# On the Spectral Asymptotics of Operators on Manifolds with Ends

Authors: Sandro Coriasco; Lidia Maniccia
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We deal with the asymptotic behaviour, forλ→+∞, of the counting
function NP(λ) of certain positive self-adjoint operators P with double order (m,μ), m,μ > 0,m≠μ
, defined on a manifold with ends M. The structure of this class
of noncompact manifolds allows to make use of calculi of pseudodifferential operators and Fourier integral operators associated with weighted symbols globally defined on ℝn. By means of these tools, we improve known results concerning the remainder terms of the Weyl Formulae for NP(λ) and show how their behaviour depends on the ratio m/μ and the dimension of M.


---

## Body

## 1. Introduction

The aim of this paper is to study the asymptotic behaviour, forλ→+∞, of the counting function
(1)NP(λ)=∑λj≤λ‍1,
where λ1≤λ2≤⋯ is the sequence of the eigenvalues, repeated according to their multiplicities, of a positive order, self-adjoint, classical, elliptic SG-pseudodifferential operator P on a manifold with ends. Explicitly, SG-pseudodifferential operators P=p(x,D)=Op(p) on ℝn can be defined via the usual left-quantization
(2)Pu(x)=1(2π)n∫eix·ξp(x,ξ)u^(ξ)dξ,u∈𝒮(ℝn),
starting from symbols p(x,ξ)∈C∞(ℝn×ℝn) with the property that, for arbitrary multiindices α,β, there exist constants Cαβ≥0 such that the estimates
(3)|DξαDxβp(x,ξ)|≤Cαβ〈ξ〉m-|α|〈x〉μ-|β|
hold for fixed m,μ∈ℝ and all (x,ξ)∈ℝn×ℝn, where 〈y〉=1+|y|2, y∈ℝn. Symbols of this type belong to the class denoted by Sm,μ(ℝn), and the corresponding operators constitute the class Lm,μ(ℝn)=Op(Sm,μ(ℝn)). In the sequel we will sometimes write Sm,μ and Lm,μ, respectively, fixing once and for all the dimension of the (noncompact) base manifold to n.These classes of operators, introduced onℝn by Cordes [1] and Parenti [2], see also Melrose [3] and Shubin [4], form a graded algebra, that is, Lr,ρ∘Lm,μ⊆Lr+m,ρ+μ. The remainder elements are operators with symbols in S-∞,-∞(ℝn)=⋂(m,μ)∈ℝ2Sm,μ(ℝn)=𝒮(ℝ2n); that is, those having kernel in 𝒮(ℝ2n), continuously mapping 𝒮′(ℝn) to 𝒮(ℝn). An operator P=Op(p)∈Lm,μ and its symbol p∈Sm,μ are called SG-elliptic if there exists R≥0 such that p(x,ξ) is invertible for |x|+|ξ|≥R and
(4)p(x,ξ)-1=O(〈ξ〉-m〈x〉-μ).In such case we will usually writeP∈ELm,μ. Operators in Lm,μ act continuously from 𝒮(ℝn) to itself and extend as continuous operators from 𝒮′(ℝn) to itself and from Hs,σ(ℝn) to Hs-m,σ-μ(ℝn), where Hs,σ(ℝn), s,σ∈ℝ, denotes the weighted Sobolev space
(5)Hs,σ(ℝn)={u∈𝒮′(ℝn):∥u∥s,σ=∥Op(πs,σ)u∥L2<∞},πs,σ(x,ξ)=〈ξ〉s〈x〉σ.Continuous inclusionsHs,σ(ℝn)↪Hr,ρ(ℝn) hold when s≥r and σ≥ρ, compact when both inequalities are strict, and
(6)𝒮(ℝn)=⋂(s,σ)∈ℝ2Hs,σ(ℝn),𝒮′(ℝn)=⋃(s,σ)∈ℝ2Hs,σ(ℝn).An elliptic SG-operatorP∈Lm,μ admits a parametrix E∈L-m,-μ such that
(7)PE=I+K1,EP=I+K2,
for suitable K1,K2∈L-∞,-∞=Op(S-∞,-∞), and it turns out to be a Fredholm operator. In 1987, Schrohe [5] introduced a class of noncompact manifolds, the so-called SG-manifolds, on which it is possible to transfer from ℝn the whole SG-calculus. In short, these are manifolds which admit a finite atlas whose changes of coordinates behave like symbols of order (0,1) (see [5] for details and additional technical hypotheses). The manifolds with cylindrical ends are a special case of SG-manifolds, on which also the concept of SG-classical operator makes sense; moreover, the principal symbol of an SG-classical operator P on a manifold with cylindrical ends M, in this case a triple σ(P)=(σψ(P),σe(P),σψe(P))=(pψ,pe,pψe), has an invariant meaning on M, see Egorov and Schulze [6], Maniccia and Panarese [7], Melrose [3], and Section 2. We indicate the subspaces of classical symbols and operators adding the subscriptcl to the notation introduced above.The literature concerning the study of the eigenvalue asymptotics of elliptic operators is vast and covers a number of different situations and operator classes, see, for example, the monograph by Ivrii [8]. Then, we only mention a few of the many existing papers and books on this deeply investigated subject, which are related to the case we consider here, either by the type of symbols and underlying spaces, or by the techniques which are used. We refer the reader to the corresponding reference lists for more complete informations. On compact manifolds, well-known results were proved by Hörmander [9] and Guillemin [10], see also the book by Kumano-go [11]. On the other hand, for operators globally defined on ℝn, see Boggiatto et al. [12], Helffer [13], Hörmander [14], Mohammed [15], Nicola [16], and Shubin [4]. Many other situations have been considered, see the cited book by Ivrii. On manifolds with ends, Christiansen and Zworski [17] studied the Laplace-Beltrami operator associated with a scattering metric, while Maniccia and Panarese [7] applied the heat kernel method to study operators similar to those considered here.Here we deal with the case of manifolds with ends forP∈ELclm,μ(M), positive and self-adjoint, such that m,μ>0, m≠μ, focusing on the (invariant) meaning of the constants appearing in the corresponding Weyl formulae and on achieving a better estimate of the remainder term. Note that the situation we consider here is different from that of the Laplace-Beltrami operator investigated in [17], where continuous spectrum is present as well. In fact, in view of Theorem 14, spec(P) consists only of a sequence of real isolated eigenvalues {λj} with finite multiplicity.As recalled above, a first result concerning the asymptotic behaviour ofNP(λ) for operators including those considered in this paper was proved by Maniccia and Panarese in [7], giving, for λ→+∞,
(8)NP(λ)={C1λn/m+o(λn/m)form<μ,C01λn/mlogλ+o(λn/mlogλ)form=μ,C2λn/μ+o(λn/μ)form>μ.
Note that the constants C1, C2, C01 above depend only on the principal symbol of P, which implies that they have an invariant meaning on the manifold M, see Sections 2 and 3. On the other hand, in view of the technique used there, the remainder terms appeared in the form o(λn/min{m,μ}) and o(λn/mlogλ) for m≠μ and m=μ, respectively. An improvement in this direction for operators on ℝn had been achieved by Nicola [16], who, in the case m=μ, proved that
(9)NP(λ)=C01λn/mlogλ+O(λn/m),λ⟶+∞,
while, for m≠μ, showed that the remainder term has the form O(λ(n/min{m,μ})-ε) for a suitable ε>0. A further improvement of these results in the case m=μ has recently appeared in Battisti and Coriasco [18], where it has been shown that, for a suitable ε>0,
(10)NP(λ)=C01λn/mlogλ+C02λn/m+O(λ(n/m)-ε),NP(λ)nC01λn/mlogλ+C02λn/m+bO,λ⟶+∞.Even the constantC02 has an invariant meaning on M, and both C01 and C02 are explicitly computed in terms of trace operators defined on Lclm,m(M).In this paper the remainder estimates in the casem≠μ are further improved. More precisely, we first consider the power Q=P1/max{m,μ} of P (see Maniccia et al. [19] for the properties of powers of SG-classical operators). Then, by studying the asymptotic behaviour in λ of the trace of the operator ψ^λ(-Q), ψλ(t)=ψ(t)e-itλ, ψ∈C0∞(ℝ), defined via a Spectral Theorem and approximated in terms of Fourier Integral Operators, we prove the following.Theorem 1.
LetM be a manifold with ends of dimension n and let P∈ELclm,μ(M) be a positive self-adjoint operator such that m,μ>0, m≠μ, with domain Hm,μ(M)↪L2(M). Then, the following Weyl formulae hold for λ→+∞:
(11)NP(λ)={C1λn/m+O(λn/μ)+O(λ(n/m)-(1/μ))=C1λn/m+O(λ(n/m)-ε1)form<μ,C2λn/μ+O(λn/m)+O(λ(n/μ)-(1/m))=C2λn/μ+O(λ(n/μ)-ε2)form>μ,
where ε1=min{1/μ,n((1/m)-(1/μ))} and ε2=min{1/m,n((1/μ)-(1/m))}.The order of the remainder is then determined by the ratio ofm and μ and the dimension of M, since
(12)nm-1μ≤nμ,form<μ⟺1<μm≤1+1n,nμ-1m≤nm,form>μ⟺1<mμ≤1+1n.In particular, whenmax{m,μ}/min{m,μ}≥2, the remainder is always O(λn/max{m,μ}).Examples include operators of Schrödinger type onM, that is, P=-Δg+V, Δg the Laplace-Beltrami operator in M associated with a suitable metric g, V a smooth potential that, in the local coordinates x∈UN⊆ℝn on the cylindrical end growths as 〈x〉μ, with an appropriate μ>0 related to g. Such examples will be discussed in detail, together with the sharpness of the results in Theorem 1, in the forthcoming paper [20], see also [21].The key point in the proof of Theorem1 is the study of the asymptotic behaviour for λ→+∞ of integrals of the form
(13)I(λ)=∫‍ei(-tλ+φ(t;x,ξ)-xξ)ψ(t)a(t;x,ξ)dtđξdx,
where a and φ satisfy certain growth conditions in x and ξ (see Section 3 for more details). The integrals I(λ) represent in fact the local expressions of the trace of ψ^λ(-Q), obtained through the so-called “geometric optic method,” specialised to the SG situation, see, for example, Coriasco [22, 23], Coriasco and Rodino [24]. To treat the integrals I(λ) we proceed similarly to Grigis and Sjöstrand [25], Helffer and Robert [26], see also Tamura [27].The paper is organised as follows. Section2 is devoted to recall the definition of SG-classical operators on a manifold with ends M. In Section 3 we show that the asymptotic behaviour of NP(λ), λ→+∞, for a positive self-adjoint operator P∈Lclm,μ(M), m,μ>0, is related to the asymptotic behaviour of oscillatory integrals of the form I(λ). In Section 4 we conclude the proof of Theorem 1, investigating the behaviour of I(λ) for λ→+∞. Finally, some technical details are collected in the Appendix.


