Fourier-integral operator approximation of solutions to first-order hyperbolic pseudodifferential equations II: microlocal analysis

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Fourier-integral operator approximation of solutions to first-order hyperbolic pseudodifferential equations II:

microlocal analysis

Jérôme Le Rousseau, Gunther Hormann

To cite this version:

Jérôme Le Rousseau, Gunther Hormann. Fourier-integral operator approximation of solutions to first-

order hyperbolic pseudodifferential equations II: microlocal analysis. Journal de Mathématiques Pures

et Appliquées, Elsevier, 2006, 86, pp.403-426. �10.1016/j.matpur.2006.08.004�. �hal-00005759�

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Fourier-integral-operator approximation of solutions to first-order hyperbolic

pseudodifferential equations II: microlocal analysis

J´erˆome Le Rousseau

^∗

Laboratoire d’Analyse, Topologie, Probabilit´es, CNRS UMR 6632 Universit´e Aix-Marseille I, 39 rue Joliot-Curie

13453 Marseille , France

tel: (33) 4 91 11 36 41, fax: (33) 4 91 11 35 52 jlerous@cmi.univ-mrs.fr

and

G¨ unther H¨ormann

^†‡

Fakult¨at f¨ ur Mathematik Universit¨at Wien, Nordbergstrae 15

A-1090 Wien,Austria

tel: (43) 1 4277 50680, fax: (43) 1 4277 50620 guenther.hoermann@univie.ac.at

June 28, 2005

Abstract

An approximation Ansatz for the operator solution,U(z⁰, z), of a hyperbolic first-order pseudodifferential equation,∂z+a(z, x, Dx) with Re(a)≥ 0, is constructed as the composition of global Fourier integral operators with complex phases. We investigate the propagation of singularities for this Ansatz and prove microlocal convergence: the wavefront set of the approximated solution is shown to converge to that of the exact solution away from the region where the phase is complex.

AMS 2000 subject classification: 35L05, 35L80, 35A21, 35S10, 35S30, 86A15

∗Corresponding author.

†Supported by FWF grants P16820-N04 and START-Project No Y237.

‡Part of this work was done while G. H¨ormann was visiting professor at LATP.

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0 Introduction

We consider the Cauchy problem

∂zu+a(z, x, Dx)u= 0, 0< z≤Z (0.1)

u|z=0=u0, (0.2)

with Z > 0 and a(z, x, ξ) ∈ C([0, Z], S¹(Rⁿ×Rⁿ)), with the usual notation D = ¹_i∂. Further assumptions will be made on the symbol a(z, x, ξ). When a(z, x, ξ) is independent of x and z it is natural to treat such a problem by means of Fourier transformation:

u(z, x⁰) = Z Z

exp[ihx⁰−x|ξi −za(ξ)]u0(x)d⁻ξ dx,

where d⁻ξ:=dξ/(2π)ⁿ. For this to be well defined for allu0∈S(Rⁿ) we shall impose the real part of the principal symbol ofato be non-negative. When the symboladepends on bothxand zwe can naively expect

u(z, x⁰)≈u1(z, x⁰) :=

ZZ

exp[ihx⁰−x|ξi −za(0, x⁰, ξ)]u0(x)d⁻ξ dx, forz small, and hence approximately solve the Cauchy problem (0.1)–(0.2) for z ∈[0, z⁽¹⁾] withz⁽¹⁾ small. If we want to progress in the z direction we have to solve the Cauchy problem

∂zu+a(z, x, Dx)u= 0, z⁽¹⁾< z≤Z u(z, .)|_z=z⁽¹⁾ =u1(z⁽¹⁾, .), which we again approximatively solve by

u(z, x⁰)≈u2(z, x⁰) :=

ZZ

exp[ihx⁰−x|ξi −(z−z⁽¹⁾)a(z⁽¹⁾, x⁰, ξ)]u1(z⁽¹⁾, x)d⁻ξ dx.

This procedure can be iterated until we reachz=Z.

