On the Spectral Asymptotics of Operators on Manifolds with Ends

Volume 2013, Article ID 909782,21pages

http://dx.doi.org/10.1155/2013/909782

Research Article

On the Spectral Asymptotics of Operators on Manifolds with

Ends

Sandro Coriasco and Lidia Maniccia

Dipartimento di Matematica, Universit`a degli Studi di Torino, V. C. Alberto, n. 10, I-10123 Torino, Italy

Correspondence should be addressed to Sandro Coriasco; sandro.coriasco@unito.it Received 28 September 2012; Accepted 16 December 2012

Academic Editor: Changxing Miao

Copyright © 2013 S. Coriasco and L. Maniccia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We deal with the asymptotic behaviour, for𝜆 → +∞, of the counting function 𝑁_𝑃(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 onR𝑛. By means of these tools, we improve known results concerning the remainder terms of the Weyl

Formulae for𝑁_𝑃(𝜆) and show how their behaviour depends on the ratio 𝑚/𝜇 and the dimension of M.

1. Introduction

The aim of this paper is to study the asymptotic behaviour, for 𝜆 → +∞, of the counting function

𝑁𝑃(𝜆) = ∑

𝜆𝑗≤𝜆

1, ₍₁₎

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 operators𝑃 = 𝑝(𝑥, 𝐷) = Op(𝑝) on R𝑛can be defined via the usual left-quantization

Pu(𝑥) = 1

(2𝜋)𝑛 ∫ 𝑒𝑖𝑥⋅𝜉𝑝 (𝑥, 𝜉) ̂𝑢 (𝜉) 𝑑𝜉, 𝑢 ∈ S (R𝑛) , (2)

starting from symbols 𝑝(𝑥, 𝜉) ∈ 𝐶∞(R𝑛 × R𝑛) with the property that, for arbitrary multiindices 𝛼, 𝛽, there exist constants𝐶_𝛼𝛽≥ 0 such that the estimates

󵄨󵄨󵄨󵄨

󵄨𝐷𝛼𝜉𝐷𝑥𝛽𝑝 (𝑥, 𝜉)󵄨󵄨󵄨󵄨_{󵄨 ≤ 𝐶}𝛼𝛽⟨𝜉⟩𝑚−|𝛼|⟨𝑥⟩𝜇−|𝛽| (3)

hold for fixed𝑚, 𝜇 ∈ R and all (𝑥, 𝜉) ∈ R𝑛 × R𝑛, where ⟨𝑦⟩= √1 + |𝑦|2_{,𝑦 ∈ R}𝑛_{. Symbols of this type belong to the}

class denoted by𝑆𝑚,𝜇(R𝑛), and the corresponding operators constitute the class𝐿𝑚,𝜇(R𝑛) = Op(𝑆𝑚,𝜇(R𝑛)). In the sequel we will sometimes write𝑆𝑚,𝜇 and 𝐿𝑚,𝜇, respectively, fixing once and for all the dimension of the (noncompact) base manifold to𝑛.

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)∈R}2𝑆𝑚,𝜇(R𝑛) = S(R2𝑛); that is, those having kernel

in S(R2𝑛), continuously mapping S󸀠(R𝑛) to S(R𝑛). An operator𝑃 = Op(𝑝) ∈ 𝐿𝑚,𝜇and its symbol𝑝 ∈ 𝑆𝑚,𝜇are called SG-elliptic if there exists𝑅 ≥ 0 such that 𝑝(𝑥, 𝜉) is invertible for|𝑥| + |𝜉| ≥ 𝑅 and

𝑝(𝑥, 𝜉)−1 = 𝑂 (⟨𝜉⟩−𝑚⟨𝑥⟩−𝜇) . (4) 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 𝐻}𝑠−𝑚,𝜎−𝜇_(R𝑛_{), where 𝐻}𝑠,𝜎_(R𝑛_{), 𝑠, 𝜎 ∈ R, denotes}

the weighted Sobolev space

𝐻𝑠,𝜎(R𝑛) = {𝑢 ∈ S󸀠(R𝑛) : ‖𝑢‖𝑠,𝜎 = 󵄩󵄩󵄩󵄩Op (𝜋𝑠,𝜎) 𝑢󵄩󵄩󵄩󵄩𝐿2< ∞} ,

𝜋_𝑠,𝜎(𝑥, 𝜉) = ⟨𝜉⟩𝑠⟨𝑥⟩𝜎.

(5) Continuous inclusions𝐻𝑠,𝜎(R𝑛) 󳨅→ 𝐻𝑟,𝜌(R𝑛) hold when 𝑠 ≥ 𝑟 and 𝜎 ≥ 𝜌, compact when both inequalities are strict, and S (R𝑛) = ⋂ (𝑠,𝜎)∈R2 𝐻𝑠,𝜎(R𝑛) , S󸀠(R𝑛) = ⋃ (𝑠,𝜎)∈R2 𝐻𝑠,𝜎(R𝑛) . (6) An elliptic SG-operator𝑃 ∈ 𝐿𝑚,𝜇admits a parametrix𝐸 ∈

𝐿−𝑚,−𝜇_{such that}

𝑃𝐸 = 𝐼 + 𝐾₁, 𝐸𝑃 = 𝐼 + 𝐾₂, (7) 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 ends𝑀, in this case a triple 𝜎(𝑃) = (𝜎_𝜓(𝑃), 𝜎_𝑒(𝑃), 𝜎_𝜓𝑒(𝑃)) = (𝑝_𝜓, 𝑝_𝑒, 𝑝_𝜓𝑒), has an invariant meaning on𝑀, 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 subscript_clto 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 infor-mations. On compact manifolds, well-known results were proved by H¨ormander [9] and Guillemin [10], see also the book by Kumano-go [11]. On the other hand, for operators globally defined onR𝑛, see Boggiatto et al. [12], Helffer [13], H¨ormander [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 associ-ated 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 for 𝑃 ∈ 𝐸𝐿𝑚,𝜇_cl (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 ofTheorem 14, spec(𝑃) consists only of a sequence of real

isolated eigenvalues{𝜆_𝑗} with finite multiplicity.

As recalled above, a first result concerning the asymptotic behaviour of𝑁_𝑃(𝜆) for operators including those considered in this paper was proved by Maniccia and Panarese in [7], giving, for𝜆 → +∞, 𝑁_𝑃(𝜆) = { { { { { { { { { { { { { { { 𝐶₁𝜆𝑛/𝑚_{+ 𝑜 (𝜆}𝑛/𝑚_{) for𝑚 < 𝜇,} 𝐶1

0𝜆𝑛/𝑚log𝜆 + 𝑜 (𝜆𝑛/𝑚log𝜆) for 𝑚 = 𝜇,

𝐶₂𝜆𝑛/𝜇+ 𝑜 (𝜆𝑛/𝜇) for𝑚 > 𝜇. (8)

Note that the constants𝐶₁, 𝐶₂, 𝐶1₀ above depend only on the principal symbol of𝑃, which implies that they have an invariant meaning on the manifold𝑀, 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 onR𝑛had been achieved by Nicola [16], who, in the case𝑚 = 𝜇, proved that

𝑁_𝑃(𝜆) = 𝐶1₀𝜆𝑛/𝑚log𝜆 + 𝑂 (𝜆𝑛/𝑚) , 𝜆 󳨀→ +∞, (9) while, for𝑚 ̸= 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 case𝑚 = 𝜇 has recently appeared in Battisti and Coriasco [18], where it has been shown that, for a suitable𝜀 > 0,

𝑁_𝑃(𝜆) = 𝐶1₀𝜆𝑛/𝑚log𝜆 + 𝐶2₀𝜆𝑛/𝑚+ 𝑂 (𝜆(𝑛/𝑚)−𝜀) ,

𝜆 󳨀→ +∞. (10) Even the constant 𝐶2₀ has an invariant meaning on𝑀, and both𝐶1₀and𝐶2₀are explicitly computed in terms of trace operators defined on𝐿𝑚,𝑚_cl (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}_{of𝑃 (see Maniccia et al. [19] for the properties}

of powers of SG-classical operators). Then, by studying the asymptotic behaviour in𝜆 of the trace of the operator ̂𝜓_𝜆(−𝑄), 𝜓_𝜆(𝑡) = 𝜓(𝑡)𝑒−𝑖𝑡𝜆_{,𝜓 ∈ 𝐶}∞

0 (R), defined via a Spectral Theorem

and approximated in terms of Fourier Integral Operators, we prove the following.

Theorem 1. Let 𝑀 be a manifold with ends of dimension 𝑛 and

𝑚, 𝜇 > 0, 𝑚 ̸= 𝜇, with domain 𝐻𝑚,𝜇_{(𝑀) 󳨅→ 𝐿}2_{(𝑀). Then, the}

following Weyl formulae hold for𝜆 → +∞:

𝑁_𝑃(𝜆) = { { { { { { { { { { { { { { { { { { { 𝐶₁𝜆𝑛/𝑚_{+ 𝑂 (𝜆}𝑛/𝜇_{) + 𝑂 (𝜆}(𝑛/𝑚)−(1/𝜇)₎ = 𝐶1𝜆𝑛/𝑚+ 𝑂 (𝜆(𝑛/𝑚)−𝜀1) 𝑓𝑜𝑟 𝑚 < 𝜇, 𝐶₂𝜆𝑛/𝜇_{+ 𝑂 (𝜆}𝑛/𝑚_{) + 𝑂 (𝜆}(𝑛/𝜇)−(1/𝑚)₎ = 𝐶₂𝜆𝑛/𝜇_{+ 𝑂 (𝜆}(𝑛/𝜇)−𝜀2) 𝑓𝑜𝑟 𝑚 > 𝜇, (11)

where𝜀₁ = 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

𝑛 𝑚− 1 𝜇 ≤ 𝑛 𝜇, for 𝑚 < 𝜇 ⇐⇒ 1 < 𝜇 𝑚≤ 1 + 1 𝑛, 𝑛 𝜇− 1 𝑚 ≤ 𝑛 𝑚, for 𝑚 > 𝜇 ⇐⇒ 1 < 𝑚 𝜇 ≤ 1 + 1 𝑛. (12)

In particular, when max{𝑚, 𝜇}/ min{𝑚, 𝜇} ≥ 2, the remainder is always𝑂(𝜆𝑛/ max{𝑚,𝜇}).

Examples include operators of Schr¨odinger type on𝑀, that is,𝑃 = −Δ_𝑔+ 𝑉, Δ_𝑔the Laplace-Beltrami operator in 𝑀 associated with a suitable metric 𝑔, 𝑉 a smooth potential that, in the local coordinates𝑥 ∈ 𝑈_𝑁⊆ R𝑛on 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 inTheorem 1, in the forthcoming paper [20], see also [21].

The key point in the proof ofTheorem 1is the study of the asymptotic behaviour for𝜆 → +∞ of integrals of the form

𝐼 (𝜆) = ∫ 𝑒𝑖(−𝑡𝜆+𝜑(𝑡;𝑥,𝜉)−𝑥𝜉)𝜓 (𝑡) 𝑎 (𝑡; 𝑥, 𝜉) 𝑑𝑡L𝜉𝑑𝑥, (13) where𝑎 and 𝜑 satisfy certain growth conditions in 𝑥 and 𝜉 (seeSection 3for more details). The integrals𝐼(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 Sj¨ostrand [25], Helffer and Robert [26], see also Tamura [27].

The paper is organised as follows.Section 2is devoted to recall the definition of SG-classical operators on a manifold with ends 𝑀. In Section 3 we show that the asymptotic behaviour of𝑁_𝑃(𝜆), 𝜆 → +∞, for a positive self-adjoint operator𝑃 ∈ 𝐿𝑚,𝜇_cl (𝑀), 𝑚, 𝜇 > 0, is related to the asymp-totic behaviour of oscillatory integrals of the form𝐼(𝜆). In

Section 4we conclude the proof ofTheorem 1, investigating the behaviour of𝐼(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 ∈}

𝑆𝑚,𝜇_cl (R𝑛_{) ⊂ 𝑆}𝑚,𝜇_(R𝑛_{). We begin recalling the basic definitions}

and results (see, e.g., [6,19] for additional details and proofs).