## 2. SG-Classical Operators on Manifolds with Ends

From now on, we will be concerned with the subclass of SG-operators given by those elementsP∈Lm,μ(ℝn), (m,μ)∈ℝ2, which are SG-classical, that is, P=Op(p) with p∈Sclm,μ(ℝn)⊂Sm,μ(ℝn). We begin recalling the basic definitions and results (see, e.g., [6, 19] for additional details and proofs).Definition 2.
(i) A symbolp(x,ξ) belongs to the class Scl(ξ)m,μ(ℝn) if there exist pm-i,·(x,ξ)∈ℋ~ξm-i(ℝn), i=0,1,…, positively homogeneous functions of order m-i with respect to the variable ξ, smooth with respect to the variable x, such that, for a 0-excision function ω,
(14)p(x,ξ)-∑i=0N-1‍ω(ξ)pm-i,·(x,ξ)∈Sm-N,μ(ℝn),p(x,ξ)-∑k=0N-1‍(x)p·,μ-k(x,ξ)v∈,N=1,2,….
(ii) A symbol p(x,ξ) belongs to the class Scl(x)m,μ(ℝn) if there exist p·,μ-k(x,ξ)∈ℋ~xμ-k(ℝn), k=0,…, positively homogeneous functions of order μ-k with respect to the variable x, smooth with respect to the variable ξ, such that, for a 0-excision function ω,
(15)p(x,ξ)-∑k=0N-1‍ω(x)p·,μ-k(x,ξ)∈Sm,μ-N(ℝn),p(x,ξ)-∑k=0N-1‍(x)p·,μ-k(x,ξ)v∈,N=1,2,….Definition 3.
A symbolp(x,ξ) is SG-classical, and we write p∈Scl(x,ξ)m,μ(ℝn)=Sclm,μ(ℝn)=Sclm,μ, if
(i) there existpm-j,·(x,ξ)∈ℋ~ξm-j(ℝn) such that for a 0-excision function ω, ω(ξ)pm-j,·(x,ξ)∈Scl(x)m-j,μ(ℝn) and
(16)p(x,ξ)-∑j=0N-1‍ω(ξ)pm-j,·(x,ξ)∈Sm-N,μ(ℝn),p(x,ξ)-∑j=0N-1‍(ξ)pm-j,·(x,ξ)∈v,N=1,2,…;
(ii) there exist p·,μ-k(x,ξ)∈ℋ~xμ-k(ℝn) such that for a 0-excision function ω, ω(x)p·,μ-k(x,ξ)∈Scl(ξ)m,μ-k(ℝn) and
(17)p(x,ξ)-∑k=0N-1ω(x)p·,μ-k∈Sm,μ-N(ℝn),N=1,2,….
We set Lcl(x,ξ)m,μ(ℝn)=Lclm,μ=Op(Sclm,μ).Remark 4.
The definition could be extended in a natural way from operators acting between scalars to operators acting between (distributional sections of) vector bundles. One should then use matrix-valued symbols whose entries satisfy the estimates (3).
Note that the definition of SG-classical symbol implies a condition of compatibility for the terms of the expansions with respect tox and ξ. In fact, defining σψm-j and σeμ-i on Scl(ξ)m,μ and Scl(x)m,μ, respectively, as
(18)σψm-j(p)(x,ξ)=pm-j,·(x,ξ),j=0,1,…,σeμ-i(p)(x,ξ)=p·,μ-i(x,ξ),i=0,1,….
It is possibile to prove that
(19)pm-j,μ-i=σψem-j,μ-i(p)=σψm-j(σeμ-i(p))=σeμ-i(σψm-j(p)),j=0,1,…,i=0,1,….
Moreover, the composition of two SG-classical operators is still classical. ForP=Op(p)∈Lclm,μ the triple σ(P)=(σψ(P),σe(P),σψe(P))=(pm,·,p·,μ,pm,μ)=(pψ,pe,pψe) is called the principal symbol ofP. The three components are also called the ψ-, e- and ψe-principal symbol, respectively. This definition keeps the usual multiplicative behaviour, that is, for any R∈Lclr,ρ, S∈Lcls,σ, (r,ρ),(s,σ)∈ℝ2, σ(RS)=σ(S)σ(T), with component-wise product in the right-hand side. We also set
(20)Symp(P)(x,ξ)=Symp(p)(x,ξ)=pm(x,ξ)=ω(ξ)pψ(x,ξ)+ω(x)(pe(x,ξ)-ω(ξ)pψe(x,ξ)),
for a fixed 0-excision function ω. Theorem 5 allows to express the ellipticity of SG-classical operators in terms of their principal symbol.Theorem 5.
An operatorP∈Lclm,μ is elliptic if and only if each element of the triple σ(P) is nonvanishing on its domain of definition.As a consequence, denoting by{λj} the sequence of eigenvalues of P, ordered such that j≤k⇒λj≤λk, with each eigenvalue repeated accordingly to its multiplicity, the counting function NP(λ)=∑λj≤λ‍1 is well defined for a SG-classical elliptic self-adjoint operator P see, for example, [16, 18, 20, 21]. We now introduce the class of noncompact manifolds with which we will deal.Definition 6.
A manifold with a cylindrical end is a triple(M,X,[f]), where M=ℳ∐C𝒞 is a n-dimensional smooth manifold and(i)
ℳ is a smooth manifold, given by ℳ=(M0∖D)∪C with a n-dimensional smooth compact manifold without boundary M0, D a closed disc of M0, and C⊂D a collar neighbourhood of ∂D in M0;(ii)
𝒞 is a smooth manifold with boundary ∂𝒞=X, with X diffeomorphic to ∂D;(iii)
f:[δf,∞)×𝕊n-1→𝒞, δf>0, is a diffeomorphism, f({δf}×𝕊n-1)=X and f({[δf,δf+εf)}×𝕊n-1), εf>0, is diffeomorphic to C;(iv)
the symbol∐C means that we are gluing ℳ and 𝒞, through the identification of C and f({[δf,δf+εf)}×𝕊n-1);(v)
the symbol[f] represents an equivalence class in the set of functions
(21){g:[δg,∞)×𝕊n-1⟶𝒞:gisadiffeomorphism,g({δg}×𝕊n-1)=Xandg([δg,δg+εg)×𝕊n-1),εg>0,is  diffeomorphic  toC},where f~g if and only if there exists a diffeomorphism Θ∈Diff(𝕊n-1) such that
(22)(g-1∘f)(ρ,γ)=(ρ,Θ(γ)),
for all ρ≥max{δf,δg} and γ∈𝕊n-1.We use the following notation:(i)
Uδf={x∈ℝn:|x|>δf};(ii)
𝒞τ=f([τ,∞)×𝕊n-1), where τ≥δf. The equivalence condition (22) implies that 𝒞τ is well defined;(iii)
π:ℝn∖{0}→(0,∞)×𝕊n-1:x↦π(x)=(|x|,x/|x|);(iv)
fπ=f∘π:Uδf¯→𝒞 is a parametrisation of the end. Let us notice that, setting F=gπ-1∘fπ, the equivalence condition (22) implies
(23)F(x)=|x|Θ(x|x|).We also denote the restriction offπ mapping Uδf onto 𝒞˙=𝒞∖X by f˙π.The couple(𝒞˙,f˙π-1) is called the exit chart. If 𝒜={(Ωi,ψi)}i=1N is such that the subset {(Ωi,ψi)}i=1N-1 is a finite atlas for ℳ and (ΩN,ψN)=(𝒞˙,f˙π-1), then M, with the atlas 𝒜, is a SG-manifold (see [4]). An atlas 𝒜 of such kind is called admissible. From now on, we restrict the choice of atlases on M to the class of admissible ones. We introduce the following spaces, endowed with their natural topologies,
(24)S(Uδ)={supx∈Uδ′|xα∂βu(x)|u∈C∞(Uδ):∀α,β∈ℕn∀δ′>δsupx∈Uδ′|xα∂βu(x)|<∞},S0(Uδ)=⋂δ′↘δ‍{u∈S(ℝn):suppu⊆Uδ′¯},S(M)={u∈C∞(M):u∘f˙π∈S(Uδf)foranyexitmapfπ(Uδf)},S′(M)denotes  the  dual  space  ofS(M).Definition 7.
The setSm,μ(Uδf) consists of all the symbols a∈C∞(Uδf) which fulfill (3) for (x,ξ)∈Uδf×ℝn only. Moreover, the symbol a belongs to the subset SGclm,μ(Uδf) if it admits expansions in asymptotic sums of homogeneous symbols with respect to x and ξ as in Definitions 2 and 3, where the remainders are now given by SG-symbols of the required order on Uδf.