If we denote byG_(z⁰_,z)the operator with kernel G(z⁰,z)(x⁰, x) =

Z

exp[ihx⁰−x|ξi] exp[−(z⁰−z)a(z, x⁰, ξ)]d⁻ξ,

then combining all iteration steps above involves composition of such operators:

let 0≤z⁽¹⁾≤ · · · ≤z^(k)≤Z, we then have

uk+1(z, x) =G_(z,z(k))◦ G_(z(k),z^(k−1))◦ · · · ◦ G_(z(1),0)(u0)(x),

if z ≥ z^(k). We then define the operator WP,z for a subdivision P of [0, Z], P={z⁽⁰⁾, z⁽¹⁾, . . . , z^(N)}with 0 =z⁽⁰⁾< z⁽¹⁾<· · ·< z^(N)=Z,

WP,z:=







G_(z,0) if 0≤z≤z⁽¹⁾,

G_(z,z^(k)₎ Y1 i=k

G_(z⁽ⁱ⁾_,z⁽ⁱ⁻¹⁾₎ ifz^(k)≤z≤z^(k+1).

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According to the procedure described aboveWP,z(u0) yields an approximation Ansatz for the solution to the Cauchy problem (0.1)–(0.2) with step size ∆P= supi=1,...,N(zi−zi−1). The operatorG_(z⁰_,z) is often referred to as thethin-slab propagator (see e.g. [3, 2]).

The approximation Ansatz proposed here is a tool to compute approximations of the exact solution to the Cauchy problem (0.1)–(0.2). Such computations in application to geophysical problems have been used in [3]. In exploration seismology one is confronted with solving equations of the type

(∂z−ib(z, x, Dt, Dx) +c(z, x, Dt, Dx))v= 0, (0.3)

v(0, .) =v0(.), (0.4)

where t is time, z is the vertical coordinate and x is the lateral or transverse coordinate; b andc are first-order pseudodifferential operators, with real principal partsb1andc1, wherec1(z, x, τ, ξ) is non-negative. Note that the Cauchy problem (0.1)–(0.2) studied here is more general. The problem (0.3)–(0.4) is obtained in geophysics by a (microlocal) decoupling of the up-going and down- going wavefields in the acoustic wave equation (see Appendix A in [15] and [19]

for details). In practice, the proposed Ansatz can then be a tool to approximate the exact solution for the purpose of imaging the Earth’s interior [3, 2]. As explained in [15, Appendix A] the operatorc acts as a damping term that sup- presses singularities in the microlocal region where its symbol does not vanishes.

We show that this effect is recovered in the proposed Ansatz.

Seismic imaging aims at recovering the singularities in the subsurface (see for instance [23, 1]). Thus, seismologists are not only interested in the convergence of this Ansatz to the exact solution of the Cauchy problem (0.3)–(0.4) but they also expect the wavefront set of the approximate solution to be close, in some sense, to that of the exact solution. Therefore, we investigate the microlocal properties of the proposed Ansatz and show how the results presented here and those of [15] can be applied to seismic imaging.

In the present paper, the operatorsG(z⁰,z)and WP,z are frequently considered as Fourier integral operators (FIO) with complex phase. They could be considered as FIO with real phase but with amplitude of type ¹₂ (see [15] and below).

However, the wavefront set and the damping effect of the real part of the principal part ofa(x, ξ) would not be recovered in the same way. We follow here the terminology introduced in [10, Sections 25.4-5] for FIOs with complex phases.

We state our main results which are proved in the subsequent sections.

Theorem 1. Letz^(N)≥z^(N−1)≥ · · · ≥z⁽⁰⁾∈[0, Z]. If

∆ = max

0≤i≤N−1(z⁽ⁱ⁺¹⁾−z⁽ⁱ⁾)

is sufficiently small then G_(z^(N)_,...,z⁽⁰⁾₎ := G_(z^(N)_,z^(N−1)₎◦ · · · ◦ G_(z⁽¹⁾_,z⁽⁰⁾₎ is a global Fourier integral operator of order 0. It can be globally parameterized by

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the non-degenerate phase function of positive type

φ_(z^(N)_,...,z⁽⁰⁾₎(x^(N), x⁽⁰⁾, ξ^(N−1), x^(N−1), . . . , ξ⁽¹⁾, x⁽¹⁾, ξ⁽⁰⁾) :=

XN i=1

φ_(z⁽ⁱ⁾_,z⁽ⁱ⁻¹⁾₎(x⁽ⁱ⁾, x⁽ⁱ⁻¹⁾, ξ⁽ⁱ⁻¹⁾)

= XN i=1

hx⁽ⁱ⁾−x⁽ⁱ⁻¹⁾|ξ⁽ⁱ⁻¹⁾i+ (z⁽ⁱ⁾−z⁽ⁱ⁻¹⁾)a1(z⁽ⁱ⁻¹⁾, x⁽ⁱ⁾, ξ⁽ⁱ⁻¹⁾), wherea1 is the principal symbol of a.