Definition 2. (i) A symbol 𝑝(𝑥, 𝜉) belongs to the class

𝑆𝑚,𝜇_cl(𝜉)(R𝑛_{) if there exist 𝑝}

𝑚−𝑖,⋅(𝑥, 𝜉) ∈ ̃H𝑚−𝑖𝜉 (R𝑛), 𝑖 = 0, 1, . . .,

positively homogeneous functions of order𝑚 − 𝑖 with respect to the variable𝜉, smooth with respect to the variable 𝑥, such that, for a0-excision function 𝜔,

𝑝 (𝑥, 𝜉) −𝑁−1∑

𝑖=0𝜔 (𝜉) 𝑝𝑚−𝑖,⋅(𝑥, 𝜉) ∈ 𝑆

𝑚−𝑁,𝜇_(R𝑛_{) ,}

𝑁 = 1, 2, . . . . (14)

(ii) A symbol𝑝(𝑥, 𝜉) belongs to the class 𝑆𝑚,𝜇_cl(𝑥)(R𝑛) if there exist 𝑝_{⋅,𝜇−𝑘}(𝑥, 𝜉) ∈ ̃H𝜇−𝑘_𝑥 (R𝑛), 𝑘 = 0, . . ., positively homogeneous functions of order𝜇 − 𝑘 with respect to the variable𝑥, smooth with respect to the variable 𝜉, such that, for a0-excision function 𝜔,

𝑝 (𝑥, 𝜉) −𝑁−1∑

𝑘=0𝜔 (𝑥) 𝑝⋅,𝜇−𝑘

(𝑥, 𝜉) ∈ 𝑆𝑚,𝜇−𝑁(R𝑛) , 𝑁 = 1, 2, . . . .

(15)

Definition 3. A symbol𝑝(𝑥, 𝜉) is SG-classical, and we write

𝑝 ∈ 𝑆𝑚,𝜇_{cl(𝑥,𝜉)}(R𝑛) = 𝑆𝑚,𝜇_cl (R𝑛) = 𝑆𝑚,𝜇_cl , if

(i) there exist𝑝_{𝑚−𝑗,⋅}(𝑥, 𝜉) ∈ ̃H𝑚−𝑗_𝜉 (R𝑛) such that for a 0-excision function𝜔, 𝜔(𝜉)𝑝_{𝑚−𝑗,⋅}(𝑥, 𝜉) ∈ 𝑆_cl(𝑥)𝑚−𝑗,𝜇(R𝑛) and

𝑝 (𝑥, 𝜉) −𝑁−1∑

𝑗=0𝜔 (𝜉) 𝑝𝑚−𝑗,⋅

(𝑥, 𝜉) ∈ 𝑆𝑚−𝑁,𝜇(R𝑛) , 𝑁 = 1, 2, . . . ;

(16)

(ii) there exist𝑝_{⋅,𝜇−𝑘}(𝑥, 𝜉) ∈ ̃H𝜇−𝑘_𝑥 (R𝑛) such that for a 0-excision function𝜔, 𝜔(𝑥)𝑝_{⋅,𝜇−𝑘}(𝑥, 𝜉) ∈ 𝑆_cl(𝜉)𝑚,𝜇−𝑘(R𝑛) and

𝑝 (𝑥, 𝜉) −𝑁−1∑

𝑘=0

𝜔 (𝑥) 𝑝⋅,𝜇−𝑘∈ 𝑆𝑚,𝜇−𝑁(R𝑛) , 𝑁 = 1, 2, . . . .

(17) We set𝐿𝑚,𝜇_{cl(𝑥,𝜉)}(R𝑛) = 𝐿𝑚,𝜇_cl = Op(𝑆𝑚,𝜇_cl ).

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 𝜎𝑚−𝑗_𝜓 and𝜎𝜇−𝑖_𝑒 on𝑆𝑚,𝜇_cl(𝜉)

and𝑆𝑚,𝜇_cl(𝑥), respectively, as

𝜎_𝜓𝑚−𝑗(𝑝) (𝑥, 𝜉) = 𝑝𝑚−𝑗,⋅(𝑥, 𝜉) , 𝑗 = 0, 1, . . . ,

𝜎_𝑒𝜇−𝑖(𝑝) (𝑥, 𝜉) = 𝑝⋅,𝜇−𝑖(𝑥, 𝜉) , 𝑖 = 0, 1, . . . .

(18) It is possibile to prove that

𝑝𝑚−𝑗,𝜇−𝑖= 𝜎𝜓𝑒𝑚−𝑗,𝜇−𝑖(𝑝) = 𝜎𝜓𝑚−𝑗(𝜎𝑒𝜇−𝑖(𝑝))

= 𝜎_𝑒𝜇−𝑖(𝜎𝑚−𝑗_𝜓 (𝑝)) , 𝑗 = 0, 1, . . . , 𝑖 = 0, 1, . . . . (19) Moreover, the composition of two SG-classical operators is still classical. For𝑃 = Op(𝑝) ∈ 𝐿𝑚,𝜇_cl the triple 𝜎(𝑃) = (𝜎_𝜓(𝑃), 𝜎_𝑒(𝑃), 𝜎_𝜓𝑒(𝑃)) = (𝑝_𝑚,⋅, 𝑝_⋅,𝜇, 𝑝_𝑚,𝜇) = (𝑝_𝜓, 𝑝_𝑒, 𝑝_𝜓𝑒) is called the principal symbol of 𝑃. The three components are also called the𝜓-, 𝑒- and 𝜓𝑒-principal symbol, respectively. This definition keeps the usual multiplicative behaviour, that is, for any𝑅 ∈ 𝐿𝑟,𝜌_cl ,𝑆 ∈ 𝐿𝑠,𝜎_cl,(𝑟, 𝜌), (𝑠, 𝜎) ∈ R2, 𝜎(𝑅𝑆) = 𝜎(𝑆)𝜎(𝑇), with component-wise product in the right-hand side. We also set

Sym_𝑝(𝑃) (𝑥, 𝜉) = Sym_𝑝(𝑝) (𝑥, 𝜉) = 𝑝_m(𝑥, 𝜉) = 𝜔 (𝜉) 𝑝𝜓(𝑥, 𝜉)

+𝜔 (𝑥) (𝑝𝑒(𝑥, 𝜉) − 𝜔 (𝜉) 𝑝𝜓𝑒(𝑥, 𝜉)) ,

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for a fixed0-excision function 𝜔.Theorem 5allows to express the ellipticity of SG-classical operators in terms of their principal symbol.

Theorem 5. An operator 𝑃 ∈ 𝐿𝑚,𝜇_𝑐𝑙 is elliptic if and only if

each element of the triple𝜎(𝑃) is nonvanishing on its domain

of definition.

As a consequence, denoting by {𝜆_𝑗} the sequence of eigenvalues of𝑃, ordered such that 𝑗 ≤ 𝑘 ⇒ 𝜆_𝑗 ≤ 𝜆_𝑘, with each eigenvalue repeated accordingly to its multiplicity, the counting function𝑁_𝑃(𝜆) = ∑_{𝜆𝑗≤𝜆}1 is well defined for a SG-classical elliptic self-adjoint operator𝑃 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 : [𝛿_𝑓, ∞)×S𝑛−1 → C, 𝛿_𝑓> 0, is a diffeomorphism, 𝑓({𝛿_𝑓} × S𝑛−1_{) = 𝑋 and 𝑓({[𝛿}

𝑓, 𝛿𝑓+ 𝜀𝑓)} × S𝑛−1),

𝜀_𝑓> 0, is diffeomorphic to 𝐶;

(iv) the symbol∐_𝐶 means that we are gluingM and C, through the identification of𝐶 and 𝑓({[𝛿_𝑓, 𝛿_𝑓+𝜀_𝑓)}× S𝑛−1_);

(v) the symbol[𝑓] represents an equivalence class in the set of functions

{𝑔 : [𝛿_𝑔, ∞) × S𝑛−1󳨀→ C : 𝑔 is a diffeomorphism, 𝑔 ({𝛿_𝑔} × S𝑛−1) = 𝑋 and 𝑔 ([𝛿_𝑔, 𝛿_𝑔+ 𝜀_𝑔) × S𝑛−1) ,

𝜀𝑔> 0, is diffeomorphic to 𝐶} ,

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where𝑓 ∼ 𝑔 if and only if there exists a diffeomorphism Θ ∈ Diff(S𝑛−1_{) such that}

(𝑔−1∘ 𝑓) (𝜌, 𝛾) = (𝜌, Θ (𝛾)) , (22) for all𝜌 ≥ max{𝛿_𝑓, 𝛿_𝑔} and 𝛾 ∈ S𝑛−1.

We use the following notation: (i)𝑈_𝛿𝑓= {𝑥 ∈ R𝑛: |𝑥| > 𝛿_𝑓};

(ii)C_𝜏= 𝑓([𝜏, ∞)×S𝑛−1), where 𝜏 ≥ 𝛿_𝑓. The equivalence condition (22) implies thatC_𝜏is well defined; (iii)𝜋 : R𝑛 \ {0} → (0, ∞) × S𝑛−1 : 𝑥 󳨃→ 𝜋(𝑥) =

(|𝑥|, 𝑥/|𝑥|);

(iv)𝑓_𝜋 = 𝑓∘𝜋 : 𝑈_𝛿𝑓 → C is a parametrisation of the end. Let us notice that, setting𝐹 = 𝑔−1_𝜋 ∘𝑓_𝜋, the equivalence condition (22) implies

𝐹 (𝑥) = |𝑥| Θ (_|𝑥|𝑥 ) . (23) We also denote the restriction of 𝑓_𝜋 mapping𝑈_𝛿𝑓 onto ̇C = C \ 𝑋 by ̇𝑓_𝜋.

The couple ( ̇C, ̇𝑓_𝜋−1) is called the exit chart. If A = {(Ω_𝑖, 𝜓_𝑖)}𝑁_𝑖=1is such that the subset{(Ω_𝑖, 𝜓_𝑖)}𝑁−1_𝑖=1 is a finite atlas forM and (Ω_𝑁, 𝜓_𝑁) = ( ̇C, ̇𝑓_𝜋−1), then 𝑀, with the atlas A, is a SG-manifold (see [4]). An atlasA 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,

S (𝑈_𝛿) = {𝑢 ∈ 𝐶∞_(𝑈 𝛿) : ∀𝛼, 𝛽 ∈ N𝑛 ∀𝛿󸀠> 𝛿 sup 𝑥∈𝑈_𝛿󸀠󵄨󵄨󵄨󵄨󵄨𝑥 𝛼_𝜕𝛽_{𝑢 (𝑥)}󵄨󵄨󵄨󵄨 󵄨 < ∞} , S₀(𝑈_𝛿) = ⋂ 𝛿󸀠_↘𝛿 {𝑢 ∈ S (R𝑛) : supp 𝑢 ⊆ 𝑈_𝛿󸀠} , S (𝑀) = {𝑢 ∈ 𝐶∞(𝑀) : 𝑢 ∘ ̇𝑓_𝜋 ∈ S (𝑈_𝛿𝑓) for any exit map𝑓_𝜋} , S󸀠_{(𝑀) denotes the dual space of S (𝑀) .}

Definition 7. The set 𝑆𝑚,𝜇(𝑈_𝛿𝑓) consists of all the symbols

𝑎 ∈ 𝐶∞_(𝑈

𝛿𝑓) which fulfill (3) for(𝑥, 𝜉) ∈ 𝑈𝛿𝑓 × R

𝑛 _only.