Note that, sinceUδf is conical, the definition of homogeneous and classical symbol on Uδf makes sense. Moreover, the elements of the asymptotic expansions of the classical symbols can be extended by homogeneity to smooth functions on ℝn∖{0}, which will be denoted by the same symbols. It is a fact that, given an admissible atlas {(Ωi,ψi)}i=1N on M, there exists a partition of unity {θi} and a set of smooth functions {χi} which are compatible with the SG-structure of M, that is,(i)
suppθi⊂Ωi, suppχi⊂Ωi, χiθi=θi, i=1,…,N;(ii)
|∂α(θN∘f˙π)(x)|≤Cα〈x〉-|α| and |∂α(χN∘f˙π)(x)|≤Cα〈x〉-|α| for all x∈Uδf.
Moreover,θN and χN can be chosen so that θN∘f˙π and χN∘f˙π are homogeneous of degree 0 on Uδ. We denote by u* the composition of u:ψi(Ωi)⊂ℝn→ℂ with the coordinate patches ψi, and by v* the composition of v:Ωi⊂M→ℂ with ψi-1, i=1,…,N. It is now possible to give the definition of SG-pseudodifferential operator on M.Definition 8.
LetM be a manifold with a cylindrical end. A linear operator P:S(M)→S′(M) is an SG-pseudodifferential operator of order (m,μ) on M, and we write P∈Lm,μ(M), if, for any admissible atlas {(Ωi,ψi)}i=1N on M with exit chart (ΩN,ψN): (1)
for alli=1,…,N-1 and any θi,χi∈Cc∞(Ωi), there exist symbols pi(x,ξ)∈Sm(ψi(Ωi)) such that
(25)(χiPθiu*)*(x)=∬ei(x-y)·ξpi(x,ξ)u(y)dydx,mmmmmmmu∈C∞(ψi(Ωi));(2)
for anyθN,χN of the type described above, there exists a symbol pN(x,ξ)∈SGm,μ(Uδf) such that
(26)(χNPθNu*)*(x)=∬ei(x-y)·ξpN(x,ξ)u(y)dydx,mmmmmmmmmu∈S0(Uδf);(3)
KP, the Schwartz kernel of P, is such that
(27)KP∈C∞((M×M)∖Δ)⋂S((𝒞˙×𝒞˙)∖W),
where Δ is the diagonal of M×M and W=(f˙π×f˙π)(V) with any conical neighbourhood V of the diagonal of Uδf×Uδf.The most important local symbol ofP is pN. Our definition of SG-classical operator on M differs slightly from the one in [7].Definition 9.
LetP∈Lm,μ(M). P is an SG-classical operator on M, and we write P∈Lclm,μ(M), if pN(x,ξ)∈Sclm,μ(Uδf) and the operator P, restricted to the manifold ℳ, is classical in the usual sense.The usual homogeneous principal symbolpψ of an SG-classical operator P∈Lclm,μ(M) is of course well defined as a smooth function on T*M. In order to give an invariant definition of the principal symbols homogeneous in x of an operator P∈Lclm,μ(M), the subbundle TX*M={(x,ξ)∈T*M:x∈X,ξ∈Tx*M} was introduced. The notions of ellipticity can be extended to operators on M as well.Definition 10.
LetP∈Lclm,μ(M) and let us fix an exit map fπ. We can define local objects pm-j,μ-i,p·,μ-i as
(28)pm-j,μ-i(θ,ξ)=pm-j,μ-iN(θ,ξ),θ∈𝕊n-1,ξ∈ℝn∖{0},p·,μ-i(θ,ξ)=p·,μ-iN(θ,ξ),θ∈𝕊n-1,ξ∈ℝn.Definition 11.
An operatorP∈Lclm,μ(M) is elliptic, and we write P∈ELclm,μ(M), if the principal part of pN∈Sm,μ(Uδf) satisfies the SG-ellipticity conditions on Uδf×ℝn and the operator P, restricted to the manifold ℳ, is elliptic in the usual sense.Proposition 12.
The propertiesP∈Lm,μ(M) and P∈Lclm,μ(M), as well as the notion of SG-ellipticity, do not depend on the (admissible) atlas on M. Moreover, the local functions pe and pψe give rise to invariantly defined elements of C∞(TX*M) and C∞(TX*M∖0), respectively.Then, with anyP∈Lclm,μ(M), it is associated an invariantly defined principal symbol in three components σ(P)=(pψ,pe,pψe). Finally, through local symbols given by πs,σj(x,ξ)=〈ξ〉s, j=1,…,N-1, and πs,σN(x,ξ)=〈ξ〉s〈x〉σ, s,σ∈ℝ, we get a SG-elliptic operator Πs,σ∈Lcls,σ(M) and introduce the (invariantly defined) weighted Sobolev spaces Hs,σ(M) as
(29)Hs,σ(M)={u∈S′(M):∏s,σu∈L2(M)}.The properties of the spacesHs,σ(ℝn) extend to Hs,σ(M) without any change, as well as the continuity of the linear mappings P:Hs,σ(M)→Hs-m,σ-μ(M) induced by P∈Lm,μ(M), mentioned in Section 1.


## 3. Spectral Asymptotics for SG-Classical Elliptic Self-Adjoint Operators on Manifolds with Ends

In this section we illustrate the procedure to prove Theorem1, similar to [13, 25, 27]. The result will follow from the Trace formula (39), (41), the asymptotic behaviour (42), and the Tauberian Theorem 19. The remaining technical points, in particular the proof of the asymptotic behaviour of the integrals appearing in (41), are described in Section 4 and in the Appendix.Let the operatorP∈ELclm,μ(M) be considered as an unbounded operator P:𝒮(M)⊂H0,0(M)=L2(M)→L2(M). The following proposition can be proved by reducing to the local situation and using continuity and ellipticity of P, its parametrix, and the density of 𝒮(M) in the Hs,σ(M) spaces.Proposition 13.
EveryP∈ELclm,μ(M), considered as an unbounded operator P:𝒮(M)⊂L2(M)→L2(M), admits a unique closed extension, still denoted by P, whose domain is 𝒟(P)=Hm,μ(M).From now on, when we writeP∈ELclm,μ(M) we always mean its unique closed extension, defined in Proposition 13. As standard, we denote by ϱ(P) the resolvent set of P, that is, the set of all λ∈ℂ such that λI-P maps Hm,μ(M) bijectively onto L2(M). The spectrum of P is then spec(P)=ℂ∖ϱ(P). The next theorem was proved in [7].Theorem 14 (Spectral theorem).
LetP∈ELclm,μ(M) be regarded as a closed unbounded operator on L2(M) with dense domain Hm,μ(M). Assume also that m,μ>0 and P*=P. Then(i)
(λI-P)-1 is a compact operator on L2(M) for every λ∈ϱ(P). More precisely, (λI-P)-1 is an extension by continuity from 𝒮(M) or a restriction from 𝒮′(M) of an operator in ELcl-m,-μ(M).(ii)
spec(P) consists of a sequence of real isolated eigenvalues {λj} with finite multiplicity, clustering at infinity; the orthonormal system of eigenfunctions {ej}j≥1 is complete in L2(M)=H0,0(M). Moreover, ej∈𝒮(M) for all j.Given a positive self-adjoint operatorP∈ELclm,μ(M), m,μ>0,  μ≠m, we can assume, without loss of generality (considering, if necessary, P+c in place of P, with c∈ℝ a suitably large constant), 1≤λ1≤λ2…. Define the counting function NP(λ), λ∈ℝ, as
(30)NP(λ)=∑λj≤λ‍1=#(spec(P)∩(-∞,λ]).Clearly,NP is nondecreasing, continuous from the right and supported in [0,+∞). If we set Q=P1/l, l=max{m,μ} (see [19] for the definition of the powers of P), Q turns out to be a SG-classical elliptic self-adjoint operator with σ(Q)=(pψ1/l,pe1/l,pψe1/l). We denote by {ηj} the sequence of eigenvalues of Q, which satisfy ηj=λj1/l. We can then, as above, consider NQ(η). It is a fact that NQ(η)=O(ηn/l), see [7].From now on we focus on the caseμ>m>0. The case m>μ>0 can be treated in a completely similar way, exchanging the role of x and ξ. So we can start from a closed positive self-adjoint operator Q∈ELclm,1(M) with domain 𝒟(Q)=Hm,1(M), m∈(0,1). For u∈Hm,1(M), t∈ℝ, we set
(31)U(t)u=∑j=1∞eitηj(u,ej)L2(M)ej,
and the series converges in the L2(M) norm (cf., e.g., [25]). Clearly, for all t∈ℝ, U(t) is a unitary operator such that
(32)U(0)=I,U(t+s)=U(t)U(s),t,s∈ℝ.Moreover, ifu∈Hkm,k(M) for some k∈ℕ, U(t)u∈Ck(ℝ,H0,0(M))∩⋯∩C0(ℝ,Hkm,k(M)) and, for u∈Hm,1(M), we have DtU(t)u-QU(t)u=0, U(0)u=u, which implies that v(t,x)=U(t)u(x) is a solution of the Cauchy problem
(33)(Dt-Q)v=0,v|t=0=u.Let us fixψ∈𝒮(ℝ). We can then define the operator ψ^(-Q) either by using the formula
(34)ψ^(-Q)u=∑j=1∞ψ^(-ηj)(u,ej)L2(M)ej,
or by means of the vector-valued integral (∫ψ(t)U(t)dt)u=∫ψ(t)U(t)udt, u∈H0,0(M). Indeed, there exists N0∈ℕ such that ∑j=1∞‍ηj-N0<∞, so the definition makes sense and gives an operator in ℒ(L2(M)) with norm bounded by ∥ψ∥L1(ℝ). The following lemma, whose proof can be found in the Appendix, is an analog on M of Proposition  1.10.11 in [13].Lemma 15.
ψ
^
(
-
Q
) is an operator with kernel Kψ(x,y)=∑j‍ψ^(-ηj)ej(x)ej(y)¯∈𝒮(M×M).Clearly, we then have(35)∫MKψ(x,x)dx=∑jψ^(-ηj).By the analysis in [22–24, 28] (see also [29]), the above Cauchy problem (33) can solve modulo 𝒮(M) by means of a smooth family of operators V(t), defined for t∈(-T,T), T>0 suitably small, in the sense that (Dt-Q)∘V is a family of smoothing operators and V(0) is the identity on 𝒮′(M). More explicitly, the following theorem holds (see the Appendix for some details concerning the extension to the manifold M of the results on ℝn proved in [22–24, 28]).Theorem 16.