Corollary 2. For ∆P sufficiently small, the operator WP,z (z ∈ [0, Z]) is a global Fourier integral operator of order 0 with complex phase.

In Section 3, we shall denote by χz the bicharacteristic flow associated to

−b1(x, ξ) = Im(a1(x, ξ)).

Theorem 3. Let u0(.) ∈ H^(−∞)(Rⁿ) and u(z, .), z ∈ [0, Z], be the solution to the Cauchy problem (0.1)–(0.2). Let Z⁰ ∈[0, Z] andK be a compact set in T^∗(Rⁿ) such that for allγ⁽⁰⁾ = (x⁰, ξ⁰)∈ K\0the bicharacteristics χz(γ⁽⁰⁾) associated to−b1 originating from γ⁽⁰⁾ atz = 0remains away from the region where c1 >0for all z∈[0, Z⁰]. Then γ⁽⁰⁾ ∈K∩WF(u0) implies χZ⁰(γ⁽⁰⁾)∈ WF(u(Z⁰, .)). For a subdivision Pof[0, Z], with∆_Psufficiently small, we then have

dist

χz(γ⁽⁰⁾),WF(WP,z(u0))

→0, as∆P→0,

uniformly w.r.t. γ⁽⁰⁾ ∈K∩WF(u0) and z∈[0, Z⁰]. Furthermore, the convergence is of order α, 0< α≤1, if b(z, .) is in C^0,α([0, Z], S¹(Rⁿ,Rⁿ)), in the sense that,

b(z⁰, x, ξ)−b(z, x, ξ) = (z⁰−z)^α˜b(z⁰, z, x, ξ), 0≤z≤z⁰≤Z, where˜b(z⁰, z, x, ξ)is bounded w.r.t. z⁰ andz with values inS¹(Rⁿ×Rⁿ).

In [15], a different approximation Ansatz,WfP,z, was introduced for which the convergence rate for the Sobolev norm was improved with less regularity of the symbol az(x, ξ) w.r.t. z. Here, we also show that this phenomenon occurs and that the continuity ofaz(x, ξ) w.r.t. zimplies the convergence of order 1 of the wavefront set ofWfP,z(u0) to that of the solution of the Cauchy problem (0.1)–

(0.2) (in the sense given in the previous theorem — see Theorem 3.12).

In Section 1, we briefly recall some of the set-up and assumptions of [15] which will be used here. In Section 2 we present the geometrical properties of the AnsatzWP,z and prove that it is a global FIO with complex phase. In Section 3 we show microlocal convergence of WP,z to the exact solution of the Cauchy problem (0.1)–(0.2). In Section 4 we show how the analysis made in this paper and [15] can be applied to seismic imaging theory via the so-called ‘double- square-root’ equation. Appendix A is dedicated to some general results on FIOs with complex phases.

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In the present paper we shall generally write X, X⁰, X⁰⁰, X⁽¹⁾, . . . , X^(N) for Rⁿ, according to variables, e.g.,x,x⁰, . . . ,x^(N).

Throughout the paper, we use spaces of global symbols: a functiona∈C^∞(Rⁿ× R^p) is inS_ρ,δ^m(Rⁿ×R^p), 0< ρ≤1, 0≤δ <1, if for all multi-indicesα,β there existsCαβ>0 such that

|∂_x^α∂^β_ξa(x, ξ)| ≤Cαβ(1 +|ξ|)m−ρ|β|+δ|α|, x∈Rⁿ, ξ∈R^p. The best possible constantsCαβ, i.e.,

pαβ(a) := sup

(x,ξ)∈Rⁿ×R^p

(1 +|ξ|)−m+ρ|β|−δ|α||∂_x^α∂_ξ^βa(x, ξ)|,

define seminorms for a Fr´echet space structure on S_ρ,δ^m(Rⁿ×R^p). As usual we write Sρ^m(Rⁿ×R^p) in the case ρ= 1−δ, ¹₂ ≤ρ <1, andS^m(Rⁿ×R^p) in the caseρ= 1,δ= 0.

1 Assumptions and previous results

The symbola(z, x, ξ) is assumed to satisfy Assumption 1.1.

az(x, ξ) =a(z, x, ξ) =−i b(z, x, ξ) +c(z, x, ξ)

where b, c∈C⁰([0, Z], S¹(Rⁿ×Rⁿ)); b has real principal symbolb1 andc has non-negative principal symbol c1. The principal symbolsb1 andc1 are homogeneous of degree 1 for |ξ| ≥1.