Moreover, the symbol𝑎 belongs to the subset SG𝑚,𝜇_cl (𝑈_𝛿𝑓) if it admits expansions in asymptotic sums of homogeneous symbols with respect to𝑥 and 𝜉 as in Definitions2and 3, where the remainders are now given by SG-symbols of the required order on𝑈_𝛿𝑓.

Note that, since𝑈_𝛿𝑓is conical, the definition of homoge-neous and classical symbol on𝑈_𝛿𝑓makes sense. Moreover, the elements of the asymptotic expansions of the classical sym-bols can be extended by homogeneity to smooth functions onR𝑛\ {0}, which will be denoted by the same symbols. It is a fact that, given an admissible atlas{(Ω_𝑖, 𝜓_𝑖)}𝑁_𝑖=1on𝑀, there exists a partition of unity{𝜃_𝑖} and a set of smooth functions {𝜒_𝑖} which are compatible with the SG-structure of 𝑀, that is,

(i) supp𝜃_𝑖⊂ Ω_𝑖, supp𝜒_𝑖⊂ Ω_𝑖,𝜒_𝑖𝜃_𝑖= 𝜃_𝑖,𝑖 = 1, . . . , 𝑁; (ii)|𝜕𝛼(𝜃_𝑁∘ ̇𝑓_𝜋)(𝑥)| ≤ 𝐶_𝛼⟨𝑥⟩−|𝛼|and|𝜕𝛼(𝜒_𝑁∘ ̇𝑓_𝜋)(𝑥)| ≤

𝐶_𝛼⟨𝑥⟩−|𝛼|_{for all𝑥 ∈ 𝑈}

𝛿𝑓.

Moreover,𝜃_𝑁and𝜒_𝑁can be chosen so that𝜃_𝑁∘ ̇𝑓_𝜋and 𝜒_𝑁∘ ̇𝑓_𝜋are homogeneous of degree0 on 𝑈_𝛿. We denote by𝑢∗ the composition of𝑢 : 𝜓_𝑖(Ω_𝑖) ⊂ R𝑛 → C with the coordinate patches𝜓_𝑖, and byV_∗the composition ofV : Ω_𝑖 ⊂ 𝑀 → C with𝜓−1_𝑖 ,𝑖 = 1, . . . , 𝑁. It is now possible to give the definition of SG-pseudodifferential operator on𝑀.

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 write𝑃 ∈ 𝐿𝑚,𝜇(𝑀), if, for any admissible atlas {(Ω_𝑖, 𝜓_𝑖)}𝑁_𝑖=1 on𝑀 with exit chart (Ω_𝑁, 𝜓_𝑁):

(1) for all𝑖 = 1, . . . , 𝑁 − 1 and any 𝜃_𝑖, 𝜒_𝑖∈ 𝐶∞_𝑐 (Ω_𝑖), there exist symbols𝑝𝑖(𝑥, 𝜉) ∈ 𝑆𝑚(𝜓_𝑖(Ω_𝑖)) such that (𝜒_𝑖𝑃𝜃_𝑖𝑢∗)_∗(𝑥) = ∬ 𝑒𝑖(𝑥−𝑦)⋅𝜉𝑝𝑖(𝑥, 𝜉) 𝑢 (𝑦) 𝑑𝑦𝑑𝑥, 𝑢 ∈ 𝐶∞(𝜓_𝑖(Ω_𝑖)) ;

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(2) for any𝜃_𝑁, 𝜒_𝑁of the type described above, there exists a symbol𝑝𝑁(𝑥, 𝜉) ∈ SG𝑚,𝜇(𝑈_𝛿𝑓) such that

(𝜒_𝑁𝑃𝜃_𝑁𝑢∗)_∗(𝑥) = ∬ 𝑒𝑖(𝑥−𝑦)⋅𝜉𝑝𝑁(𝑥, 𝜉) 𝑢 (𝑦) 𝑑𝑦𝑑𝑥, 𝑢 ∈ S0(𝑈𝛿𝑓) ;

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(3)𝐾_𝑃, the Schwartz kernel of𝑃, is such that

𝐾_𝑃∈ 𝐶∞((𝑀 × 𝑀) \ Δ) ⋂ S (( ̇C × ̇C) \ 𝑊) , (27) whereΔ is the diagonal of 𝑀 × 𝑀 and 𝑊 = ( ̇𝑓_𝜋×

̇

𝑓𝜋)(𝑉) with any conical neighbourhood 𝑉 of the

diagonal of𝑈_𝛿𝑓× 𝑈_𝛿𝑓.

The most important local symbol of 𝑃 is 𝑝𝑁. Our definition of SG-classical operator on𝑀 differs slightly from the one in [7].

Definition 9. Let𝑃 ∈ 𝐿𝑚,𝜇(𝑀). 𝑃 is an SG-classical operator

on𝑀, and we write 𝑃 ∈ 𝐿𝑚,𝜇_cl (𝑀), if 𝑝𝑁(𝑥, 𝜉) ∈ 𝑆𝑚,𝜇_cl (𝑈_𝛿𝑓) and the operator𝑃, restricted to the manifold M, is classical in the usual sense.

The usual homogeneous principal symbol𝑝_𝜓of an SG-classical operator𝑃 ∈ 𝐿𝑚,𝜇_cl (𝑀) is of course well defined as a smooth function on𝑇∗𝑀. In order to give an invariant definition of the principal symbols homogeneous in𝑥 of an operator 𝑃 ∈ 𝐿𝑚,𝜇_cl (𝑀), the subbundle 𝑇_𝑋∗𝑀 = {(𝑥, 𝜉) ∈ 𝑇∗𝑀 : 𝑥 ∈ 𝑋, 𝜉 ∈ 𝑇_𝑥∗𝑀} was introduced. The notions of ellipticity can be extended to operators on𝑀 as well.

Definition 10. Let𝑃 ∈ 𝐿𝑚,𝜇_cl (𝑀) and let us fix an exit map 𝑓_𝜋.

We can define local objects𝑝_{𝑚−𝑗,𝜇−𝑖}, 𝑝_{⋅,𝜇−𝑖}as

𝑝_{𝑚−𝑗,𝜇−𝑖}(𝜃, 𝜉) = 𝑝𝑁_{𝑚−𝑗,𝜇−𝑖}(𝜃, 𝜉) , 𝜃 ∈ S𝑛−1, 𝜉 ∈ R𝑛\ {0} ,

𝑝_{⋅,𝜇−𝑖}(𝜃, 𝜉) = 𝑝𝑁_{⋅,𝜇−𝑖}(𝜃, 𝜉) , 𝜃 ∈ S𝑛−1, 𝜉 ∈ R𝑛.

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Definition 11. An operator𝑃 ∈ 𝐿𝑚,𝜇_cl (𝑀) is elliptic, and we

write𝑃 ∈ 𝐸𝐿𝑚,𝜇_cl (𝑀), if the principal part of 𝑝𝑁 ∈ 𝑆𝑚,𝜇(𝑈_𝛿𝑓) satisfies the SG-ellipticity conditions on 𝑈_𝛿𝑓 × R𝑛 and the operator𝑃, restricted to the manifold M, is elliptic in the usual sense.

Proposition 12. The properties 𝑃 ∈ 𝐿𝑚,𝜇_{(𝑀) and 𝑃 ∈}

𝐿𝑚,𝜇_𝑐𝑙 (𝑀), as well as the notion of SG-ellipticity, do not depend

on the (admissible) atlas on𝑀. Moreover, the local functions 𝑝_𝑒

and𝑝_𝜓𝑒give rise to invariantly defined elements of𝐶∞(𝑇_𝑋∗𝑀)

and𝐶∞(𝑇_𝑋∗𝑀 \ 0), respectively.

Then, with any 𝑃 ∈ 𝐿𝑚,𝜇_cl (𝑀), it is associated an invariantly defined principal symbol in three components 𝜎(𝑃) = (𝑝𝜓, 𝑝𝑒, 𝑝𝜓𝑒). Finally, through local symbols given by

𝜋_𝑠,𝜎𝑗 (𝑥, 𝜉) = ⟨𝜉⟩𝑠,𝑗 = 1, . . . , 𝑁 − 1, and 𝜋𝑁_𝑠,𝜎(𝑥, 𝜉) = ⟨𝜉⟩𝑠⟨𝑥⟩𝜎, 𝑠, 𝜎 ∈ R, we get a SG-elliptic operator Π_𝑠,𝜎 ∈ 𝐿𝑠,𝜎

cl(𝑀) and

introduce the (invariantly defined) weighted Sobolev spaces 𝐻𝑠,𝜎_{(𝑀) as}

𝐻𝑠,𝜎(𝑀) = {𝑢 ∈ S󸀠(𝑀) : ∏

𝑠,𝜎𝑢 ∈ 𝐿

2_{(𝑀)} . (29)}

The properties of the spaces𝐻𝑠,𝜎(R𝑛) extend to 𝐻𝑠,𝜎(𝑀) without any change, as well as the continuity of the linear mappings𝑃 : 𝐻𝑠,𝜎(𝑀) → 𝐻𝑠−𝑚,𝜎−𝜇(𝑀) induced by 𝑃 ∈ 𝐿𝑚,𝜇(𝑀), mentioned inSection 1.

3. Spectral Asymptotics for

SG-Classical Elliptic Self-Adjoint

Operators on Manifolds with Ends

In this section we illustrate the procedure to proveTheorem 1, 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 inSection 4and in the Appendix.

Let the operator 𝑃 ∈ 𝐸𝐿𝑚,𝜇_cl (𝑀) be considered as an unbounded operator𝑃 : 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. Every 𝑃 ∈ 𝐸𝐿𝑚,𝜇_𝑐𝑙 (𝑀), considered as an

unbounded operator𝑃 : S(𝑀) ⊂ 𝐿2(𝑀) → 𝐿2(𝑀), admits

a unique closed extension, still denoted by𝑃, whose domain is

D(𝑃) = 𝐻𝑚,𝜇_(𝑀).

From now on, when we write𝑃 ∈ 𝐸𝐿𝑚,𝜇_cl (𝑀) we always mean its unique closed extension, defined inProposition 13. As standard, we denote by󰜚(𝑃) the resolvent set of 𝑃, that is, the set of all𝜆 ∈ C such that 𝜆𝐼−𝑃 maps 𝐻𝑚,𝜇(𝑀) bijectively onto𝐿2(𝑀). The spectrum of 𝑃 is then spec(𝑃) = C \ 󰜚(𝑃). The next theorem was proved in [7].

Theorem 14 (Spectral theorem). Let 𝑃 ∈ 𝐸𝐿𝑚,𝜇_𝑐𝑙 (𝑀) be

regarded as a closed unbounded operator on𝐿2(𝑀) with dense

domain𝐻𝑚,𝜇(𝑀). Assume also that 𝑚, 𝜇 > 0 and 𝑃∗ = 𝑃.

Then

(i)(𝜆𝐼 − 𝑃)−1 is a compact operator on𝐿2(𝑀) for every

𝜆 ∈ 󰜚(𝑃). More precisely, (𝜆𝐼 − 𝑃)−1is an extension by

continuity fromS(𝑀) or a restriction from S󸀠(𝑀) of

an operator in𝐸𝐿−𝑚,−𝜇_𝑐𝑙 (𝑀).

(ii) spec(𝑃) consists of a sequence of real isolated

eigenval-ues{𝜆_𝑗} with finite multiplicity, clustering at infinity;

the orthonormal system of eigenfunctions {𝑒_𝑗}_𝑗≥1 is

complete in𝐿2(𝑀) = 𝐻0,0(𝑀). Moreover, 𝑒_𝑗 ∈ S(𝑀)

for all𝑗.