DefineV(t)u=∑k=1N‍χkAk(t)(θku), where θk and χk are as in Definition 8, with χkθk=θk, k=1,…,N, while the Ak(t) are SG FIOs which, in the local coordinate open set Uk=ψk(Ωk) and with v∈𝒮(ℝn), are given by
(36)(Ak(t)v)(x)=∫eiφk(t;x,ξ)ak(t;x,ξ)v^(ξ)đξ.
EachAk(t) solves a local Cauchy problem (Dt-Qk)∘Ak∈C∞((-T,T),L-∞,-∞(ℝn)), Ak(0)=I, with Qk=Op(qk) and {qk}⊂SGclm,1(ℝn) local (complete) symbol of Q associated with {θk}, {χk}, with phase and amplitude functions such that
(37)∂tφk(t;x,ξ)-qk(x,dxφk(t;x,ξ))=0,φk(0;x,ξ)=xξ,ak∈C∞((-T,T),SGcl0,0(ℝn)),ak(0;x,ξ)=1.
Then, V(t) satisfies
(38)(Dt-Q)∘V∈C∞((-T,T),L-∞,-∞(M)),V(0)=I,
and U-V∈C∞((-T,T),L-∞,-∞(M)).Remark 17.
Trivially, fork=1,…,N-1, qk and ak can be considered SG-classical, since, in those cases, they actually have order -∞ with respect to x, by the fact that qk(x,ξ) vanishes for x outside a compact set.Remark 18.
Notation likeb∈C∞((-T,T),Sr,ρ(ℝn)), B∈C∞((-T,T),Lr,ρ(M)), and similar, in Theorem 16 and in the sequel, also mean that the seminorms of the involved elements in the corresponding spaces (induced, in the mentioned cases, by (3)), are uniformly bounded with respect to t∈(-T,T).If we writeψλ(t)=ψ(t)e-itλ in place of ψ(t), for a chosen ψ∈C0∞((-T,T)), the trace formula (35) becomes
(39)∫MKψλ(x,x)dx=∑ψ^(λ-ηj).Let us denote the kernel ofU-V by r(t;x,y)∈C∞((-T,T),𝒮(M×M)). Then, the distribution kernel of ∫‍e-itλψ(t)U(t)dt=ψ^λ(-Q) is
(40)Kψλ(x,y)=∑k=1Nχk(x)∬ψ(t)ei(-tλ+φk(t;x,ξ)-yξ)×ak(t;x,ξ)dtđξθk(y)+∫e-itλψ(t)r(t;x,y)dt,
where the local coordinates in the right-hand side depend on k and, to simplify the notation, we have omitted the corresponding coordinate maps. By the choices of ψ, θk and χk we obtain
(41)∑jψ^(λ-ηj)=∑k=1N∭ψ(t)ei(-tλ+φk(t;x,ξ)-xξ)asa∑jψ^(λ-ηj)s×ak(t;x,ξ)θk(x)dtđξdx∑jψ^(λ-ssηj)+∬e-itλψ(t)r(t;x,x)dtdx∑jψ^(λ-ηj)=∑k=1N∭ψ(t)ei(-tλ+φk(t;x,ξ)-xξ)asa∑jψ^(λ-ηj)sasa×ak(t;x,ξ)θk(x)dtđξdx∑jψ^(λ-ηj)+O(|λ|-∞).Letψ∈C0∞((-T,T)), T>0, be such that ψ(0)=1 and ψ^≥0, ψ^(0)>0 (e.g., set ψ=χ*χ˘¯ with a suitable χ∈C0∞((-T,T))). By the analysis of the asymptotic behaviour of the integrals appearing in (41), described in Section 4, we finally obtain
(42)∑jψ^(λ-ηj)={nmd0λ(n/m)-1+O(λn*-1)O(|λ|-∞)forλ⟶+∞,O(|λ|-∞)forλ⟶-∞,
with n*=min{n,(n/m)-1}. The following Tauberian theorem is a slight modification of Theorem  4.2.5 of [13] (see the Appendix).Theorem 19.
Assume that(i)
ψ∈C0∞(ℝ) is an even function satisfying ψ(0)=1, ψ^≥0, ψ^(0)>0;(ii)
NQ(λ) is a nondecreasing function, supported in [0,+∞), continuous from the right, with polynomial growth at infinity and isolated discontinuity points of first kind {ηj}, j∈ℕ, such that ηj→+∞;(iii)
there existsd0≥0 such that
(43)∑j‍ψ^(λ-ηj)=∫‍ψ^(λ-η)dNQ(η)={nmd0λ(n/m)-1+O(λn*-1)O(|λ|-∞)forλ⟶+∞,O(|λ|-∞)forλ⟶-∞,with m∈(0,1), n*=min{n,(n/m)-1}.
Then(44)NQ(λ)=d02πλn/m+O(λn*),forλ⟶+∞.Remark 20.
The previous statement can be modified as follows: withψ, NQ, and m as in Theorem 19, when
(45)∫ψ^(λ-η)dNQ(η)={nmd0λ(n/m)-1+O(λ(n/m)-2)+O(λn-1)O(|λ|-∞)forλ⟶+∞,O(|λ|-∞)forλ⟶-∞,
with m∈(0,1), then NQ(λ)=(d0/2π)λn/m+O(λ(n/m)-1)+O(λn), for λ→+∞.


## 4. Proof of Theorem1

In view of Theorem19 and Remark 20, to complete the proof of Theorem 1 we need to show that (42) holds. To this aim, as explained previously, this section will be devoted to studying the asymptotic behaviour for |λ|→+∞ of
(46)I(λ)=∫eiΦ(t;x,ξ;λ)ψ(t)a(t;x,ξ)dtđξdx,
where ψ∈C0∞((-T,T)), ψ(0)=1, a∈C∞((-T,T),  S0,0(ℝn)), a(0;x,ξ)=1, and
(47)Φ(t;x,ξ;λ)=φ(t;x,ξ)-xξ-tλ,φ∈C∞((-T,T),Scl1,1(ℝn)),
such that(i)
∂tφ(t;x,ξ)=q(x,dxφ(t;x,ξ)), φ(0;x,ξ)=xξ;(ii)
C-1〈ξ〉≤〈dxφ(t;x,ξ)〉≤C〈ξ〉, for a suitable constant C>1;(iii)
q∈Sclm,1(ℝn), 0<m<1, SG-elliptic.Sinceq-1(x,ξ)∈O(〈x〉-1〈ξ〉-m) for |x|+|ξ|≥R>0, it is not restrictive to assume that this estimate holds on the whole phase space, so that, for a certain constant A>1,
(48)A-1〈x〉〈ξ〉m≤q(x,ξ)≤A〈x〉〈ξ〉m.Remark 21.
The assumption onq-1 above amounts, at most, to modifying q by adding and subtracting a compactly supported symbol, that is, an element of S-∞,-∞(ℝn). The corresponding solutions φ and a of the eikonal and transport equations, respectively, would then change, at most, by an element of C∞((-T,T),S-∞,-∞(ℝn)), see [23, 24, 28]. It is immediate, by integration by parts with respect to t, that an integral as (46) is O(|λ|-∞) for a∈C∞((-T,T),S-∞,-∞(ℝn)). Then, the modified q obviously keeps the same sign everywhere.For two functionsf,g, defined on a common subset X of ℝd1 and depending on parameters y∈Y⊆ℝd2, we will write f≺g or f(x,y)≺g(x,y) to mean that there exists a suitable constant c>0 such that |f(x,y)|≤c|g(x,y)| for all (x,y)∈X×Y. The notation f~g or f(x,y)~g(x,y) means that both f≺g and g≺f hold.Remark 22.
The ellipticity ofq yields, for λ<0,
(49)∂tΦ(t;x,ξ;λ)=q(x,dxφ(t;x,ξ))-λ≻〈x〉〈ξ〉m+|λ|
which, by integration by parts, implies I(λ)=O(|λ|-∞) when λ→-∞.From now on any asymptotic estimate is to be meant forλ→+∞.We will make use of a partition of unity on the phase space. The supports of its elements will depend on suitably large positive constantsk1,k2>1. We also assume, as it is possible, λ≥λ0, again with an appropriate λ0≫1. As we will see below, the values of k1, k2, and λ0 depend only on q and its associated seminorms.Proposition 23.
LetH1 be any function in C0∞(ℝ) such that suppH1⊆[(2k1)-1,2k1], 0≤H1≤1 and H1≡1 on [k1-1,k1], where k1>1 is a suitably chosen, large positive constant. Then
(50)I(λ)=O(λ-∞)+∫eiΦ(t;x,ξ;λ)ψ(t)H1(〈x〉〈ξ〉mλ)×a(t;x,ξ)dtđξdx.Proof.
Write(51)I(λ)=∫eiΦ(t;x,ξ;λ)ψ(t)[1-H1(〈x〉〈ξ〉mλ)]×a(t;x,ξ)dtđξdx+∫‍eiΦ(t;x,ξ;λ)ψ(t)H1(〈x〉〈ξ〉mλ)×a(t;x,ξ)dtđξdx,
and observe that, by A-1〈x〉〈ξ〉m≤q(x,ξ)≤A〈x〉〈ξ〉m, x,ξ∈ℝn, we find
(52)|∂tΦ(t;x,ξ;λ)|≥λ2+(k12-AC)〈x〉〈ξ〉mnnwhen〈x〉〈ξ〉mλ≤k1-1,(53)|∂tΦ(t;x,ξ;λ)|≥(AC)-12〈x〉〈ξ〉m+[(AC)-12k1-1]λbbbbbbbbbbbbbwhen〈x〉〈ξ〉mλ≥k1.