We denote by a1=−ib1+c1 the principal symbol of aand write b=b1+b0

andc=c1+c0 withb0, c0∈C⁰([0, Z], S⁰(Rⁿ×Rⁿ)). Assumption 1.1 ensures that the hypotheses (i)–(iii) of Theorem 23.1.2 in [11] are satisfied. Then there exists a unique solution inC⁰([0, Z], H^(s+1)(Rⁿ))∩C¹([0, Z], H^(s)(Rⁿ)) to the Cauchy problem (0.1)–(0.2) ifu0∈H^(s+1)(Rⁿ).

By Proposition 9.3 in [5, Chapter VI] the family of operators (az)_z∈[0,Z] generates a strongly continuous evolution system, U(z⁰, z), on the Sobolev space H^(s+1)(Rⁿ),s∈R,

U(z⁰⁰, z⁰)◦U(z⁰, z) =U(z⁰⁰, z), Z≥z⁰⁰≥z⁰≥z≥0.

and

∂zU(z, z0)u0+a(z, x, Dx)U(z, z0)u0= 0, 0≤z0< z≤Z, U(z0, z0)u0=u0∈H^(s+1)(Rⁿ) whileU(z, z0)u0∈H^(s+1)(Rⁿ) for all z∈[z0, Z].

We now recall some results obtained in [15]. Let z⁰, z∈[0, Z] withz⁰ ≥z and let ∆ :=z⁰−z. Defineφ(z⁰,z)∈C^∞(X⁰×X×Rⁿ) by

(1.5) φ(z⁰,z)(x⁰, x, ξ) :=hx⁰−x|ξi+i∆a1(z, x⁰, ξ)

=hx⁰−x|ξi+ ∆b1(z, x⁰, ξ) +i∆c1(z, x⁰, ξ).

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Lemma 1.2. φ(z⁰,z)is a non-degenerate complex phase function of positive type (at any point (x⁰₀, x0, ξ0)where∂ξφ(z⁰,z)= 0).

We put

g(z⁰,z)(x, ξ) := exp[−∆a0(z, x, ξ)]∈S⁰(X×Rⁿ).

(1.6)

and define a distribution kernel G(z⁰,z)(x⁰, x) ∈D⁰(X⁰×X) by the oscillatory integral

G(z⁰,z)(x⁰, x) = Z

exp[ihx⁰−x|ξi] exp[−∆a(z, x⁰, ξ)]d⁻ξ

= Z

exp[iφ_(z⁰_,z)(x⁰, x, ξ)]g_(z⁰_,z)(x⁰, ξ)d⁻ξ.

We denote the associated operator byG(z⁰,z). (This corresponds to the thin-slab propagator (see e.g. [3, 2]).)

LetJ(z⁰,z)be the canonical ideal locally generated by the phase functionφ(z⁰,z). Proposition 1.3. There exists ∆1 > 0, such that, for all z⁰, z ∈ [0, Z], with z⁰ > z and ∆ =z⁰−z ≤∆1, the phase functionφ(z⁰,z) globally generates the canonical ideal J_(z⁰_,z). Alternatively, it is also generated by the functions

vξj(x⁰, x, ξ⁰, ξ) =∂x⁰_jφ(z⁰,z)(x⁰, x, ξ)−ξ_j⁰ =ξj−ξ_j⁰ +i∆∂xja1(z, x⁰, ξ), (1.7)

vxj(x⁰, x, ξ⁰, ξ) =∂ξjφ(z⁰,z)(x⁰, x, ξ) =x⁰_j−xj+i∆∂ξja1(z, x⁰, ξ), j= 1, . . . , n.

Proposition 1.4. If 0 ≤ ∆ = z⁰ −z ≤ ∆1 then the operator G(z⁰,z) is a global Fourier integral operator with complex phase and kernelG(z⁰,z)∈I⁰(X⁰× X,(J_(z⁰_,z))⁰,Ω^1/2_X0×X).

We denote the half density bundle onX⁰×X by Ω^1/2_X0×X and note that (J(z⁰,z))⁰ stands for the twisted canonical ideal, i.e. a Lagrangian ideal (see Section 25.5 in [10]).

Proposition 1.5. Let s ∈ R. There exists ∆2 >0 such that if z⁰, z ∈ [0, Z]

with 0≤∆ :=z⁰−z≤∆2 then G(z⁰,z) continuously maps S intoS, S⁰ into S⁰, and H^(s)(Rⁿ)intoH^(s)(Rⁿ).