Given a positive self-adjoint operator 𝑃 ∈ 𝐸𝐿𝑚,𝜇_cl (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 function𝑁_𝑃(𝜆), 𝜆 ∈ R, as

𝑁_𝑃(𝜆) = ∑

𝜆𝑗≤𝜆

1 = # (spec (𝑃) ∩ (−∞, 𝜆]) . ₍₃₀₎ Clearly,𝑁_𝑃is 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) = (𝑝1/𝑙

𝜓 , 𝑝1/𝑙𝑒 , 𝑝1/𝑙𝜓𝑒). We denote by {𝜂𝑗} the sequence of

eigenvalues of𝑄, which satisfy 𝜂_𝑗 = 𝜆1/𝑙_𝑗 . We can then, as above, consider𝑁_𝑄(𝜂). It is a fact that 𝑁_𝑄(𝜂) = 𝑂(𝜂𝑛/𝑙), see [7].

From now on we focus on the case 𝜇 > 𝑚 > 0. The case𝑚 > 𝜇 > 0 can be treated in a completely similar way, exchanging the role of𝑥 and 𝜉. So we can start from a closed positive self-adjoint operator𝑄 ∈ 𝐸𝐿𝑚,1_cl (𝑀) with domain D(𝑄) = 𝐻𝑚,1(𝑀), 𝑚 ∈ (0, 1). For 𝑢 ∈ 𝐻𝑚,1(𝑀), 𝑡 ∈ R, we set 𝑈 (𝑡) 𝑢 =∑∞ 𝑗=1𝑒 𝑖𝑡𝜂𝑗 _{(𝑢, 𝑒} 𝑗)_𝐿2_(𝑀)𝑒𝑗, (31)

and the series converges in the𝐿2(𝑀) norm (cf., e.g., [25]). Clearly, for all𝑡 ∈ R, 𝑈(𝑡) is a unitary operator such that

𝑈 (0) = 𝐼, 𝑈 (𝑡 + 𝑠) = 𝑈 (𝑡) 𝑈 (𝑠) , 𝑡, 𝑠 ∈ R. (32) Moreover, if 𝑢 ∈ 𝐻𝑘𝑚,𝑘(𝑀) for some 𝑘 ∈ N, 𝑈(𝑡)𝑢 ∈ 𝐶𝑘_{(R, 𝐻}0,0_{(𝑀)) ∩ ⋅ ⋅ ⋅ ∩ 𝐶}0_{(R, 𝐻}𝑘𝑚,𝑘_{(𝑀)) and, for 𝑢 ∈}

𝐻𝑚,1_{(𝑀), we have 𝐷}

𝑡𝑈(𝑡)𝑢 − 𝑄𝑈(𝑡)𝑢 = 0, 𝑈(0)𝑢 = 𝑢, which

implies thatV(𝑡, 𝑥) = 𝑈(𝑡)𝑢(𝑥) is a solution of the Cauchy problem

(𝐷_𝑡− 𝑄) V = 0, V|𝑡=0= 𝑢. (33)

Let us fix𝜓 ∈ S(R). We can then define the operator ̂𝜓(−𝑄) either by using the formula

̂𝜓 (−𝑄) 𝑢 =∑∞

𝑗=1̂𝜓 (−𝜂𝑗

) (𝑢, 𝑒_𝑗)_𝐿2_(𝑀)𝑒𝑗, (34)

or by means of the vector-valued integral(∫ 𝜓(𝑡)𝑈(𝑡)𝑑𝑡)𝑢 = ∫ 𝜓(𝑡)𝑈(𝑡)𝑢 𝑑𝑡, 𝑢 ∈ 𝐻0,0(𝑀). Indeed, there exists 𝑁₀ ∈ N such that ∑∞_𝑗=1𝜂−𝑁0

𝑗 < ∞, so the definition makes sense

and gives an operator inL(𝐿2(𝑀)) with norm bounded by ‖𝜓‖_𝐿1_(R). The following lemma, whose proof can be found in

the Appendix, is an analog on𝑀 of Proposition 1.10.11 in [13].

Lemma 15. ̂𝜓(−𝑄) is an operator with kernel 𝐾_𝜓(𝑥, 𝑦) = ∑_𝑗 ̂𝜓(−𝜂𝑗)𝑒𝑗(𝑥)𝑒𝑗(𝑦) ∈ S(𝑀 × 𝑀).

Clearly, we then have ∫

𝑀𝐾𝜓(𝑥, 𝑥) 𝑑𝑥 = ∑_𝑗 ̂𝜓 (−𝜂𝑗) . (35)

By the analysis in [22–24,28] (see also [29]), the above Cauchy problem (33) can solve moduloS(𝑀) by means of a smooth family of operators𝑉(𝑡), defined for 𝑡 ∈ (−𝑇, 𝑇), 𝑇 > 0 suitably small, in the sense that (𝐷_𝑡−𝑄)∘𝑉 is a family of smoothing operators and𝑉(0) is the identity on S󸀠(𝑀). More explicitly, the following theorem holds (see the Appendix for some details concerning the extension to the manifold𝑀 of the results onR𝑛proved in [22–24,28]).

Theorem 16. Define 𝑉(𝑡)𝑢 = ∑𝑁_𝑘=1𝜒_𝑘𝐴_𝑘(𝑡)(𝜃_𝑘𝑢), where 𝜃_𝑘

and𝜒_𝑘are as inDefinition 8, with𝜒_𝑘𝜃_𝑘 = 𝜃_𝑘,𝑘 = 1, . . . , 𝑁,

while the𝐴_𝑘(𝑡) are SG FIOs which, in the local coordinate open

set𝑈_𝑘= 𝜓_𝑘(Ω_𝑘) and with V ∈ S(R𝑛), are given by

(𝐴_𝑘(𝑡) V) (𝑥) = ∫ 𝑒𝑖𝜑𝑘(𝑡;𝑥,𝜉)_𝑎

𝑘(𝑡; 𝑥, 𝜉) ̂V (𝜉) L𝜉. (36) Each𝐴_𝑘(𝑡) solves a local Cauchy problem (𝐷_𝑡− 𝑄_𝑘) ∘ 𝐴_𝑘∈

𝐶∞_{((−𝑇, 𝑇), 𝐿}−∞,−∞_(R𝑛_{)), 𝐴}

𝑘(0) = 𝐼, with 𝑄𝑘= 𝑂𝑝(𝑞𝑘) and

{𝑞_𝑘} ⊂ 𝑆𝐺𝑚,1_𝑐𝑙 (R𝑛) local (complete) symbol of 𝑄 associated with {𝜃_𝑘}, {𝜒_𝑘}, with phase and amplitude functions such that

𝜕_𝑡𝜑_𝑘(𝑡; 𝑥, 𝜉) − 𝑞_𝑘(𝑥, 𝑑_𝑥𝜑_𝑘(𝑡; 𝑥, 𝜉)) = 0, 𝜑_𝑘(0; 𝑥, 𝜉) = 𝑥𝜉, 𝑎_𝑘∈ 𝐶∞((−𝑇, 𝑇) , 𝑆𝐺0,0_𝑐𝑙 (R𝑛)) , 𝑎_𝑘(0; 𝑥, 𝜉) = 1. (37) Then,𝑉(𝑡) satisfies (𝐷_𝑡− 𝑄) ∘ 𝑉 ∈ 𝐶∞((−𝑇, 𝑇) , 𝐿−∞,−∞(𝑀)) , 𝑉 (0) = 𝐼, (38) and𝑈 − 𝑉 ∈ 𝐶∞((−𝑇, 𝑇), 𝐿−∞,−∞(𝑀)).

Remark 17. Trivially, for𝑘 = 1, . . . , 𝑁 − 1, 𝑞_𝑘 and𝑎_𝑘 can be

considered SG-classical, since, in those cases, they actually have order −∞ with respect to 𝑥, by the fact that 𝑞_𝑘(𝑥, 𝜉) vanishes for𝑥 outside a compact set.

Remark 18. Notation like𝑏 ∈ 𝐶∞((−𝑇, 𝑇), 𝑆𝑟,𝜌(R𝑛)), 𝐵 ∈

𝐶∞_{((−𝑇, 𝑇), 𝐿}𝑟,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 men-tioned cases, by (3)), are uniformly bounded with respect to 𝑡 ∈ (−𝑇, 𝑇).

If we write𝜓_𝜆(𝑡) = 𝜓(𝑡)𝑒−𝑖𝑡𝜆in place of𝜓(𝑡), for a chosen 𝜓 ∈ 𝐶∞

0 ((−𝑇, 𝑇)), the trace formula (35) becomes

∫

𝑀𝐾𝜓𝜆(𝑥, 𝑥) 𝑑𝑥 = ∑ ̂𝜓 (𝜆 − 𝜂𝑗) . (39)

Let us denote the kernel of 𝑈 − 𝑉 by 𝑟(𝑡; 𝑥, 𝑦) ∈ 𝐶∞_{((−𝑇, 𝑇), S(𝑀 × 𝑀)). Then, the distribution kernel of}

∫ 𝑒−𝑖𝑡𝜆_{𝜓(𝑡) 𝑈(𝑡)𝑑𝑡 = ̂𝜓} 𝜆(−𝑄) is 𝐾_𝜓𝜆(𝑥, 𝑦) =∑𝑁 𝑘=1 𝜒_𝑘(𝑥) ∬ 𝜓 (𝑡) 𝑒𝑖(−𝑡𝜆+𝜑𝑘(𝑡;𝑥,𝜉)−𝑦𝜉) × 𝑎_𝑘(𝑡; 𝑥, 𝜉) 𝑑𝑡L𝜉𝜃𝑘(𝑦) + ∫ 𝑒−𝑖𝑡𝜆𝜓 (𝑡) 𝑟 (𝑡; 𝑥, 𝑦) 𝑑𝑡, (40)

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 of𝜓, 𝜃_𝑘and 𝜒_𝑘we obtain ∑ 𝑗 ̂𝜓 (𝜆 − 𝜂𝑗 ) =∑𝑁 𝑘=1∭ 𝜓 (𝑡) 𝑒 𝑖(−𝑡𝜆+𝜑𝑘(𝑡;𝑥,𝜉)−𝑥𝜉) × 𝑎_𝑘(𝑡; 𝑥, 𝜉) 𝜃_𝑘(𝑥) 𝑑𝑡L𝜉𝑑𝑥 + ∬ 𝑒−𝑖𝑡𝜆𝜓 (𝑡) 𝑟 (𝑡; 𝑥, 𝑥) 𝑑𝑡𝑑𝑥 =∑𝑁 𝑘=1 ∭ 𝜓 (𝑡) 𝑒𝑖(−𝑡𝜆+𝜑𝑘(𝑡;𝑥,𝜉)−𝑥𝜉) × 𝑎𝑘(𝑡; 𝑥, 𝜉) 𝜃𝑘(𝑥) 𝑑𝑡L𝜉𝑑𝑥 + 𝑂 (|𝜆|−∞) . (41)

Let𝜓 ∈ 𝐶∞₀ ((−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 ∈ 𝐶∞

0 ((−𝑇, 𝑇))). By the analysis of the asymptotic behaviour

of the integrals appearing in (41), described inSection 4, we finally obtain ∑ 𝑗 ̂𝜓 (𝜆 − 𝜂𝑗 ) = { { { { { { { { { { { { { 𝑛 𝑚𝑑0𝜆(𝑛/𝑚)−1+ 𝑂 (𝜆𝑛 ∗₋₁ ) for𝜆 󳨀→ +∞, 𝑂 (|𝜆|−∞) for 𝜆 󳨀→ −∞, (42)

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 ∈ 𝐶∞₀ (R) is an even function satisfying 𝜓(0) = 1, ̂𝜓 ≥ 0, ̂𝜓(0) > 0;