Thus, ifk1>2AC we have |∂tΦ(t;x,ξ;λ)|~λ+〈x〉〈ξ〉m on the support of 1-H1(〈x〉〈ξ〉m/λ), and the assertion follows integrating by parts with respect to t in the first integral of (51).Remark 24.
We actually choosek1>4AC>2AC, since this will be needed in the proof of Proposition 28; see also Section C in the Appendix.
Let us now pickH2∈C0∞(ℝ) such that 0≤H2(υ)≤1, H2(υ)=1 for |υ|≤k2 and H2(υ)=0 for |υ|≥2k2, where k2>1 is a constant which we will choose big enough (see below). We can then write
(54)(λ)=O(λ-∞)+∫eiΦ(t;x,ξ;λ)ψ(t)H1(〈x〉〈ξ〉mλ)=O(λ-∞)bb+×H2(|ξ|)a(t;x,ξ)dtđξdx+∫‍eiΦ(t;x,ξ;λ)ψ(t)H1(〈x〉〈ξ〉mλ)×[1-H2(|ξ|)]a(t;x,ξ)dtđξdx=O(λ-∞)+I1(λ)+I2(λ).
In what follows, we will systematically use the notationSr,ρ=Sr,ρ(y,η), y∈ℝk, η∈ℝl, to generally denote functions depending smoothly on y and η and satisfying SG-type estimates of order r,ρ in y,η. In a similar fashion, STr,ρ=C∞((-T,T),Sr,ρ(y,η)) will stand for some function of the same kind which, additionally, depends smoothly on t∈(-T,T), and, for all s∈ℤ+, DtsC∞((-T,T),Sr,ρ(y,η)) satisfies SG-type estimates of order r,ρ in y,η, uniformly with respect to t∈(-T,T).
To estimateI1(λ), we will apply the stationary phase theorem. We begin by rewriting the integral I1(λ), using the fact that φ is solution of the eikonal equation associated with q and that q is a classical SG-symbol. Note that then ∂t2φ∈C∞((-T,T),Scl2m-1,1(ℝn))⊆C∞((-T,T),Sclm,1(ℝn)), since
(55)∂t2φ(t;x,ξ)=∑i=1n(∂ξiq)(x,dxφ(t;x,ξ))asasas×∂xi(q(x,dxφ(t;x,ξ))).
In view of the Taylor expansion ofφ at t=0, recalling the property q(x,ξ)=ω(x)qe(x,ξ)+Sm,0(x,ξ), ω a fixed 0-excision function, we have, for some 0<δ1<1,
(56)Φ(t;x,ξ;λ)=-λt-xξ+φ(0;x,ξ)+t∂tφ(0;x,ξ)+t22∂t2φ(tδ1;x,ξ)=-λt+tq(x,ξ)+t2ST2m-1,1(x,ξ)=-λt+tω(x)qe(x,ξ)+tSm,0(x,ξ)+t2ST2m-1,1(x,ξ)=-λt+tω(x)qe(x,ξ)+tSm,0(x,ξ)+t2ω(x)ST,e2m-1,1(x,ξ)+t2ST2m-1,0(x,ξ),
where the subscript e denotes the x-homogeneous (exit) principal parts of the involved symbols, which are all SG-classical and real-valued, see [28].Observe that|x|~λ on the support of the integrand in I1(λ), so that we can, in fact, assume ω(x)≡1 there. Indeed, recalling that, by definition, ω∈C∞(ℝn), ω(υ)≡0 for |υ|≤B, ω(υ)≡1 for |υ|≥2B, with a fixed constant B>0, it is enough to observe that
(57)|ξ|≺1,〈x〉〈ξ〉m~λ⟹〈x〉~λ,
which of course implies 〈x〉~|x|, provided that λ0≤λ is large enough. Moreover, by the ellipticity of q, writing x=|x|ς, ς∈𝕊n-1, with the constant A>1 of (48),
(58)A-1〈x〉〈ξ〉m≤q(x,ξ)=ω(x)qe(x,ξ)+Sm,0(x,ξ)≤A〈x〉〈ξ〉m⟹A-1〈x〉|x|〈ξ〉m≤ω(x)qe(ς,ξ)+Sm,0(x,ξ)|x|≤A〈x〉|x|〈ξ〉m⟹A-1〈ξ〉m≤qe(ς,ξ)≤A〈ξ〉m,⟹A-1〈ξ〉mnn≤ς∈𝕊n-1,ξ∈ℝn
taking the limit for |x|→+∞. Then, setting x=λζς, ζ∈[0,+∞), ς∈𝕊n-1, λ≥λ0≫1, in I1(λ), by homogeneity and the previous remarks, we can write
(59)Φ(t;λζς,ξ;λ)=-λt+tω(λζς)qe(λζς,ξ)+tSm,0(λζς,ξ)+t2ω(λζς)ST,e2m-1,1(λζς,ξ)+t2ST2m-1,0(λζς,ξ)=-λt+λζtqe(ς,ξ)+λζt2ST,e2m-1,1(ς,ξ)+tSm,0(λζς,ξ)+t2ST2m-1,0(λζς,ξ)=λ[-t+ζtqe(ς,ξ)+ζt2ST,e2m-1,1(ς,ξ)]+G1(λ;t,ζ;ς,ξ)=λF1(t,ζ;ς,ξ)+G1(λ;t,ζ;ς,ξ),
and find, in view of the compactness of the support of the integrand (see the proof of Proposition 25 below) and the hypotheses
(60)I1(λ)=λn∫eiλF1(t,ζ;ς,ξ)eiG1(λ;t,ζ;ς,ξ)ψ(t)×a(t;λζς,ξ)H1(〈λζ〉〈ξ〉mλ)×H2(|ξ|)ζn-1dtdζđξdς=λn(2π)n∫eiλF1(X,Y)U1(X,Y;λ)dXdY,
with X=(t,ζ), Y=(ς,ξ). We can now prove the following.Proposition 25.
Choosing the constantsk1,λ0>1 large enough and T>0 suitably small, one has, for any k2>1 and for a certain sequence cj, j=0,1,…,
(61)I1(λ)~∑j=0+∞cjλn-1-j,
that is, I1(λ)=c0λn-1+O(λn-2), with
(62)c0=1(2π)n-1∫ℝn∫𝕊n-1H2(|ξ|)qe(ς,ξ)ndςdξ.Proof.
It is easy to see that, on the support ofU1, the phase function F1(X,Y) admits a unique, nondegenerate, stationary point X0=X0(Y)=(0,qe(ς,ξ)-1), that is, F1,X′(X0(Y),Y)=0 for all Y such that (X,Y)∈suppU1, provided that T>0 is chosen suitably small (see, e.g., [25, page 136]), and the Hessian det(F1,X′′(X0(Y),Y)) equals -qe(ς,ξ)2<0. Moreover, the amplitude function
(63)U1(X,Y;λ)=ψ(t)H1(〈λζ〉〈ξ〉mλ)H2(|ξ|)×a(t;λζς,ξ)ζn-1eiG(λ;t,ζ;ς,ξ)
is compactly supported with respect to the variables X and Y and satisfies, for all γ∈ℤ+2,
(64)DXγU1(X,Y;λ)≺1
for all X, Y, λ≥λ0. In fact,(1)
ψ∈C0∞((-T,T)), ς∈𝕊n-1, supp[H2(|ξ|)]⊆{ξ:|ξ|≤2k2}, and
(65)(2k1)-1≤〈ξ〉m1λ2+ζ2≤2k1⟹0<14k12〈2k2〉2m-1λ02≤ζ≤2k1,
where λ0>2k1〈2k2〉m;(2)
all the factors appearing in the expression ofU1 are uniformly bounded, together with all their X-derivatives, for X∈SX=suppψ×[ζ0,ζ1], Y∈SY=𝕊n-1×{ξ:|ξ|≤2k2}, and λ≥λ0.
Of course, (2) trivially holds for the cutoff functionsψ(t) and H2(|ξ|), and for the factor ζn-1. Since a(t;x,ξ)∈ST0,0(x,ξ), on SX×SY we have, for all γ∈ℤ+2 and λ≥λ0>1,
(66)DXγa(t;λζς,ξ)≺〈λζ〉-γ2λγ2〈ξ〉m≺1((1/λ2)+ζ2)γ2/2<1ζγ2≺1.
Moreover, sinceG1∈STm,0(x,ξ) is actually in ST-∞,0(x,ξ)⊂ST0,0(x,ξ) on SX×SY, the same holds for exp(iG1), by an application of the Faà di Bruno formula for the derivatives of compositions of functions, so also this factor fulfills the desired estimates. Finally, another straightforward computation shows that, for all γ2∈ℤ+ and λ≥λ0>1,
(67)Dζγ2H1(〈λζ〉〈ξ〉mλ)≺1
on SX×SY.  The proposition is then a consequence of the stationary phase theorem (see [30, Proposition  1.2.4], [31, Theorem  7.7.6]), applied to the integral with respect to X=(t,ζ). In particular, the leading term is given by λn/(2π)n-1 times the integral with respect to Y of λ-1|det(F1,X′′(X0(Y),Y))|-1/2U1(X0(Y),Y;λ), that is
(68)I1(λ)=λn-1(2π)n-1∫ℝn∫𝕊n-11qe(ς,ξ)ψ(0)×H1(〈λ/qe(ς,ξ)〉〈ξ〉mλ)×H2(|ξ|)qe(ς,ξ)n-1a(0;λςqe(ς,ξ),ξ)dςdξ+O(λn-2)=λn-1(2π)n-1∫ℝn∫𝕊n-1H1(〈λ/qe(ς,ξ)〉〈ξ〉mλ)×H2(|ξ|)qe(ς,ξ)ndςdξ+O(λn-2)=λn-1(2π)n-1∫ℝn‍∫𝕊n-1‍H2(|ξ|)qe(ς,ξ)ndςdξ+O(λn-2),
recalling that ψ(0)=1, a(0;x,ξ)=1 for all x,ξ∈ℝn.
