λχ,δm-1,γ(Z)w(4.59)(λχ,δm-1,γ(Z)w)∣x1=0(x′)=(2π)-n∫λδm-1,γ(ξ)(∫eiξ⋅yχ(y)e-y1/2(w∣x1=0)(x′-y′)dy)dξ.
Now we substitute (4.53) into the y-integral appearing in the last expression above; then Fubini's theorem gives
(4.60)∫eiξ⋅yχ1(y1)χ̃(y′)e-y1/2(w∣x1=0)(x′-y′)dy=∫eiξ′⋅y′(∫eiξ1y1e-y1/2χ1(y1)dy1)χ̃(y′)(w∣x1=0)(x′-y′)dy′=(e(⋅)1/2χ1)∧1(ξ1)∫eiξ′⋅y′χ̃(y′)(w∣x1=0)(x′-y′)dy′,
where ∧1 is used to denote the one-dimensional Fourier transformation with respect to y1. Writing, by the inversion formula, (w∣x1=0)(x′-y′)=(2π)-n+1∫ei(x′-y′)·η′w∣x1=0̂(η′)dη′ and using once more Fubini's theorem, we further obtain
(4.61)∫eiξ′⋅y′χ̃(y′)(w∣x1=0)(x′-y′)dy′=(2π)-n+1∫eiξ′⋅y′χ̃(y′)(∫ei(x′-y′)⋅η′w∣x1=0̂(η′)dη′)dy′=∫eix′⋅η′((2π)-n+1∫ei(ξ′-η′)⋅y′χ̃(y′)dy′)w∣x1=0̂(η′)dη′=(2π)-n+1∫eix′⋅η′χ̃̂(ξ′-η′)w∣x1=0̂(η′)dη′;∧ is used here to denote the (n-1)-dimensional Fourier transformation with respect to x′. Inserting (4.60), (4.61) into (4.59) then leads to
(4.62)(λχ,δm-1,γ(Z)w)∣x1=0(x′)=(2π)-n∫λδm-1,γ(ξ)(e(⋅)1/2χ1)∧1(ξ1)((2π)-n+1∫eix′⋅η′χ̃̂(ξ′-η′)w∣x1=0̂(η′)dη′)dξ.
Because (e(·)1/2χ1)∧1∈𝒮(ℝ), χ̃̂∈𝒮(ℝn-1), and w∣x1=0̂∈𝒮(ℝn-1), the double integral
(4.63)∬eix′⋅η′λδm-1,γ(ξ)(e(⋅)1/2χ1)∧1(ξ1)χ̃̂(ξ′-η′)w∣x1=0̂(η′)dη′dξ
converges absolutely; hence Fubini's theorem allows to exchange the order of the integrations in (4.62) and find
(4.64)(λχ,δm-1,γ(Z)w)∣x1=0(x′)=(2π)-n+1∫eix′⋅η′bm,δ′(η′,γ)w∣x1=0̂(η′)dη′,
where bm,δ′(η′,γ) is defined by (4.54). This shows the identity (4.55).
The proof of estimates (4.56) is similar to that of estimates (4.32) in Lemma 4.9 (see Appendix A); so we will omit it.Let us now illustrate how formula (4.55) can be used to derive the desired boundary condition satisfied by λχ,δm-1,γ(Z)u.Again, letu be the L2-solution to the original BVP (1.2)-(1.3) and {uν}ν=1+∞ the corresponding sequence in C(0)∞(ℝ+n), approximating u in the sense of (4.36).The last convergence in (4.36) and the Sobolev continuity of standard pseudodifferential operators gives in particular that(4.65)bm,δ′(D′,γ)(uν∣x1=0I)⟶bm,δ′(D′,γ)(u∣x1=0I),inH-m+1/2(Rn-1).
On the other hand, (4.36) and (4.52) (written for u and uν) can be used to prove that (λχ,δm-1,γ(Z)uI)∣x1=0 and (λχ,δm-1,γ(Z)uνI)∣x1=0, for each ν, are traces well defined in H-1/2(ℝn-1) and the convergence(4.66)(λχ,δm-1,γ(Z)uνI)∣x1=0⟶(λχ,δm-1,γ(Z)uI)∣x1=0
holds true, at least in 𝒟′(ℝn-1). The proof of this assertion is postponed to Appendix A (see Lemma A.2).Sinceuν are smooth functions, from Proposition 4.10, it comes that for each ν:(4.67)(λχ,δm-1,γ(Z)uνI)∣x1=0=bm,δ′(D′,γ)(uν∣x1=0I).
Then, letting ν→+∞, (4.65) and (4.66) yield(4.68)(λχ,δm-1,γ(Z)uI)∣x1=0=bm,δ′(D′,γ)(u∣x1=0I).