(ii)𝑁_𝑄(𝜆) is a nondecreasing function, supported in [0, +∞), continuous from the right, with polynomial

growth at infinity and isolated discontinuity points of

first kind{𝜂_𝑗}, 𝑗 ∈ N, such that 𝜂_𝑗 → +∞;

(iii) there exists𝑑₀≥ 0 such that ∑ 𝑗 ̂𝜓 (𝜆 − 𝜂𝑗 ) = ∫ ̂𝜓 (𝜆 − 𝜂) 𝑑𝑁_𝑄(𝜂) = { { { { { { { { { { { { { { { 𝑛 𝑚𝑑0 𝜆(𝑛/𝑚)−1+ 𝑂 (𝜆𝑛 ∗₋₁ ) 𝑓𝑜𝑟 𝜆 󳨀→ +∞, 𝑂 (|𝜆|−∞) 𝑓𝑜𝑟 𝜆 󳨀→ −∞, (43) with𝑚 ∈ (0, 1), 𝑛∗= min{𝑛, (𝑛/𝑚) − 1}. Then 𝑁_𝑄(𝜆) = 𝑑0 2𝜋𝜆𝑛/𝑚+ 𝑂 (𝜆𝑛 ∗ ) , 𝑓𝑜𝑟 𝜆 󳨀→ +∞. (44)

Remark 20. The previous statement can be modified as

follows: with𝜓, 𝑁_𝑄, and𝑚 as inTheorem 19, when ∫ ̂𝜓 (𝜆 − 𝜂) 𝑑𝑁_𝑄(𝜂) = { { { { { { { { { { { { { 𝑛 𝑚𝑑0 𝜆(𝑛/𝑚)−1+ 𝑂 (𝜆(𝑛/𝑚)−2) + 𝑂 (𝜆𝑛−1) for 𝜆 󳨀→ +∞, 𝑂 (|𝜆|−∞_{) for 𝜆 󳨀→ −∞,} (45) with𝑚 ∈ (0, 1), then 𝑁_𝑄(𝜆) = (𝑑₀/2𝜋)𝜆𝑛/𝑚+ 𝑂(𝜆(𝑛/𝑚)−1) + 𝑂(𝜆𝑛_{), for 𝜆 → +∞.}

4. Proof of

Theorem 1

In view ofTheorem 19andRemark 20, to complete the proof ofTheorem 1we 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

𝐼 (𝜆) = ∫ 𝑒𝑖Φ(𝑡;𝑥,𝜉;𝜆)𝜓 (𝑡) 𝑎 (𝑡; 𝑥, 𝜉) 𝑑𝑡L𝜉𝑑𝑥, (46) where 𝜓 ∈ 𝐶₀∞((−𝑇, 𝑇)), 𝜓(0) = 1, 𝑎 ∈ 𝐶∞((−𝑇, 𝑇), 𝑆0,0_(R𝑛_{)), 𝑎(0; 𝑥, 𝜉) = 1, and} Φ (𝑡; 𝑥, 𝜉; 𝜆) = 𝜑 (𝑡; 𝑥, 𝜉) − 𝑥𝜉 − 𝑡𝜆, 𝜑 ∈ 𝐶∞((−𝑇, 𝑇) , 𝑆1,1_cl (R𝑛)) , (47) such that (i)𝜕_𝑡𝜑(𝑡; 𝑥, 𝜉) = 𝑞(𝑥, 𝑑_𝑥𝜑(𝑡; 𝑥, 𝜉)), 𝜑(0; 𝑥, 𝜉) = 𝑥𝜉; (ii)𝐶−1⟨𝜉⟩≤ ⟨𝑑_𝑥𝜑(𝑡; 𝑥, 𝜉)⟩≤ 𝐶⟨𝜉⟩, for a suitable

constant𝐶 > 1;

(iii)𝑞 ∈ 𝑆𝑚,1_cl (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 constant𝐴 > 1,

𝐴−1⟨𝑥⟩ ⟨𝜉⟩𝑚≤ 𝑞 (𝑥, 𝜉) ≤ 𝐴 ⟨𝑥⟩ ⟨𝜉⟩𝑚. (48)

Remark 21. The assumption on𝑞−1above amounts, at most,

to modifying 𝑞 by adding and subtracting a compactly supported symbol, that is, an element of𝑆−∞,−∞(R𝑛). The corresponding solutions𝜑 and 𝑎 of the eikonal and transport equations, respectively, would then change, at most, by an element of𝐶∞((−𝑇, 𝑇), 𝑆−∞,−∞(R𝑛)), see [23, 24, 28]. It is immediate, by integration by parts with respect to t, that an integral as (46) is𝑂(|𝜆|−∞) for 𝑎 ∈ 𝐶∞((−𝑇, 𝑇), 𝑆−∞,−∞(R𝑛)). Then, the modified𝑞 obviously keeps the same sign every-where.

For two functions𝑓, 𝑔, defined on a common subset 𝑋 ofR𝑑1_{and depending on parameters}𝑦 ∈ 𝑌 ⊆ R𝑑2_{, we will}

write𝑓 ≺ 𝑔 or 𝑓(𝑥, 𝑦) ≺ 𝑔(𝑥, 𝑦) to mean that there exists a suitable constant𝑐 > 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 both𝑓 ≺ 𝑔 and 𝑔 ≺ 𝑓 hold.

Remark 22. The ellipticity of𝑞 yields, for 𝜆 < 0,

𝜕_𝑡Φ (𝑡; 𝑥, 𝜉; 𝜆) = 𝑞 (𝑥, 𝑑𝑥𝜑 (𝑡; 𝑥, 𝜉)) − 𝜆 ≻ ⟨𝑥⟩ ⟨𝜉⟩𝑚+ |𝜆|

(49) which, by integration by parts, implies𝐼(𝜆) = 𝑂(|𝜆|−∞) when 𝜆 → −∞.

From now on any asymptotic estimate is to be meant for 𝜆 → +∞.

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 appropriate𝜆₀ ≫ 1. As we will see below, the values of𝑘₁,𝑘₂, and𝜆₀depend only on𝑞 and its associated seminorms.

Proposition 23. Let 𝐻1be any function in𝐶∞0 (R) such that

supp𝐻₁⊆ [(2𝑘₁)−1, 2𝑘₁], 0 ≤ 𝐻₁≤ 1 and 𝐻₁≡ 1 on [𝑘−1₁ , 𝑘₁],

where𝑘₁> 1 is a suitably chosen, large positive constant. Then

𝐼 (𝜆) = 𝑂 (𝜆−∞) + ∫ 𝑒𝑖Φ(𝑡;𝑥,𝜉;𝜆)𝜓 (𝑡) 𝐻1(⟨𝑥⟩ ⟨𝜉⟩ 𝑚 𝜆 ) × 𝑎 (𝑡; 𝑥, 𝜉) 𝑑𝑡L𝜉𝑑𝑥. (50) Proof. Write 𝐼 (𝜆) = ∫ 𝑒𝑖Φ(𝑡;𝑥,𝜉;𝜆)𝜓 (𝑡) [1 − 𝐻1(⟨𝑥⟩ ⟨𝜉⟩ 𝑚 𝜆 )] × 𝑎 (𝑡; 𝑥, 𝜉) 𝑑𝑡L𝜉𝑑𝑥 + ∫ 𝑒𝑖Φ(𝑡;𝑥,𝜉;𝜆)𝜓 (𝑡) 𝐻1(⟨𝑥⟩ ⟨𝜉⟩ 𝑚 𝜆 ) × 𝑎 (𝑡; 𝑥, 𝜉) 𝑑𝑡L𝜉𝑑𝑥, (51)

and observe that, by 𝐴−1⟨𝑥⟩⟨𝜉⟩𝑚 ≤ 𝑞(𝑥, 𝜉) ≤ 𝐴⟨𝑥⟩⟨𝜉⟩𝑚, 𝑥, 𝜉 ∈ R𝑛_{, we find} 󵄨󵄨󵄨󵄨𝜕𝑡Φ (𝑡; 𝑥, 𝜉; 𝜆)󵄨󵄨󵄨󵄨 ≥ 𝜆₂ + (𝑘₂1 − 𝐴𝐶) ⟨𝑥⟩ ⟨𝜉⟩𝑚 when ⟨𝑥⟩ ⟨𝜉⟩ 𝑚 𝜆 ≤ 𝑘−11 , (52) 󵄨󵄨󵄨󵄨𝜕𝑡Φ (𝑡; 𝑥, 𝜉; 𝜆)󵄨󵄨󵄨󵄨 ≥ (𝐴𝐶) −1 2 ⟨𝑥⟩⟨𝜉⟩ 𝑚_{+ [(}𝐴𝐶)−1 2 𝑘1− 1] 𝜆 when ⟨𝑥⟩ ⟨𝜉⟩ 𝑚 𝜆 ≥ 𝑘1. (53) Thus, if𝑘₁> 2𝐴𝐶 we have |𝜕_𝑡Φ(𝑡; 𝑥, 𝜉; 𝜆)| ∼ 𝜆 + ⟨𝑥⟩⟨𝜉⟩𝑚 on the support of1−𝐻₁(⟨𝑥⟩⟨𝜉⟩𝑚/𝜆), and the assertion follows integrating by parts with respect to𝑡 in the first integral of (51).

Remark 24. We actually choose𝑘₁> 4𝐴𝐶 > 2𝐴𝐶, since this

will be needed in the proof ofProposition 28; see also Section C in the Appendix.

Let us now pick𝐻₂ ∈ 𝐶∞₀ (R) such that 0 ≤ 𝐻₂(𝜐) ≤ 1, 𝐻₂(𝜐) = 1 for |𝜐| ≤ 𝑘₂and𝐻₂(𝜐) = 0 for |𝜐| ≥ 2𝑘₂, where 𝑘₂ > 1 is a constant which we will choose big enough (see below). We can then write

(𝜆) = 𝑂 (𝜆−∞) + ∫ 𝑒𝑖Φ(𝑡;𝑥,𝜉;𝜆)_{𝜓 (𝑡) 𝐻} 1(⟨𝑥⟩ ⟨𝜉⟩ 𝑚 𝜆 ) × 𝐻₂(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨)𝑎(𝑡;𝑥,𝜉)𝑑𝑡L𝜉𝑑𝑥 + ∫ 𝑒𝑖Φ(𝑡;𝑥,𝜉;𝜆)𝜓 (𝑡) 𝐻1(⟨𝑥⟩ ⟨𝜉⟩ 𝑚 𝜆 ) × [1 − 𝐻₂(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨)]𝑎(𝑡;𝑥,𝜉)𝑑𝑡L𝜉𝑑𝑥 = 𝑂 (𝜆−∞) + 𝐼1(𝜆) + 𝐼2(𝜆) . (54)

In what follows, we will systematically use the notation 𝑆𝑟,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 order𝑟, 𝜌 in 𝑦, 𝜂. In a similar fashion, 𝑆𝑟,𝜌_𝑇 = 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₊,𝐷_𝑡𝑠𝐶∞((−𝑇, 𝑇), 𝑆𝑟,𝜌(𝑦, 𝜂)) satisfies SG-type estimates of order𝑟, 𝜌 in 𝑦, 𝜂, uniformly with respect to𝑡 ∈ (−𝑇, 𝑇).