Indeed, having chosenk1>2A, λ0>2k1〈2k2〉m, (58) implies
(69)k1-1<A-1<〈λ/qe(ς,ξ)〉〈ξ〉mλ=(〈ξ〉mλ)2+(〈ξ〉mqe(ς,ξ))2<14k12+A2<k1,
uniformly for ς∈𝕊n-1, ξ∈supp[H2(|ξ|)], λ≥λ0. This concludes the proof.Let us now considerI2(λ). We follow a procedure close to that used in the proof of Theorem  7.7.6 of [31]. However, since here we lack the compactness of the support of the amplitude with respect to x, we need explicit estimates to show that all the involved integrals are convergent, so we give the argument in full detail in what follows.We initially proceed as in the analysis ofI1(λ) mentioned previously. In view of the presence of the factor 1-H2(|ξ|) in the integrand, we can now assume |ξ|≥k2>max{B,1}, B>0 the radius of the smallest ball in ℝn including supp(1-ω), so that q(x,ξ)=ω(ξ)qψ(x,ξ)+Sm-1,1(x,ξ)=qψ(x,ξ)+Sm-1,1(x,ξ). Then, with some 0<δ2<1,
(70)Φ(t;x,ξ;λ)=-λt-xξ+φ(0;x,ξ)+t∂tφ(0;x,ξ)+t22∂t2φ(tδ2;x,ξ)=-λt+tq(x,ξ)+t2ST2m-1,1(x,ξ)=-λt+tqψ(x,ξ)+tSm-1,1(x,ξ)+t2ST2m-1,1(x,ξ).Settingξ=(λζ)1/mς, ζ∈[0,+∞), ς∈𝕊n-1, λ≥λ0, we can rewrite I2(λ) as
(71)I2(λ)=nmλn/m(2π)n×∫eiλ(-t+ζtqψ(x,ς)+tλ-1Sm-1,1(x,(λζ)1/mς)+t2λ-1ST2m-1,1(x,(λζ)1/mς))×ψ(t)a(t;x,(λζ)1/mς)×H1(〈x〉〈(λζ)1/mς〉mλ)×[1-H2((λζ)1/m)]ζ(n/m)-1dtdζdςdx=nmλn/m(2π)n∫eiλF2(X,Y;λ)U2(X,Y;λ)dXdY,X=(t,ζ), Y=(ς,x), where we have set
(72)F2(X,Y;λ)=-t+ζtqψ(x,ς)+tλ-1Sm-1,1(x,(λζ)1/mς)+t2λ-1ST2m-1,1(x,(λζ)1/mς)U2(X,Y;λ)=ψ(t)H1(〈x〉〈(λζ)1/mς〉mλ)×[1-H2((λζ)1/m)]×a(t;x,(λζ)1/mς)ζ(n/m)-1.On the support ofU2, we have
(73)〈x〉〈(λζ)1/mς〉mλ~1,(λζ)1/m≻1⟹〈(λζ)1/mς〉m=〈(λζ)1/m〉m~λζ,
so that
(74)〈x〉λζλ~1⟺ζ~〈x〉-1,|x|<〈x〉≤2k1(k2)-mλ=ϰ~λ.For any fixedY∈𝕊n-1×ℝn, we then have X belonging to a compact set, uniformly with respect to λ≥λ0, say suppψ×[c-1〈x〉-1,c〈x〉-1], for a suitable c>1.Remark 26.
Incidentally, we observe that a rough estimate ofλn/mI2(λ) is
(75)∫eiλF2(X,Y;λ)U2(X,Y;λ)dX≺〈x〉-(n/m)+1∫c-1〈x〉-1c〈x〉-1dζ≺〈x〉-n/m⟹λn/m∫eiλF2(X,Y;λ)U2(X,Y;λ)dXdY≺λn,⟹λn/m∫eiλF2(X,Y;λ)U2bbbdXdY≺λnλ⟶+∞.
An even less precise result would be the bound λn/m, using the convergence of the integral with respect to x in the whole ℝn, given by -(n/m)+n<0.The next lemma is immediate, and we omit the proof.Lemma 27.
S
T
s
,
σ
(
x
,
(
λ
ζ
)
1
/
m
ς
)
=
S
T
s
,
σ
(
x
,
(
λ
ζ
)
1
/
m
) for any ζ∈[0,+∞), x∈ℝn, ς∈𝕊n-1, λ≥λ0, m∈(0,1), and, for all γ∈ℤ+2,
(76)DXγSTs,σ(x,(λζ)1/m)=ζ-γ2STs,σ(x,(λζ)1/m).The main result of this section is as follows.Proposition 28.
Ifk1,k2,λ0>1 are chosen large enough, one has
(77)I2(λ)=nmd0λ(n/m)-1+O(λn-1)+O(λ(n/m)-2).
Explicitly,
(78)d0=1(2π)n-1∫ℝn∫𝕊n-11qψ(x,ς)n/mdςdx.We will prove Proposition28 through various intermediate steps. First of all, arguing as in the proof of (58), exchanging the role of x and ξ, we note that, for all x∈ℝn, ς∈𝕊n-1,
(79)A-1〈x〉≤qψ(x,ς)≤A〈x〉,(x,ς)∈ℝn×𝕊n-1. We now study(80)F2,X′(X,Y;λ)=(∂tF2(X,Y;λ)∂ζF2(X,Y;λ))=(-1+ζζ0+λ-1Sm-1,1(x,(λζ)1/m)+tλ-1ST2m-1,1(x,(λζ)1/m)t(qψ(x,ς)+λ-1ζ-1Sm-1,1(x,(λζ)1/m)+tλ-1ζ-1ST2m-1,1(x,(λζ)1/m))),X=(t,ζ)∈SX=suppψ×[c-1〈x〉-1,c〈x〉-1], Y=(ς,x)∈SY=𝕊n-1×ℝn, λ≥λ0, where we have used Lemma 27. By the symbolic calculus, remembering that λζ≥k2m>1 on suppU2, we can rewrite the expressions mentioned previously as
(81)∂tF2(X,Y;λ)=-1+ζζ0+ζ(λζ)-1Sm-1,1(x,(λζ)1/m)+tζ(λζ)-1ST2m-1,1(x,(λζ)1/m)=-1+ζζ0+ζ[(λζ)1/m]-mSm-1,1(x,(λζ)1/m)+tζ[(λζ)1/m]-mST2m-1,1(x,(λζ)1/m)=-1+ζζ0+ζS-1,1(x,(λζ)1/m)+tζSTm-1,1(x,(λζ)1/m),∂ζF2(X,Y;λ)=t(qψ(x,ς)+S-1,1(x,(λζ)1/m)+tSTm-1,1(x,(λζ)1/m)).
It is clear that ζ~〈x〉-1 implies ζS-1,1(x,(λζ)1/m)=S-1,0(x,(λζ)1/m)and ζSTm-1,1(x,(λζ)1/m)=STm-1,0(x,(λζ)1/m), so that we finally have
(82)∂tF2(X,Y;λ)=-1+ζζ0+S-1,0(x,(λζ)1/m)+tSTm-1,0(x,(λζ)1/m),∂ζF2(X,Y;λ)=t(qψ(x,ς)+S-1,1(x,(λζ)1/m)tb+tSTm-1,1(x,(λζ)1/m)).We now prove that, modulo anO(|λ|-∞) term, we can consider an amplitude such that, on its support, the ration ζ/ζ0 is very close to 1. To this aim, take H3∈C0∞(ℝ) such that 0≤H3(υ)≤1, H3(υ)=1 for |υ|≤(3/2)ε and H3(υ)=0 for |υ|≥2ε, with an arbitrarily fixed, small enough ε∈(0,1/2), and set
(83)V1(X,Y;λ)=U2(X,Y;λ)·[1-H3(ζζ0-1)],V2(X,Y;λ)=U2(X,Y;λ)·H3(ζζ0-1),J1(λ)=∫eiλF2(X,Y;λ)V1(X,Y;λ)dXdY,J2(λ)=∫eiλF2(X,Y;λ)V2(X,Y;λ)dXdY.Proposition 29.
With the choices ofT,k1,λ0, for any ε∈(0,1/2), one can find k2>1 large enough such that J1(λ)=O(λ-∞).Proof.
Since0<m<1, in view of (3), (74), and (79), we can choose k2>1 so large that, for an arbitrarily fixed ε∈(0,1/2), for any λ≥λ0, ζ∈(0,+∞) satisfying |ξ|=(λζ)1/m≥k2,(84a)in∂tF2(X,Y;λ),in∂tF2(X,Y;λ),|S-1,0(x,(λζ)1/m)|≤ε2,in∂tF2(X,Y;λ),|tSTm-1,0(x,(λζ)1/m)|≤ε2,|ζ0ddζS-1,0(x,(λζ)1/m)|=|ζ0ζ-1S-1,0(x,(λζ)1/m))||ζ0ddζS-1,0(x,(λζ)1/m)|≤k0<1,(84b)in∂ζF2(X,Y;λ),in∂ζF2(X,Y;λ),|S-1,1(x,(λζ)1/m)+tSTm-1,1(x,(λζ)1/m)|in∂ζF2(X,Y;λ),≤A-12〈x〉,uniformly with respect to (X,Y)∈SX×SY⊇suppU2(·;λ). Then, F2 is nonstationary on suppV1, since there we have |(ζ/ζ0)-1|≥(3/2)ε, while
(85)|S-1,0(x,(λζ)1/m)+tSTm-1,0(x,(λζ)1/m)|≤ε,
which implies ∂tF2(X,Y;λ)≻1. Observing that, on suppV1, ∂tF2(X,Y;λ)=ST0,0(x,(λζ)1/m), as well as V1(X,Y;λ)=ST0,0(x,(λζ)1/m), the assertion follows by repeated integrations by parts with respect to t, using the operator
(86)Lt=1λ∂tF2(X,Y;λ)Dt⟹L1eiλF2(X,Y;λ)=eiλF2(X,Y;λ),
and recalling Remark 26.Proposition 30.