Recalling that M=(Id,0), from the boundary condition (1.3) and (4.68), we immediately find(4.69)(M(λχ,δm-1,γ(Z)u))∣x1=0=M(λχ,δm-1,γ(Z)uI)∣x1=0=Mbm,δ′(D′,γ)(u∣x1=0I)=bm,δ′(D′,γ)(Mu∣x1=0)=bm,δ′(D′,γ)G.


## 4.5. Derivation of the Conormal Regularity at the Orderm

We are now in the position to get the desired conormal regularity of the solutionu to (1.2)-(1.3), under the assumptions that F∈ℋtan,γs+m,r+m(ℝ+n), G∈Hγs+m(ℝn-1). From now on, assume that γ≥γm-1; so far, from the inductive hypothesis we know that, for such a γ, u belongs to Htan,γm-1(ℝ+n), u∣x1=0I∈Hγm-1(ℝn-1) and the estimate (4.1) is satisfied (see the end of Section 4.1). Because of the calculations made in Sections 4.3 and 4.4, it follows that λχ,δm-1,γ(Z)u is an L2 solution of the problem(4.70)(γ+L+B+ρm,δ(x,Z,γ))(λχ,δγ,m-1(Z)u)=(λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F-τm,δ(x,Z,γ)u,inR+n,(4.71)M(λχ,δm-1,γ(Z)u)=bm,δ′(D′,γ)G,onRn-1.
The previous one reads as the original BVP (1.2)-(1.3) solved by u, where the role of the lower-order term ℬ is played here by the conormal operator B+ρm,δ(x,Z,γ). As already discussed in the end of Section 4.3, in view of Proposition 3.8 and Lemma 4.5, the symbol of B+ρm,δ(x,Z,γ) belongs to Γ0, and {ρm,δ}0<δ≤1 is a bounded subset of Γ0.As regards to the termsτm,δ(x,Z,γ)u and ηm,δ(x,Z,γ)F appearing into the right-hand side of (4.70), they can be regarded as a part of the source term in the interior equation (4.70) (this is the reason why they have been moved in the right-hand side of (4.70)).Let us firstly focus onτm,δ(x,Z,γ)u. After Lemma 4.5 and Proposition 3.8 (see also Proposition 4.8, Remark 4.7, and formulas (4.16), (4.51)), we know that for any k∈ℕ the operators [λχ,δm-1,γ(Z),Ltan+B]λ̃δ-m+1,γ(Z) and qm,δ(x,Z,γ)ℒIλ̃δ-m+1,γ(Z) extend as linear-bounded mappings from Htan,γk(ℝ+n) into itself, and their operator norms are uniformly bounded with respect to γ and δ. Combining with the result of Lemma 4.2, it follows that a positive constant Cs>0, independent of γ and δ, can be found in such a way that(4.72)‖τm,δ(x,Z,γ)u‖Htan,γs,r(R+n)≤‖τm,δ(x,Z,γ)u‖Htan,γs(R+n)≤Csγs‖u‖L2(R+n).
Concerning now the term (λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F, after Lemma 4.5 and Proposition 4.8 we already know that the symbol ηm,δ(x,ξ,γ) has order m-2 and obeys the same decay estimates as in (4.19). From (3.17) it then follows that {ηm,δ}0<δ≤1 is a bounded subset of Γm-1; because {λχ,δm-1,γ}0<δ≤1 is also a bounded subset of Γm (as a consequence of (4.4) and (3.17) again), after Proposition 3.10 we conclude that there exists a positive constant Cm,s,r such that(4.73)‖(λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F‖Htan,γs,r(R+n)≤Cm,s,r‖F‖Htan,γs+m,r+m(R+n).
As regards to the boundary datum bm,δ′(D′,γ)G in (4.71), the family of symbols {bm,δ′}0<δ≤1 in ℝn-1 defines a bounded subset of Γm; this follows from estimates (4.56) and the inequality (3.17) (in dimension n-1). Therefore, the Sobolev continuity of ordinary pseudodifferential operators in ℝn-1 implies the existence of a positive constant Cm,s, independent of γ and δ, such that:(4.74)‖bm,δ′(D′,γ)G‖Hγs(Rn-1)≤Cm,s‖G‖Hγs+m(Rn-1).
From the assumptions made about the BVP (1.2)-(1.3) in Theorem 1.1, we find some constants γ̃m≥1,  C̃m>0 such that for all γ≥γm≔max{γm-1,γ̃m} and every δ∈]0,1], λχ,δm-1,γ(Z)u is the only L2-solution of (4.70)-(4.71) and obeys the estimate(4.75)γ‖λχ,δm-1,γ(Z)u‖L2(R+n)2+‖(λχ,δm-1,γ(Z)uI)∣x1=0‖L2(Rn-1)2≤C̃m(1γ2s+1‖(λχ,δm-1,γ(Z)-ηm,δ(x,Z,γ))F-τm,δ(x,Z,γ)u‖Htan,γs,r(R+n)2+1γ2s‖bm,δ′(D′,γ)G‖Hγs(Rn-1)2).