To estimate 𝐼₁(𝜆), we will apply the stationary phase theorem. We begin by rewriting the integral𝐼₁(𝜆), using the fact that𝜑 is solution of the eikonal equation associated with 𝑞 and that 𝑞 is a classical SG-symbol. Note that then 𝜕_𝑡2𝜑 ∈ 𝐶∞((−𝑇, 𝑇), 𝑆2𝑚−1,1_cl (R𝑛)) ⊆ 𝐶∞((−𝑇, 𝑇), 𝑆𝑚,1_cl (R𝑛)), since 𝜕_𝑡2𝜑 (𝑡; 𝑥, 𝜉) =∑𝑛 𝑖=1 (𝜕_𝜉𝑖𝑞) (𝑥, 𝑑_𝑥𝜑 (𝑡; 𝑥, 𝜉)) × 𝜕_𝑥𝑖(𝑞 (𝑥, 𝑑_𝑥𝜑 (𝑡; 𝑥, 𝜉))) . (55)

In view of the Taylor expansion of𝜑 at 𝑡 = 0, recalling the property𝑞(𝑥, 𝜉) = 𝜔(𝑥)𝑞_𝑒(𝑥, 𝜉) + 𝑆𝑚,0(𝑥, 𝜉), 𝜔 a fixed 0-excision function, we have, for some0 < 𝛿₁< 1,

Φ (𝑡; 𝑥, 𝜉; 𝜆) = −𝜆𝑡 − 𝑥𝜉 + 𝜑 (0; 𝑥, 𝜉) + 𝑡𝜕𝑡𝜑 (0; 𝑥, 𝜉) +𝑡2 2 𝜕𝑡2𝜑 (𝑡𝛿1; 𝑥, 𝜉) = −𝜆𝑡 + 𝑡𝑞 (𝑥, 𝜉) + 𝑡2𝑆2𝑚−1,1_𝑇 (𝑥, 𝜉) = −𝜆𝑡 + 𝑡𝜔 (𝑥) 𝑞𝑒(𝑥, 𝜉) + 𝑡𝑆𝑚,0(𝑥, 𝜉) + 𝑡2𝑆2𝑚−1,1_𝑇 (𝑥, 𝜉) = −𝜆𝑡 + 𝑡𝜔 (𝑥) 𝑞𝑒(𝑥, 𝜉) + 𝑡𝑆𝑚,0(𝑥, 𝜉) + 𝑡2𝜔 (𝑥) 𝑆2𝑚−1,1𝑇,𝑒 (𝑥, 𝜉) + 𝑡2𝑆2𝑚−1,0𝑇 (𝑥, 𝜉) , (56) 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

󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨 ≺ 1, ⟨𝑥⟩ ⟨𝜉⟩𝑚∼ 𝜆 󳨐⇒ ⟨𝑥⟩ ∼ 𝜆, (57) which of course implies⟨𝑥⟩∼ |𝑥|, provided that 𝜆₀ ≤ 𝜆 is large enough. Moreover, by the ellipticity of𝑞, writing 𝑥 = |𝑥|𝜍, 𝜍 ∈ S𝑛−1_{, with the constant𝐴 > 1 of (48),}

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) 𝑞} 𝑒(𝜍, 𝜉) +𝑆𝑚,0(𝑥, 𝜉) |𝑥| ≤ 𝐴⟨ 𝑥⟩ |𝑥|⟨𝜉⟩ 𝑚 󳨐⇒ 𝐴−1⟨𝜉⟩𝑚≤ 𝑞_𝑒(𝜍, 𝜉) ≤ 𝐴⟨𝜉⟩𝑚, 𝜍 ∈ S𝑛−1, 𝜉 ∈ R𝑛 (58)

taking the limit for|𝑥| → +∞. Then, setting 𝑥 = 𝜆𝜁𝜍, 𝜁 ∈ [0, +∞), 𝜍 ∈ S𝑛−1_{,𝜆 ≥ 𝜆}

0≫ 1, in 𝐼1(𝜆), by homogeneity and

the previous remarks, we can write

Φ (𝑡; 𝜆𝜁𝜍, 𝜉; 𝜆) = −𝜆𝑡 + 𝑡𝜔 (𝜆𝜁𝜍) 𝑞𝑒(𝜆𝜁𝜍, 𝜉) + 𝑡𝑆𝑚,0(𝜆𝜁𝜍, 𝜉) + 𝑡2𝜔 (𝜆𝜁𝜍) 𝑆2𝑚−1,1_𝑇,𝑒 (𝜆𝜁𝜍, 𝜉) + 𝑡2𝑆2𝑚−1,0_𝑇 (𝜆𝜁𝜍, 𝜉) = −𝜆𝑡 + 𝜆𝜁𝑡𝑞𝑒(𝜍, 𝜉) + 𝜆𝜁𝑡2𝑆2𝑚−1,1𝑇,𝑒 (𝜍, 𝜉) + 𝑡𝑆𝑚,0_{(𝜆𝜁𝜍, 𝜉) + 𝑡}2_𝑆2𝑚−1,0 𝑇 (𝜆𝜁𝜍, 𝜉) = 𝜆 [−𝑡 + 𝜁𝑡𝑞_𝑒(𝜍, 𝜉) + 𝜁𝑡2𝑆_𝑇,𝑒2𝑚−1,1(𝜍, 𝜉)] + 𝐺₁(𝜆; 𝑡, 𝜁; 𝜍, 𝜉) = 𝜆𝐹₁(𝑡, 𝜁; 𝜍, 𝜉) + 𝐺1(𝜆; 𝑡, 𝜁; 𝜍, 𝜉) , (59) and find, in view of the compactness of the support of the integrand (see the proof ofProposition 25 below) and the hypotheses 𝐼₁(𝜆) = 𝜆𝑛∫ 𝑒𝑖𝜆𝐹1(𝑡,𝜁;𝜍,𝜉)_𝑒𝑖𝐺1(𝜆;𝑡,𝜁;𝜍,𝜉)_{𝜓 (𝑡)} × 𝑎 (𝑡; 𝜆𝜁𝜍, 𝜉) 𝐻1(⟨𝜆𝜁⟩ ⟨𝜉⟩ 𝑚 𝜆 ) × 𝐻₂(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨)𝜁𝑛−1_{𝑑𝑡𝑑𝜁L𝜉𝑑𝜍} = 𝜆𝑛 (2𝜋)𝑛∫ 𝑒𝑖𝜆𝐹1(𝑋,𝑌)𝑈1(𝑋, 𝑌; 𝜆) 𝑑𝑋𝑑𝑌, (60)

Proposition 25. Choosing the constants 𝑘₁, 𝜆₀ > 1 large

enough and𝑇 > 0 suitably small, one has, for any 𝑘₂ > 1 and

for a certain sequence𝑐_𝑗,𝑗 = 0, 1, . . .,

𝐼₁(𝜆) ∼+∞∑

𝑗=0

𝑐_𝑗𝜆𝑛−1−𝑗, (61)

that is,𝐼₁(𝜆) = 𝑐₀𝜆𝑛−1+ 𝑂(𝜆𝑛−2), with

𝑐₀= 1

(2𝜋)𝑛−1∫_R𝑛∫_S𝑛−1

𝐻₂(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨)

𝑞_𝑒(𝜍, 𝜉)𝑛𝑑𝜍𝑑𝜉. (62)

Proof. It is easy to see that, on the support of𝑈₁, the phase

function𝐹₁(𝑋, 𝑌) admits a unique, nondegenerate, stationary point𝑋₀ = 𝑋₀(𝑌) = (0, 𝑞_𝑒(𝜍, 𝜉)−1), that is, 𝐹_1,𝑋󸀠 (𝑋₀(𝑌), 𝑌) = 0 for all 𝑌 such that (𝑋, 𝑌) ∈ supp 𝑈1, provided that𝑇 >

0 is chosen suitably small (see, e.g., [25, page 136]), and the Hessian det(𝐹󸀠󸀠

1,𝑋(𝑋0(𝑌), 𝑌)) equals −𝑞𝑒(𝜍, 𝜉)2< 0. Moreover,

the amplitude function

𝑈₁(𝑋, 𝑌; 𝜆) = 𝜓 (𝑡) 𝐻₁(⟨𝜆𝜁⟩ ⟨𝜉⟩

𝑚

𝜆 ) 𝐻2(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨) × 𝑎 (𝑡; 𝜆𝜁𝜍, 𝜉) 𝜁𝑛−1𝑒𝑖𝐺(𝜆;𝑡,𝜁;𝜍,𝜉)

(63)

is compactly supported with respect to the variables𝑋 and 𝑌 and satisfies, for all𝛾 ∈ Z2₊,

𝐷𝛾_𝑋𝑈1(𝑋, 𝑌; 𝜆) ≺ 1 (64)

for all𝑋, 𝑌, 𝜆 ≥ 𝜆₀. In fact,

(1)𝜓 ∈ 𝐶₀∞((−𝑇, 𝑇)), 𝜍 ∈ S𝑛−1, supp[𝐻₂(|𝜉|)] ⊆ {𝜉 : |𝜉| ≤ 2𝑘₂}, and (2𝑘₁)−1≤ ⟨𝜉⟩𝑚_{√ 1} 𝜆2 + 𝜁2≤ 2𝑘1 󳨐⇒ 0 < √ 1 4𝑘2 1⟨2𝑘2⟩2𝑚 − 1 𝜆2 0 ≤ 𝜁 ≤ 2𝑘1, (65) where𝜆₀> 2𝑘₁⟨2𝑘₂⟩𝑚;

(2) all the factors appearing in the expression of 𝑈₁ are uniformly bounded, together with all their 𝑋-derivatives, for𝑋 ∈ 𝑆_𝑋 = supp 𝜓 × [𝜁₀, 𝜁₁], 𝑌 ∈ 𝑆_𝑌= S𝑛−1_{× {𝜉 : |𝜉| ≤ 2𝑘}

2}, and 𝜆 ≥ 𝜆0.

Of course, (2) trivially holds for the cutoff functions 𝜓(𝑡) and 𝐻₂(|𝜉|), and for the factor 𝜁𝑛−1_{. Since 𝑎(𝑡; 𝑥, 𝜉) ∈}

𝑆0,0_𝑇 (𝑥, 𝜉), on 𝑆𝑋× 𝑆𝑌we have, for all𝛾 ∈ Z2+and𝜆 ≥ 𝜆0> 1,

𝐷𝛾_𝑋𝑎 (𝑡; 𝜆𝜁𝜍, 𝜉) ≺ ⟨𝜆𝜁⟩−𝛾2_𝜆𝛾2_⟨𝜉⟩𝑚 _≺ 1

((1/𝜆2_{) + 𝜁}2₎𝛾2/2

< 1 𝜁𝛾2 ≺ 1.

(66)

Moreover, since𝐺₁∈ 𝑆𝑚,0_𝑇 (𝑥, 𝜉) is actually in 𝑆−∞,0_𝑇 (𝑥, 𝜉) ⊂ 𝑆0,0_𝑇 (𝑥, 𝜉) on 𝑆𝑋 × 𝑆𝑌, the same holds for exp(𝑖𝐺1), by an

application of the Fa`a di Bruno formula for the derivatives of compositions of functions, so also this factor fulfills the desired estimates. Finally, another straightforward computa-tion shows that, for all𝛾₂∈ Z₊and𝜆 ≥ 𝜆₀> 1,

𝐷𝛾2

𝜁𝐻1(⟨𝜆𝜁⟩ ⟨𝜉⟩ 𝑚

𝜆 ) ≺ 1 (67)

on 𝑆_𝑋 × 𝑆_𝑌. 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}

(𝐹_1,𝑋󸀠󸀠 (𝑋₀(𝑌), 𝑌))|−1/2𝑈₁(𝑋₀(𝑌), 𝑌; 𝜆), that is 𝐼1(𝜆) = 𝜆 𝑛−1 (2𝜋)𝑛−1∫R𝑛∫_S𝑛−1 1 𝑞_𝑒(𝜍, 𝜉) 𝜓 (0) × 𝐻₁(⟨𝜆/𝑞𝑒(𝜍, 𝜉)⟩ ⟨𝜉⟩ 𝑚 𝜆 ) × 𝐻2(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨) 𝑞_𝑒(𝜍, 𝜉)𝑛−1𝑎 (0; 𝜆𝜍 𝑞_𝑒(𝜍, 𝜉), 𝜉) 𝑑𝜍𝑑𝜉 + 𝑂 (𝜆𝑛−2) = 𝜆𝑛−1 (2𝜋)𝑛−1∫_R𝑛∫_S𝑛−1𝐻1( ⟨𝜆/𝑞_𝑒(𝜍, 𝜉)⟩ ⟨𝜉⟩𝑚 𝜆 ) × 𝐻2(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨) 𝑞_𝑒(𝜍, 𝜉)𝑛𝑑𝜍𝑑𝜉 + 𝑂 (𝜆𝑛−2) = 𝜆𝑛−1 (2𝜋)𝑛−1∫_R𝑛∫_S𝑛−1 𝐻₂(󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨) 𝑞_𝑒(𝜍, 𝜉)𝑛𝑑𝜍𝑑𝜉 + 𝑂 (𝜆𝑛−2) , (68) recalling that𝜓(0) = 1, 𝑎(0; 𝑥, 𝜉) = 1 for all 𝑥, 𝜉 ∈ R𝑛.