With the choices ofε,T>0, k1,k2,λ0>1, one can assume, modulo an O(λn-1) term, that the integral with respect to x in J2(λ) is extended to the set {x∈ℝn:〈x〉≤ϰλ}, with
(87)ϰ=(1-ε2)[A(2k2)m]-1.Proof.
Indeed ifϰ<ϰ~=2k1〈k2〉-m, we can split J2(λ) into the sum
(88)∫ϰλ≤〈x〉≤ϰ~λ∫𝕊n-1∫eiλF2V2dXdςdx+∫〈x〉≤ϰλ∫𝕊n-1∫eiλF2V2dXdςdx,
since the inequality ϰ<ϰ~ is true when k2 is sufficiently large.
Observing that, on suppU2,
(89)〈x〉~λ⟹〈ξ〉m=〈x〉〈ξ〉mλλ〈x〉~1⟹|ξ|≤k3,
switching back to the original variables, the first integral in (88) can be treated as I1(λ), and gives, in view of Proposition 25, an O(λn-1) term, as stated.Now we can show thatF2(X,Y;λ) admits a unique, nondegenerate stationary point X0*=X0*(Y,λ) belonging to suppV2 for 〈x〉≤ϰλ. Under the same hypotheses, X0* lies in a circular neighbourhood of X0=(0,ζ0)=(0,qψ(x,ς)-1) of arbitrarily small radius.Proposition 31.
Withε∈(0,1/2), T>0, k1,k2,λ0>1 fixed previously, F2,X′(X,Y;λ) vanishes on suppV2 only for X=X0*(Y;λ)=(0,ζ0*(Y;λ)), that is, F2,X′(X0*(Y;λ),Y;λ)=0 for all Y such that (X,Y;λ)∈suppV2. Moreover,
(90)det(F2,X′′(X0*(Y;λ),Y))~〈x〉2,|X0*(Y;λ)-X0(Y)|=|ζ0*(Y;λ)-ζ0(Y)|≤Aε2〈x〉-1
holds on suppV2.Proof.
We have to solve(91)0=-1+ζζ0+S-1,0(x,(λζ)1/m)+tSTm-1,0(x,(λζ)1/m)0=t((x,(λζ)1/m)qψ(x,ς)+S-1,1(x,(λζ)1/m)+tSTm-1,1(x,(λζ)1/m)),(X,Y;λ)∈suppV2. By (79) and (84a) and (84b), with the choices of ε,T>0, k1,k2,λ0, the coefficient of t in the second equation does not vanish at any point of suppV2. Then t=0, and ζ must satisfy
(92)-1+ζζ0+S-1,0(x,(λζ)1/m)=0⟺ζ=ζ0(1+S-1,0(x,(λζ)1/m))=G(ζ;Y;λ).
Since, by the choice of k2, |∂ζG(ζ;Y;λ)|≤k0<1, uniformly with respect to Y∈𝕊n-1×{x∈ℝn:〈x〉≤ϰλ}, λ≥λ0, G has a unique fixed point ζ0*=ζ0*(Y;λ), smoothly depending on the parameters; see the Appendix for more details. Since
(93)∂t2F2(X,Y;λ)=STm-1,0(x,(λζ)1/m),∂t∂ζF2(X,Y;λ)=qψ(x,ς)(+tSTm-1,0(x,(λζ)1/m))1+ζ0ζ-1((x,(λζ)1/m)S-1,0(x,(λζ)1/m)bbbbbbbbbbbbbbbbbbbbbbb+tSTm-1,0(x,(λζ)1/m))),∂ζ2F2(X,Y;λ)=tζ-1(S-1,1(x,(λζ)1/m)vvvvvvvvvvvvvvvvvv+tSTm-1,1(x,(λζ)1/m)),
we can assume that λζ≥k2m and the choices of the other parameters imply, on suppV2,
(94)∂t2F2(X,Y;λ)≺ε2,∂t∂ζF2(X,Y;λ)~〈x〉,∂ζ2F2(X,Y;λ)≺ε2〈x〉2.
So we have proved that, on suppV2,(95)M=F2,X′′(X0*(Y;λ),Y;λ)=(M11M12M120)=(STm-1,0(x,(λζ0*)1/m)qψ(x,ς)[1+ζ0ζ0*S-1,0(x,(λζ0*)1/m)]qψ(x,ς)[1+ζ0ζ0*S-1,0(x,(λζ0*)1/m)]0)⟹det(M)=-qψ(x,ς)2[1+ζ0ζ0*(Y;λ)S-1,0(x,(λζ0*(Y;λ))1/m)]2~〈x〉2,∥M∥~〈x〉.
By (3), (92), and ζ0*=G(ζ0*;Y;λ), (X,Y)∈SX×SY⊇suppV2(·;λ), we also find
(96)|X0*(Y;λ)-X0(Y)|=|ζ0*(Y;λ)-ζ0(Y)|=|ζ0S-1,0(x,(λζ0*(Y;λ))1/m)|≤Aε2〈x〉-1,
uniformly with respect to λ≥λ0. The proof is complete.Remark 32.
The choice ofk2 depends only on the properties of q and on the values of k1 and ε; that is, we first fix k1>4AC>2AC>2 and ε∈(0,1/2), then T>0 small enough as explained at the beginning of the proof of Proposition 25, then k2>1 as explained in the proofs of Propositions 29 and 31, then, finally, λ0>2k1〈2k2〉m.The next lemma says that the presence in the amplitude of factors which vanish atX=X0* implies the gain of negative powers of λ.Lemma 33.
Assumeα∈ℤ+2, |α|>0,
(97)W=W(X,Y;λ)≺V2(X,Y;λ)tα1×[Wα1+α2(X,Y;λ)(ζ-ζ0*(Y;λ))α1+α2]orW=W(X,Y;λ)≺V2(X,Y;λ)tα1+α2×[Wα2(X,Y;λ)(ζ-ζ0*(Y;λ))α2],W is smooth, Wk(X,Y;λ)≺〈x〉k, k∈ℤ+, and has a SG-behaviour as the factors appearing in the expression of V2. Then
(98)∫eiλF2(X,Y;λ)W(X,Y;λ)dX=λ-|α|∫eiλF2(X,Y;λ)W~(X,Y;λ)dX,
where W~ has the same SG-behaviour, support and x-order of V2, including the powers of ζ.Proof.
By arguments similar to those used in the proof of Proposition29, on suppW(99)∂ζF2(X,Y;λ)≻〈x〉|t|,∂tF2(X,Y;λ)≻〈x〉|ζ-ζ0*(Y;λ)|.
Assume that the first condition in (97) holds. Under the hypotheses, if α1>0, we can first insert eiλF2(X,Y;λ)=Lζα1eiλF2(X,Y;λ) in the left-hand side of (98), where Lζ=Dζ/λ∂ζF2(X,Y;λ), and integrate by parts α1 times. Similarly, if α2>0, we subsequently use eiλF2(X,Y;λ)=Ltα2eiλF2(X,Y;λ), Lt=Dt/λ∂tF2(X,Y;λ), and integrate by parts α2 times. The assertion then follows, remembering that ζ-derivatives of W produce either an additional ζ-1 factor or a lowering of the exponent of ζ-ζ0*, and that ζ,ζ0*~〈x〉-1 on suppW. The proof in the case that the second condition in (97) holds is the same, using first Lζ and then Lt.Proof of Proposition28.
Define,(100)𝒬=𝒬(X,Y;λ)=〈M(X-X0*(Y;λ)),X-X0*(Y;λ)〉,
and, for s∈[0,1],
(101)ℱs(X,Y;λ)=𝒬(X,Y;λ)+s𝒢(X,Y;λ),𝒢(X,Y;λ)=F2(X,Y;λ)-𝒬(X,Y;λ).
Remembering that F2(X0*(Y),Y;λ)=0, F2,X′(X0*(Y),Y;λ)=0, 𝒬 is the Taylor polynomial of degree two of F2 at X=X0*, so that 𝒢 vanishes of order 3 at X=X0*. Obviously, ℱ0(X,Y;λ)=𝒬(X,Y;λ) and ℱ1(X,Y;λ)=F2(X,Y;λ). Write
(102)𝒥τ(s)=∫eiλℱs(X,Y;τ-1)V2(X,Y;τ-1)dX,τ∈(0,λ0-1], and consider the Taylor expansion of 𝒥τ(s) of order 2𝒩-1, 𝒩>1, so that
(103)|𝒥τ(1)-∑k=02𝒩-1𝒥τ(k)(0)k!|≤sup0<s<1|𝒥τ(2𝒩)(s)|(2𝒩)!.
Since(104)𝒥τ(2𝒩)(s)=(iλ)2𝒩∫eiλℱs(X,Y;τ-1)𝒢(X,Y;τ-1)2𝒩×V2(X,Y;τ-1)dX.
Remark 26 and Lemma 33 imply that |𝒥τ(2𝒩)(s)|≺λ-𝒩〈x〉-n/m, τ∈(0,λ0-1], s∈[0,1]. Indeed, it is easy to see, by direct computation, that 𝒢 can be bounded by linear combinations of expressions of the form
(105)t3,t2[W1(X,Y;τ)(ζ-ζ0*(Y;τ))],t[W2(X,Y;τ)(ζ-ζ0*(Y;τ))2],t[W3(X,Y;λ)(ζ-ζ0*(Y;τ))3],
with Wk, k∈ℤ+, having the required properties. Then, the bound of 𝒢2𝒩 will always contain a term of the type t3𝒩[W3𝒩(X,Y;λ)(ζ-ζ0*(Y;λ))3𝒩], which corresponds to the (minimum) value |α|=3𝒩 in (97).
Each term𝒥τ(k)(0), k=0,…,2𝒩-1, has the quadratic phase function 𝒬, which of course also satisfies
(106)∂ζ𝒬(X,Y;τ-1)≻〈x〉|t|,∂t𝒬(X,Y;τ-1)≻〈x〉|ζ-ζ0*(Y;τ-1)|.