The approximation Ansatz is defined by

Definition 1.6. Let P = {z⁽⁰⁾, z⁽¹⁾, . . . , z^(N)} be a subdivision of [0, Z] with 0 = z⁽⁰⁾ < z⁽¹⁾ <· · · < z^(N) =Z such thatz⁽ⁱ⁺¹⁾−z⁽ⁱ⁾ = ∆P. The operator WP,z is defined as

WP,z :=







G_(z,0) if 0≤z≤z⁽¹⁾,

G_(z,z^(k)₎ Y1 i=k

G_(z⁽ⁱ⁾_,z⁽ⁱ⁻¹⁾₎ if z^(k)≤z≤z^(k+1).

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In the sequel we shall need the following lemma [15].

Lemma 1.7. Consider H : Rⁿ → Rⁿ, H(∆,z,x⁰,x)(ξ) = ξ + ∆h(z, x⁰, x, ξ), wherehis continuous w.r.t. z with values inS¹(R²ⁿ×Rⁿ). If∆ is sufficiently small, uniformly w.r.t. z ∈ [0, Z], then H(∆,z,x⁰,x) is a global diffeomorphism.

Furthermore, ξ(∆, z, x˜ ⁰, x, ξ) = H_(∆,z,x⁻¹ 0,x)(ξ) is homogeneous of degree 1 in ξ, for |ξ| ≥1, continuous w.r.t. z, and C^∞w.r.t.∆ with values inS¹(R²ⁿ×Rⁿ), when∆ is sufficiently small, i.e.,

∃∆3>0, ξ˜∈C⁰([0, Z],C^∞([0,∆3], S¹(R²ⁿ×Rⁿ))).

Recall the smoothness (or differentiability) of a map with values in a Frechet space is to be understood in the sense of Definition 40.2 in [25].

In the applications we have in mind, the principal part of the damping term, c1, will affect only certain parts of phase-space (see Appendix A in [15]). In this paper, where the propagation of singularities is analyzed, we shall make the additional assumption

Assumption 1.8. The open set Ω =

[0, Z]×(T^∗(Rⁿ)\0)

\supp(c1)is not empty.

2 Geometrical and FIO properties of W

P

In this section we investigate the microlocal properties ofWP. To do so we need to analyze how the product

WP,z=G_(z,z(k))

Y1 i=k

G_(z(i),z⁽ⁱ⁻¹⁾)

for z^(k) ≤ z ≤ z^(k+1), k ≥ 1, can be understood as a composition of FIOs and yields in turn an FIO. Let z⁰, z∈ [0, Z] with z⁰ ≥z and put ∆ =z⁰−z.

We recall that the global phase function of G(z⁰,z) is given by (1.5). As in [10, Sections 25.4 and 25.5], ifI is an ideal of complex valued functions onT^∗(Rⁿ), we denote byI_R the subset ofT^∗(Rⁿ) where all the functions inI vanish. By Lemma 1.3 the following holdsglobally:

(2.8) J(z⁰,z)R=

(x⁰, ∂x⁰φ(z⁰,z)(x⁰, x, ξ), x, ξ)|∂ξφ(z⁰,z)(x⁰, x, ξ) = 0,

(x⁰, x, ξ)∈X⁰×X×(Rⁿ\0) ⊂T^∗(X⁰×X)\0.

Remark 2.1. (i). The phase functionφ_(z⁰_,z)is homogeneous of degree 1 for

|ξ| ≥ 1. With a cut-off function ψ ∈ C_c^∞(R) such that ψ(ξ) = 1 when

|ξ| ≤1 andψ(ξ) = 0 when |ξ| ≥2 we can writeG(z⁰,z) =G_(z⁽¹⁾0,z)+G_(z⁽²⁾0,z)

with respective amplitudesψ(ξ)g(z⁰,z)(x⁰, ξ) and (1−ψ(ξ))g(z⁰,z)(x⁰, ξ). We can now assume thatφ(z⁰,z) is homogeneous of degree 1 in the expression of the kernel ofG_(z⁽²⁾0,z)andG_(z⁽¹⁾0,z)is a regularizing operator. For the study of the microlocal properties of G_(z⁰_,z), and WP,z, we may thus consider G_(z⁽²⁾0,z)in place ofG(z⁰,z). Note thatG_(z⁽²⁾0,z)mapsS intoS andS⁰intoS⁰,

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H^(s)(Rⁿ) intoH^(s)(Rⁿ), for anys∈R, continuously, as doesG_(z⁰_,z). In the sequel, we may therefore assume thatφ(z⁰,z), b1 and c1 are homogeneous of degree 1 inξ.