Indeed, having chosen𝑘₁ > 2𝐴, 𝜆₀ > 2𝑘₁⟨2𝑘₂⟩𝑚, (58) implies 𝑘−1₁ < 𝐴−1 < ⟨𝜆/𝑞𝑒(𝜍, 𝜉)⟩ ⟨𝜉⟩ 𝑚 𝜆 = √(⟨𝜉⟩𝑚 𝜆 ) 2 + ( ⟨𝜉⟩𝑚 𝑞_𝑒(𝜍, 𝜉)) 2 < √_4𝑘1₂ 1 + 𝐴2_{< 𝑘} 1, (69)

uniformly for𝜍 ∈ 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 Theorem7.7.6 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 factor1 − 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)𝑞_𝜓(𝑥, 𝜉) + 𝑆𝑚−1,1_{(𝑥, 𝜉) = 𝑞}

𝜓(𝑥, 𝜉) +

𝑆𝑚−1,1_{(𝑥, 𝜉). Then, with some 0 < 𝛿}

2< 1, 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) + 𝑡2𝑆2𝑚−1,1_𝑇 (𝑥, 𝜉) = −𝜆𝑡 + 𝑡𝑞_𝜓(𝑥, 𝜉) + 𝑡𝑆𝑚−1,1(𝑥, 𝜉) + 𝑡2𝑆2𝑚−1,1_𝑇 (𝑥, 𝜉) . (70) Setting𝜉 = (𝜆𝜁)1/𝑚𝜍, 𝜁 ∈ [0, +∞), 𝜍 ∈ S𝑛−1,𝜆 ≥ 𝜆₀, we can rewrite𝐼₂(𝜆) as 𝐼₂(𝜆) = _𝑚𝑛_(2𝜋)𝜆𝑛/𝑚_𝑛 × ∫ 𝑒𝑖𝜆(−𝑡+𝜁𝑡𝑞𝜓(𝑥,𝜍)+𝑡𝜆−1𝑆𝑚−1,1(𝑥,(𝜆𝜁)1/𝑚𝜍)+𝑡2𝜆−1𝑆2𝑚−1,1𝑇 (𝑥,(𝜆𝜁)1/𝑚𝜍)) × 𝜓 (𝑡) 𝑎 (𝑡; 𝑥, (𝜆𝜁)1/𝑚𝜍) × 𝐻₁(⟨𝑥⟩ ⟨(𝜆𝜁) 1/𝑚_𝜍⟩𝑚 𝜆 ) × [1 − 𝐻₂((𝜆𝜁)1/𝑚)] 𝜁(𝑛/𝑚)−1𝑑𝑡𝑑𝜁𝑑𝜍𝑑𝑥 = _𝑚𝑛_(2𝜋)𝜆𝑛/𝑚_𝑛∫ 𝑒𝑖𝜆𝐹2(𝑋,𝑌;𝜆)_𝑈 2(𝑋, 𝑌; 𝜆) 𝑑𝑋𝑑𝑌, (71) 𝑋 = (𝑡, 𝜁), 𝑌 = (𝜍, 𝑥), where we have set

𝐹₂(𝑋, 𝑌; 𝜆) = − 𝑡 + 𝜁𝑡𝑞_𝜓(𝑥, 𝜍) + 𝑡𝜆−1𝑆𝑚−1,1(𝑥, (𝜆𝜁)1/𝑚𝜍) + 𝑡2𝜆−1𝑆2𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚𝜍) 𝑈₂(𝑋, 𝑌; 𝜆) = 𝜓 (𝑡) 𝐻₁(⟨𝑥⟩ ⟨(𝜆𝜁) 1/𝑚_𝜍⟩𝑚 𝜆 ) × [1 − 𝐻₂((𝜆𝜁)1/𝑚)] × 𝑎 (𝑡; 𝑥, (𝜆𝜁)1/𝑚𝜍) 𝜁(𝑛/𝑚)−1. (72) On the support of𝑈₂, we have

⟨𝑥⟩ ⟨(𝜆𝜁)1/𝑚𝜍⟩𝑚 𝜆 ∼ 1, (𝜆𝜁)1/𝑚≻ 1 󳨐⇒ ⟨(𝜆𝜁)1/𝑚𝜍⟩𝑚= ⟨(𝜆𝜁)1/𝑚⟩𝑚∼ 𝜆𝜁, (73) so that ⟨𝑥⟩ 𝜆𝜁 𝜆 ∼ 1 ⇐⇒ 𝜁 ∼ ⟨𝑥⟩−1, |𝑥| < ⟨𝑥⟩ ≤ 2𝑘₁(𝑘₂)−𝑚𝜆 = ̃𝜘𝜆. (74)

For any fixed𝑌 ∈ S𝑛−1× R𝑛, we then have𝑋 belonging to a compact set, uniformly with respect to𝜆 ≥ 𝜆₀, say supp𝜓 × [𝑐−1_⟨𝑥⟩−1_{, 𝑐⟨𝑥⟩}−1_{], for a suitable 𝑐 > 1.}

Remark 26. Incidentally, we observe that a rough estimate of

𝜆𝑛/𝑚_𝐼 2(𝜆) is ∫ 𝑒𝑖𝜆𝐹2(𝑋,𝑌;𝜆)𝑈 2(𝑋, 𝑌; 𝜆) 𝑑𝑋 ≺ ⟨𝑥⟩−(𝑛/𝑚)+1∫𝑐⟨𝑥⟩ −1 𝑐−1_⟨𝑥⟩−1𝑑𝜁 ≺ ⟨𝑥⟩ −𝑛/𝑚 󳨐⇒ 𝜆𝑛/𝑚∫ 𝑒𝑖𝜆𝐹2(𝑋,𝑌;𝜆)_𝑈 2(𝑋, 𝑌; 𝜆) 𝑑𝑋𝑑𝑌 ≺ 𝜆𝑛, 𝜆 󳨀→ +∞. (75)

An even less precise result would be the bound𝜆𝑛/𝑚, using the convergence of the integral with respect to𝑥 in the whole R𝑛_{, given by−(𝑛/𝑚) + 𝑛 < 0.}

The next lemma is immediate, and we omit the proof.

Lemma 27. 𝑆𝑠,𝜎

𝑇 (𝑥, (𝜆𝜁)1/𝑚𝜍) = 𝑆𝑇𝑠,𝜎(𝑥, (𝜆𝜁)1/𝑚) for any 𝜁 ∈

[0, +∞), 𝑥 ∈ R𝑛,𝜍 ∈ S𝑛−1,𝜆 ≥ 𝜆₀,𝑚 ∈ (0, 1), and, for all

𝛾 ∈ Z2

+,

𝐷𝛾_𝑋𝑆𝑠,𝜎_𝑇 (𝑥, (𝜆𝜁)1/𝑚) = 𝜁−𝛾2_𝑆𝑠,𝜎

𝑇 (𝑥, (𝜆𝜁)1/𝑚) . (76)

The main result of this section is as follows.

Proposition 28. If 𝑘₁, 𝑘₂, 𝜆₀> 1 are chosen large enough, one

has 𝐼₂(𝜆) = 𝑛 𝑚 𝑑0𝜆(𝑛/𝑚)−1+ 𝑂 (𝜆𝑛−1) + 𝑂 (𝜆(𝑛/𝑚)−2) . (77) Explicitly, 𝑑0=_(2𝜋)1𝑛−1∫_R𝑛∫_S𝑛−1 1 𝑞_𝜓(𝑥, 𝜍)𝑛/𝑚 𝑑𝜍𝑑𝑥. (78) We will prove Proposition 28through various interme-diate steps. First of all, arguing as in the proof of (58), exchanging the role of𝑥 and 𝜉, we note that, for all 𝑥 ∈ R𝑛, 𝜍 ∈ S𝑛−1_,

𝐴−1⟨𝑥⟩ ≤ 𝑞𝜓(𝑥, 𝜍) ≤ 𝐴 ⟨𝑥⟩ , (79)

𝐹_2,𝑋󸀠 (𝑋, 𝑌; 𝜆) = (𝜕𝑡𝐹2(𝑋, 𝑌; 𝜆) 𝜕𝜁𝐹2(𝑋, 𝑌; 𝜆) ) = ( −1 + 𝜁 𝜁₀ + 𝜆−1𝑆𝑚−1,1(𝑥, (𝜆𝜁)1/𝑚) + 𝑡𝜆−1𝑆2𝑚−1,1𝑇 (𝑥, (𝜆𝜁)1/𝑚) 𝑡 (𝑞_𝜓(𝑥, 𝜍) + 𝜆−1𝜁−1_𝑆𝑚−1,1_{(𝑥, (𝜆𝜁)}1/𝑚_{) +𝑡𝜆}−1_𝜁−1_𝑆2𝑚−1,1 𝑇 (𝑥, (𝜆𝜁)1/𝑚)) ) , (80) 𝑋 = (𝑡, 𝜁) ∈ 𝑆_𝑋 = supp 𝜓 × [𝑐−1_⟨𝑥⟩−1_{, 𝑐⟨𝑥⟩}−1_{], 𝑌 =} (𝜍, 𝑥) ∈ 𝑆_𝑌 = S𝑛−1 _{× R}𝑛_{, 𝜆 ≥ 𝜆}

0, where we have used

Lemma 27. By the symbolic calculus, remembering that𝜆𝜁 ≥ 𝑘𝑚

2 > 1 on supp 𝑈2, we can rewrite the expressions mentioned

previously as 𝜕_𝑡𝐹₂(𝑋, 𝑌; 𝜆) = − 1 +_𝜁𝜁 0 + 𝜁(𝜆𝜁) −1_𝑆𝑚−1,1_{(𝑥, (𝜆𝜁)}1/𝑚₎ + 𝑡𝜁(𝜆𝜁)−1𝑆2𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚) = − 1 +_𝜁𝜁 0 + 𝜁[(𝜆𝜁) 1/𝑚_]−𝑚_𝑆𝑚−1,1_{(𝑥, (𝜆𝜁)}1/𝑚₎ + 𝑡𝜁[(𝜆𝜁)1/𝑚]−𝑚𝑆2𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚) = − 1 + 𝜁 𝜁₀ + 𝜁𝑆−1,1(𝑥, (𝜆𝜁)1/𝑚) + 𝑡𝜁𝑆𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚) , 𝜕_𝜁𝐹₂(𝑋, 𝑌; 𝜆) = 𝑡 (𝑞𝜓(𝑥, 𝜍) + 𝑆−1,1(𝑥, (𝜆𝜁)1/𝑚) +𝑡𝑆𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚)) . (81)