Then, denoting by Γ the Taylor expansion of 𝒢 at X0* of order 3𝒩, we observe that 𝒢k-Γk can be bounded by polynomial expressions in X-X0* of the kind appearing in the right-hand side of (97), with |α|=𝒩+k (cf. the proof of Theorem 7.7.5 in [31]). Setting
(107)𝒯τk=∫eiλ𝒬(X,Y;τ-1)(iλΓ(X,Y;τ-1))kV2(X,Y;τ-1)dX.
Lemma 33 implies
(108)𝒥τ(k)(0)-𝒯τk≺λ-𝒩〈x〉-n/m.
We now apply the stationary phase method to𝒯τk and prove that
(109)𝒥τ(1)~∑j=0+∞‍dj(Y;τ)λ-1-j,
which is a consequence of
(110)𝒯τk~λ-1det(M2πi)-1/2∑l‍Ll,k,Y,τV2,Ll,k,Y,τV2=∑l(2iλ)-l〈M-1DX,DX〉l[(iλΓ)kV2]×(X0*(Y;τ-1),Y;τ-1)/l!,
with M evaluated with τ-1 in place of λ. Recalling (95), it follows that the inverse matrix M-1 satisfies, on suppV2,
(111)M-1=(01M121M12-M11M122),1M12≺〈x〉-1,M11M122≺ε〈x〉-2,∥M-1∥~〈x〉-1,
in view of the ellipticity of the involved symbols. Then, the operators Lj,k,Y,τ, j,k∈ℤ+, do not increase the x-order of the resulting function with respect to that of their arguments, (iλΓ)kV2, which is the same of V2, uniformly with respect to τ. The proof of (110) then follows by Theorem 7.6.1, the proof of Lemma 7.7.3 and formula (7.6.7) in [31]; see also [26, 32]. Indeed, by the mentioned results,
(112)𝒥τk-λ-1det(M2πi)-1/2∑l≤k+𝒩Ll,k,Y,τV2=[𝒥τk-λ-1det(M2πi)-1/2∑l≤k+𝒩+1Ll,k,Y,τV2]+λ-1det(M2πi)-1/2Lk+𝒩+1,k,Y,τV2≺λ-𝒩-3〈x〉-1∑|β|≤2∥DXβ〈M-1DX,DX〉k+𝒩+3×[𝒢kV2](X,Y;τ-1)DXβ〈M-1DX,DX〉k+𝒩+3∥L2(ℝX2)+λ-𝒩-2〈x〉-1|Lk+𝒩+1,k,Y,τV2(X0*(Y;τ-1),Y;τ-1)|≺λ-𝒩-3〈x〉-1[∫c-1〈x〉-1c〈x〉-1ζ2((n/m)-3)dζ]1/2+λ-𝒩-2〈x〉-n/m≺λ-𝒩-3〈x〉(-n/m)+(3/2)+λ-𝒩-2〈x〉-n/m≺λ-𝒩-1-(1/2)〈x〉-n/m,λ⟶+∞,
since 〈x〉≺λ on suppV2. It is then enough to sum all the expansions of 𝒥τk/k!, k=0,…,2𝒩-1, and sort the terms by decreasing exponents of λ (as in the proof of Theorem 7.7.5 in [31]) to obtain (109) with the usual expression
(113)d~j(Y;τ)=det(M2πi)-1/2×∑k-l=j∑2k≥3li-j2-k〈M-1DX,DX〉k×[(iΓ)lV2](X0*(Y;τ-1),Y;τ-1),
so that, in particular,
(114)d~j(Y;τ)≺〈x〉-n/m,
for any j∈ℤ+, τ∈(0,λ0-1]. We can then integrate 𝒥τ(1) and its asymptotic expansions with respect to Y∈𝕊n-1×{x∈ℝn:〈x〉≤ϰλ} and find
(115)J2(λ)=∫〈x〉≤ϰλ∫𝕊n-1𝒥λ-1(1)dY~∑jλ-1-j∫〈x〉≤ϰλ∫𝕊n-1d~j(Y;λ-1)dY,λ⟶+∞.
Recall that ψ(0)=1 and a(0,x,ξ)=1, for all x,ξ∈ℝn. Moreover, for ζ=ζ0*(Y;λ), the factors H1, H2, and H3 are identically equal to 1 (see the Appendix). Then, the coefficient of the leading term in (115) is given by
(116)∫d~0(Y;λ-1)dY=∫〈x〉≤ϰλ∫𝕊n-1det(M2πi)-1/2×V2(X0*(Y;λ),Y;λ)dY=2π∫〈x〉≤ϰλ∫𝕊n-1H1(〈x〉〈(λζ0*(ς,x;λ))1/m〉mλ)2π∫〈x〉≤ϰλ∫𝕊n-1b×H2((λζ0*(ς,x;λ))1/m)2π∫〈x〉≤ϰλ∫𝕊n-1b×H3(ζ0*(ς,x;λ)ζ0(ς,x)-1)2π∫〈x〉≤ϰλ∫𝕊n-1b×|det(M)|-1/2ζ0*(ς,x;λ)(n/m)-1dςdx=2π∫〈x〉≤ϰλ∫𝕊n-1|det(M)|-1/2ζ0*(ς,x;λ)(n/m)-1dςdx,
with M evaluated in ζ=ζ0*. We say that
(117)∫d~0(Y;λ-1)dY=2π∫ℝn∫𝕊n-11qψ(x,ς)n/mdςdx+O(λmax{-1/m,n-(n/m),-1})=2πd0+O(λmax{-1/m,n-(n/m),-1}),asasasasλ⟶+∞.
To confirm this, first note that ζ0*(Y;λ)→ζ0(Y), λ→+∞, for any (Y;λ) belonging to the support of the integrand, see the Appendix. Moreover, the integrand is uniformly bounded by the summable function 〈x〉-n/m, and its support is included in the set S. Then, recalling (95) and setting H~=|ζ02det(M)|-1/2,
(118)R=∫〈x〉≤ϰλ∫𝕊n-1|det(M)|-1/2ζ0*(Y;λ)(n/m)-1dY-∫ℝn∫𝕊n-1ζ0(Y)n/mdY=∫〈x〉≤ϰλ∫𝕊n-1ζ0[H~(ζ0*)(n/m)-1-ζ0(n/m)-1]dςdx-∫〈x〉≥ϰλ∫𝕊n-1ζ0n/mdςdx.
The second integral is alwaysO(λn-(n/m)), since qψ(x,ς)~〈x〉 implies
(119)R2=∫〈x〉≥ϰλ∫𝕊n-1ζ0n/mdςdx~∫ϰλ+∞rn-(n/m)-1dr=(ϰλ)n-(n/m)(n/m)-n,λ⟶+∞.
The first integral can be estimated as follows. Since(120)ζ0*-ζ0=ζ0S-1,0(x,(λζ0*)1/m)=ζ0O((λζ0*)-1/m),
by the properties of ζ0* (see the appendix) we find
(121)(ζ0*ζ0)(n/m)-1-1=(1+O((λζ0*)-1/m))(n/m)-1-1=O((λζ0*)-1/m)=O(〈x〉1/mλ-1/m),
since S-1,0(x,(λζ0*)1/m)≪1. By (95), we similarly have H~=1+O(〈x〉1/mλ-1/m), so that
(122)R1=∫〈x〉≤ϰλ∫𝕊n-1ζ0[H~(ζ0*)(n/m)-1-ζ0(n/m)-1]dςdx=∫〈x〉≤ϰλ∫𝕊n-1ζ0n/m[H~(ζ0*ζ0)(n/m)-1-1]dςdx≺λ-1/m∫〈x〉≤ϰλ〈x〉-(n-1)/mdx.
If n>1/(1-m)⇔n-1-((n-1)/m)<-1, n∈ℕ, m∈(0,1), the integral in R1 is convergent for λ→+∞ and R1=O(λ-1/m). In this case, R1 contributes an O(λ(n/m)-1-(1/m)) term to the expansion of I2(λ), which is of lower order than the O(λ(n/m)-2) term, which is one of the remainders appearing in (77). On the other hand, if n<1/(1-m), the integral in R1 is divergent, and R1 itself is O(λn-(n/m)), since, trivially
(123)limλ→+∞λ-1/m∫0ϰλ(rn-1/(1+r2)(n-1)/2m)drλn-(n/m)as=limλ→+∞∫0ϰλ(rn-1/(1+r2)(n-1)/2m)drλ(n-((n-1)/m))as=ϰn-1-((n-1)/m)n-((n-1)/m).
Finally, if n=1/(1-m), R1 is O(λ-1/mlnλ), by
(124)limλ→+∞∫0ϰλ(r1/(1-m)-1/(1+r2)1/2(1-m))drlnλ=ϰ-1,
and again contributes a term of lower order than the remainder O(λ(n/m)-2). Similar conclusions can be obtained for the subsequent terms of the expansion of J2(λ).
The proof is complete, combining the contributions of the remainders likeR with the other terms in the expansion of J2(λ), and remembering that
(125)I2(λ)=nmλn/m(2π)nJ2(λ)+O(|λ|-∞)=nmλn/m∑j=0+∞‍(djλ-1-j+O(λn-(n/m)-1-j))+O(|λ|-∞).Remark 34.
The same conclusions concerning the behaviour ofR1 in the final step of the proof of Proposition 28 could have been obtained studying the Taylor expansion of the extension of ζ0*(Y;τ-1), τ=λ-1, to the interval [0,λ0-1]τ, similarly to [32].Proof of Theorem1.
The statement forμ>m follows by the arguments in Section 3 and Propositions 23, 25 and 28, summing up the contribution of the local symbol on the exit chart to the contributions of the remaining local symbols, which gives the desired multiple of the integral of qψ-n/m on the cosphere bundle as coefficient of the leading term λn/m. The remainder has then order equal to the maximum between (n/m)-1 and n, as claimed. The proof for μ<m is the same, by exchanging step by step the role of x and ξ.



---

*Source: 909782-2013-04-11.xml*