We remark that, according to the statement of Theorem 1.1, the constants γ̃m, C̃m are only dependent on a δ,γ-uniform upper bound of the kth-order norm (3.14), computed on the symbol of B+ρm,δ(x,Z,γ), besides the coefficients Aj, 1≤j≤n, and the integer numbers r,s.Then, using (4.72)–(4.74) and (1.7) to estimate the right-hand side of (4.75), we get(4.76)γ‖λχ,δm-1,γ(Z)u‖L2(R+n)2+‖(λχ,δm-1,γ(Z)uI)∣x1=0‖L2(Rn-1)2≤C̃m,s,r(1γ2s+1‖F‖Htan,γs+m,r+m(R+n)2+1γ2s‖G‖Hγs+m(Rn-1)2),
where C̃m,s,r is a suitable positive constant independent of γ≥γm and δ. Since the L2-norms ∥λχ,δm-1,γ(Z)u∥L2(ℝ+n) are bounded by (4.76), uniformly with respect to δ∈]0,1], Corollary 4.4 gives u∈Htan,γm(ℝ+n).As to the Sobolev regularity of the trace on the boundary of the noncharacteristic componentuI of the solution, the estimate (4.76) gives a bound of ∥(λχ,δm-1,γ(Z)uI)∣x1=0∥L2(ℝn-1)=∥bm,δ′(D′,γ)(u∣x1=0I)∥L2(ℝn-1) uniform with respect to δ∈]0,1] (cf. (4.68)). Then u∣x1=0I∈Hγm(ℝn-1) can be derived from the next result, the proof of which will be given in Appendix A.Lemma 4.11.
Form∈ℕ and δ∈]0,1], let bm,δ′(ξ′,γ) be defined by (4.54). Then there exists a symbol βm,δ(ξ′,γ)∈Γm-2 such that
(4.77)bm,δ′(ξ′,γ)=λδm-1,γ(ξ′)+βm,δ(ξ′,γ),∀ξ′∈Rn-1.
In addition, the symbol βm,δ satisfies the following estimates: for every α′∈ℕn-1, there exists a positive constant Cm,α′, independent of γ and δ, such that
(4.78)|∂ξ′α′βm,δ(ξ′,γ)|≤Cm,α′λδm-2-|α′|,γ(ξ′),∀ξ′∈Rn-1.Arguing as was done to derive Corollary4.4 from Lemma 4.2, from Lemma 4.11 we deduce the following.Corollary 4.12.
For every positive integerm and γ≥1, v∈Hγm(ℝn-1) if and only if v∈Hγm-1(ℝn-1) and the set {∥bm,δ′(D′,γ)v∥L2(ℝn-1)}0<δ≤1 is bounded.After the result of Corollary4.12, we conclude that u∣x1=0I∈Hγm(ℝn-1).It remains to prove that the solution of (1.2)-(1.3) satisfies the a priori estimate (1.8) of order m. From estimate (4.76) and the use of the identities (4.7), (4.68), (4.77), we also deduce
(4.79)γ‖λδm-1,γ(Z)u‖L2(R+n)2+‖λδm-1,γ(D′)(u∣x1=0I)‖L2(Rn-1)2≤Cm,s,r′(1γ2s+1‖F‖Htan,γs+m,r+m(R+n)2+1γ2s‖G‖Hγs+m(Rn-1)2)+2γ‖rm,δ(Z,γ)u‖L2(R+n)2+2‖βm,δ(D′,γ)(u∣x1=0I)‖L2(Rn-1)2,
where the positive constant Cm,s,r′ is again independent of γ≥γm and δ. On the other hand, using Lemma 4.2 and that {βm,δ}0<δ≤1 is a bounded subset of Γm-1 (that follows at once from Lemma 4.11 and inequality (3.17)), one can estimate(4.80)γ‖rm,δ(Z,γ)u‖L2(R+n)2+‖βm,δ(D′,γ)(u∣x1=0I)‖L2(Rn-1)2≤Cm(γ‖u‖L2(R+n)2+‖u∣x1=0I‖Hγm-1(Rn-1)2)
with positive constant Cm independent, once again, of γ and δ.In the end, combining (4.79), (4.80) and using the a priori estimate (4.1) of order m-1 on u, which holds true by the inductive assumption, we conclude that there exists a constant Cm,s,r′′>0 such that(4.81)γ‖λδm-1,γ(Z)u‖L2(R+n)2+‖λδm-1,γ(D′)(u∣x1=0I)‖L2(Rn-1)2≤Cm,s,r′′(1γ2s+1‖F‖Htan,γs+m,r+m(R+n)2+1γ2s‖G‖Hγs+m(Rn-1)2)
for all γ≥γm and δ∈]0,1].The energy estimate (1.8) of order m hence follows by letting δ→0 into the left-hand side of (4.81) (for an arbitrarily fixed γ≥γm) and exploiting the results of Propositions 3.1 and 3.2.



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*Source: 524736-2010-12-02.xml*