It is clear that𝜁 ∼ ⟨𝑥⟩−1 implies𝜁𝑆−1,1(𝑥, (𝜆𝜁)1/𝑚) = 𝑆−1,0 (𝑥, (𝜆𝜁)1/𝑚) and 𝜁𝑆𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚) = 𝑆𝑚−1,0_𝑇 (𝑥, (𝜆𝜁)1/𝑚), so that we finally have

𝜕_𝑡𝐹₂(𝑋, 𝑌; 𝜆) = − 1 + 𝜁 𝜁₀ + 𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚) + 𝑡𝑆𝑚−1,0_𝑇 (𝑥, (𝜆𝜁)1/𝑚) , 𝜕𝜁𝐹2(𝑋, 𝑌; 𝜆) = 𝑡 (𝑞𝜓(𝑥, 𝜍) + 𝑆−1,1(𝑥, (𝜆𝜁)1/𝑚) + 𝑡𝑆𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚)) . (82)

We now prove that, modulo an𝑂(|𝜆|−∞) term, we can consider an amplitude such that, on its support, the ration 𝜁/𝜁0is very close to1. To this aim, take 𝐻3∈ 𝐶∞0 (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 𝑉₁(𝑋, 𝑌; 𝜆) = 𝑈₂(𝑋, 𝑌; 𝜆) ⋅ [1 − 𝐻₃(_𝜁𝜁 0 − 1)] , 𝑉₂(𝑋, 𝑌; 𝜆) = 𝑈₂(𝑋, 𝑌; 𝜆) ⋅ 𝐻₃(_𝜁𝜁 0 − 1) , 𝐽₁(𝜆) = ∫ 𝑒𝑖𝜆𝐹2(𝑋,𝑌;𝜆)𝑉 1(𝑋, 𝑌; 𝜆) 𝑑𝑋𝑑𝑌, 𝐽₂(𝜆) = ∫ 𝑒𝑖𝜆𝐹2(𝑋,𝑌;𝜆)_𝑉 2(𝑋, 𝑌; 𝜆) 𝑑𝑋𝑑𝑌. (83)

Proposition 29. With the choices of 𝑇, 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

choose𝑘₂> 1 so large that, for an arbitrarily fixed 𝜀 ∈ (0, 1/2), for any𝜆 ≥ 𝜆₀,𝜁 ∈ (0, +∞) satisfying |𝜉| = (𝜆𝜁)1/𝑚≥ 𝑘₂,

in𝜕_𝑡𝐹₂(𝑋, 𝑌; 𝜆) , 󵄨󵄨󵄨󵄨 󵄨𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚)󵄨󵄨󵄨󵄨󵄨 ≤ ₂𝜀, 󵄨󵄨󵄨󵄨 󵄨𝑡𝑆𝑇𝑚−1,0(𝑥, (𝜆𝜁)1/𝑚)󵄨󵄨󵄨󵄨󵄨 ≤ 𝜀₂, 󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨𝜁0_𝑑𝜁𝑑 𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚)󵄨󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨 󵄨󵄨󵄨󵄨_󵄨𝜁0𝜁−1𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚))󵄨󵄨󵄨󵄨󵄨 ≤ 𝑘₀< 1, (84a) in𝜕_𝜁𝐹₂(𝑋, 𝑌; 𝜆) , 󵄨󵄨󵄨󵄨 󵄨𝑆−1,1(𝑥, (𝜆𝜁)1/𝑚) + 𝑡𝑆𝑚−1,1𝑇 (𝑥, (𝜆𝜁)1/𝑚)󵄨󵄨󵄨󵄨󵄨 ≤ 𝐴−1 2 ⟨𝑥⟩ , (84b)

uniformly with respect to(𝑋, 𝑌) ∈ 𝑆_𝑋× 𝑆_𝑌 ⊇ supp 𝑈₂(⋅; 𝜆). Then,𝐹₂ is nonstationary on supp𝑉₁, since there we have |(𝜁/𝜁₀) − 1| ≥ (3/2)𝜀, while

󵄨󵄨󵄨󵄨

󵄨𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚) + 𝑡𝑆𝑚−1,0𝑇 (𝑥, (𝜆𝜁)1/𝑚)󵄨󵄨󵄨󵄨󵄨 ≤ 𝜀, (85)

which implies𝜕_𝑡𝐹₂(𝑋, 𝑌; 𝜆) ≻ 1. Observing that, on supp 𝑉₁, 𝜕_𝑡𝐹₂(𝑋, 𝑌; 𝜆) = 𝑆0,0

𝑇 (𝑥, (𝜆𝜁)1/𝑚), as well as 𝑉1(𝑋, 𝑌; 𝜆) =

𝑆0,0_𝑇 (𝑥, (𝜆𝜁)1/𝑚), the assertion follows by repeated integrations by parts with respect to𝑡, using the operator

𝐿𝑡=_𝜆𝜕 1

𝑡𝐹2(𝑋, 𝑌; 𝜆)𝐷𝑡󳨐⇒ 𝐿1𝑒

𝑖𝜆𝐹2(𝑋,𝑌;𝜆)

= 𝑒𝑖𝜆𝐹2(𝑋,𝑌;𝜆)_,

(86)

and recallingRemark 26.

Proposition 30. With the choices of 𝜀, 𝑇 > 0, 𝑘₁, 𝑘₂, 𝜆₀ > 1,

one can assume, modulo an𝑂(𝜆𝑛−1) term, that the integral with

respect to𝑥 in 𝐽₂(𝜆) is extended to the set {𝑥 ∈ R𝑛 : ⟨𝑥⟩ ≤ 𝜘𝜆},

with

𝜘 = (1 − 𝜀₂) [𝐴(2𝑘2)𝑚]−1. (87)

Proof. Indeed if𝜘 < ̃𝜘 = 2𝑘₁⟨𝑘₂⟩−𝑚, we can split𝐽₂(𝜆) into

the sum ∫ 𝜘𝜆≤⟨𝑥⟩≤̃𝜘𝜆∫S𝑛−1∫ 𝑒 𝑖𝜆𝐹2_𝑉 2𝑑𝑋𝑑𝜍𝑑𝑥 + ∫ ⟨𝑥⟩≤𝜘𝜆∫S𝑛−1∫ 𝑒 𝑖𝜆𝐹2_𝑉 2𝑑𝑋𝑑𝜍𝑑𝑥, (88)

since the inequality𝜘 < ̃𝜘 is true when 𝑘₂is sufficiently large. Observing that, on supp𝑈₂,

⟨𝑥⟩ ∼ 𝜆 󳨐⇒ ⟨𝜉⟩𝑚= ⟨𝑥⟩ ⟨𝜉⟩

𝑚

𝜆 𝜆

⟨𝑥⟩∼ 1 󳨐⇒ 󵄨󵄨󵄨󵄨𝜉󵄨󵄨󵄨󵄨 ≤ 𝑘3, (89) switching back to the original variables, the first integral in (88) can be treated as 𝐼₁(𝜆), and gives, in view of

Proposition 25, an𝑂(𝜆𝑛−1) term, as stated.

Now we can show that 𝐹₂(𝑋, 𝑌; 𝜆) admits a unique, nondegenerate stationary point𝑋∗₀ = 𝑋∗₀(𝑌, 𝜆) belonging to supp𝑉₂for⟨𝑥⟩≤ 𝜘𝜆. Under the same hypotheses, 𝑋∗₀ lies in a circular neighbourhood of𝑋₀ = (0, 𝜁₀) = (0, 𝑞_𝜓(𝑥, 𝜍)−1) of arbitrarily small radius.

Proposition 31. With 𝜀 ∈ (0, 1/2), 𝑇 > 0, 𝑘1, 𝑘2, 𝜆0 > 1

fixed previously,𝐹_2,𝑋󸀠 (𝑋, 𝑌; 𝜆) vanishes on supp 𝑉₂only for𝑋 =

𝑋∗

0(𝑌; 𝜆) = (0, 𝜁0∗(𝑌; 𝜆)), that is, 𝐹2,𝑋󸀠 (𝑋∗0(𝑌; 𝜆), 𝑌; 𝜆) = 0 for all𝑌 such that (𝑋, 𝑌; 𝜆) ∈ supp 𝑉₂. Moreover,

det(𝐹_2,𝑋󸀠󸀠 (𝑋∗₀(𝑌; 𝜆) , 𝑌)) ∼ ⟨𝑥⟩2, 󵄨󵄨󵄨󵄨𝑋∗ 0(𝑌; 𝜆) − 𝑋0(𝑌)󵄨󵄨󵄨󵄨 =󵄨󵄨󵄨󵄨𝜁0∗(𝑌; 𝜆) − 𝜁0(𝑌)󵄨󵄨󵄨󵄨 ≤ 𝐴𝜀₂ ⟨𝑥⟩−1 (90) holds on supp𝑉₂.

Proof. We have to solve

0 = − 1 +_𝜁𝜁 0 + 𝑆 −1,0_{(𝑥, (𝜆𝜁)}1/𝑚₎ + 𝑡𝑆𝑚−1,0_𝑇 (𝑥, (𝜆𝜁)1/𝑚) 0 = 𝑡 (𝑞_𝜓(𝑥, 𝜍) + 𝑆−1,1(𝑥, (𝜆𝜁)1/𝑚) +𝑡𝑆𝑚−1,1_𝑇 (𝑥, (𝜆𝜁)1/𝑚)) , (91) (𝑋, 𝑌; 𝜆) ∈ supp 𝑉₂. By (79) and (84a) and (84b), with the choices of𝜀, 𝑇 > 0, 𝑘₁, 𝑘₂, 𝜆₀, the coefficient of𝑡 in the second equation does not vanish at any point of supp𝑉₂. Then𝑡 = 0, and𝜁 must satisfy

− 1 + 𝜁

𝜁₀ + 𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚) = 0

⇐⇒ 𝜁 = 𝜁0(1 + 𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚))

= 𝐺 (𝜁; 𝑌; 𝜆) .

(92)

Since, by the choice of𝑘₂,|𝜕_𝜁𝐺(𝜁; 𝑌; 𝜆)| ≤ 𝑘₀< 1, uniformly with respect to𝑌 ∈ S𝑛−1× {𝑥 ∈ R𝑛 : ⟨𝑥⟩ ≤ 𝜘𝜆}, 𝜆 ≥ 𝜆₀,𝐺 has a unique fixed point𝜁∗₀ = 𝜁₀∗(𝑌; 𝜆), smoothly depending on the parameters; see the Appendix for more details. Since

𝜕2_𝑡𝐹₂(𝑋, 𝑌; 𝜆) = 𝑆𝑚−1,0_𝑇 (𝑥, (𝜆𝜁)1/𝑚) , 𝜕_𝑡𝜕_𝜁𝐹₂(𝑋, 𝑌; 𝜆) = 𝑞𝜓(𝑥, 𝜍) (1 + 𝜁0𝜁−1(𝑆−1,0(𝑥, (𝜆𝜁)1/𝑚) +𝑡𝑆𝑚−1,0 𝑇 (𝑥, (𝜆𝜁)1/𝑚))) , 𝜕2_𝜁𝐹₂(𝑋, 𝑌; 𝜆) = 𝑡𝜁−1(𝑆−1,1(𝑥, (𝜆𝜁)1/𝑚) +𝑡𝑆_𝑇𝑚−1,1(𝑥, (𝜆𝜁)1/𝑚)) , (93)

we can assume that𝜆𝜁 ≥ 𝑘𝑚₂ and the choices of the other parameters imply, on supp𝑉₂,

𝜕_𝑡2𝐹₂(𝑋, 𝑌; 𝜆) ≺ 𝜀₂, 𝜕_𝑡𝜕_𝜁𝐹₂(𝑋, 𝑌; 𝜆) ∼ ⟨𝑥⟩ , 𝜕_𝜁2𝐹₂(𝑋, 𝑌; 𝜆) ≺ 𝜀

2⟨𝑥⟩2.

(94)