(ii). Observe that the composition of the two FIOsG(z⁰⁰,z⁰)andG(z⁰,z)is natural as operators onS,S⁰, orH^(s)(Rⁿ), without further requirement such as having the operators properly supported.

(iii). If∂ξφ(z⁰,z)= 0 then∂ξc1(z, x⁰, ξ) = 0. Sincec1 is homogeneous of degree 1 in ξ, Euler’s identity then yields c1(z, x⁰, ξ) = 0. Conversely, since c1(z, x⁰, ξ) is non-negative, c1(z, x⁰, ξ) = 0 implies∂xc1(z, x⁰, ξ) = 0 and

∂ξc1(z, x⁰, ξ) = 0. Thus if (x⁰, ξ⁰, x, ξ)∈J(z⁰,z)Rthen∂xc(z, x⁰, ξ) = 0 and

∂ξc(z, x⁰, ξ) = 0 which is equivalent to having c1(z, x⁰, ξ) = 0. Observe that∂x⁰φ(z⁰,z)(x⁰, x, ξ) is thus real in (2.8).

Lemma 2.2. There exists ∆4 > 0 such that for all z⁰, z ∈ [0, Z] with ∆ = z⁰−z∈[0,∆4]we haveJ(z⁰,z)R⊂T^∗(X⁰)\0×T^∗(X)\0.

Proof. Let (x⁰, ξ⁰, x, ξ)∈J(z⁰,z)R. Then by Proposition 1.3 we have ξ−ξ⁰+i∆∂xa1(z, x⁰, ξ) = 0, x⁰−x+i∆∂ξa1(z, x⁰, ξ) = 0.

(2.9)

Remark 2.1-(iii) (or only considering the real part in (2.9)) yields ξ−ξ⁰+ ∆∂xb1(z, x⁰, ξ) = 0, x⁰−x+ ∆∂ξb1(z, x⁰, ξ) = 0.

By Lemma 1.7 the mapξ 7→ξ+ ∆∂xb1(z, x⁰, ξ) is a global diffeomorphism for

∆ sufficiently small and its inverse map is also homogeneous of degree 1. We thus obtain that ξ = 0⇔ ξ⁰ = 0. Since J(z⁰,z)R ⊂T^∗(X⁰×X)\0 the result follows.

Letz^(N)≥z^(N−1)· · · ≥z⁽⁰⁾ ∈[0, Z]. We define

Je(z^(N),...,z⁽⁰⁾)R:=J(z^(N),z^(N−1))R◦ · · · ◦J(z⁽¹⁾,z⁽⁰⁾)R. (2.10)

By induction onN one proves

Lemma 2.3. For all z^(N) ≥z^(N−1)≥ · · · ≥z⁽⁰⁾ ∈[0, Z], with z⁽ⁱ⁺¹⁾−z⁽ⁱ⁾≤

∆4, i= 0, . . . , N−1, we have

Je_(z^(N)_,...,z⁽⁰⁾₎_R⊂T^∗(X^(N))\0×T^∗(X⁽⁰⁾)\0.

Lemma 2.4. There exists ∆5>0such that withz⁰⁰≥z⁰≥z∈[0, Z]the map π:J(z⁰⁰,z⁰)R×J(z⁰,z)R∩T^∗(X⁰⁰)×diag(T^∗(X⁰))×T^∗(X)→T^∗(X⁰⁰×X)\0

(x⁰⁰, ξ⁰⁰, x⁰, ξ⁰, x⁰, ξ⁰, x, ξ)7→(x⁰⁰, ξ⁰⁰, x, ξ) is injective and proper if max(z⁰⁰−z⁰, z⁰−z)≤∆5.

We write diag(T^∗(X⁰)) for the diagonal of T^∗(X⁰)×T^∗(X⁰). Here we give a direct proof of the lemma but it follows in fact from results on the real part of the phase function (see (2.20) and Remark 2.13-(i) below) and Proposition 3.13 in [13, Chapter 10].

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Proof. Letγ = (x⁰⁰, ξ⁰⁰, x, ξ) be in the range ofπ, that is in J(z⁰⁰,z⁰)R◦J(z⁰,z)R. With Lemma 1.3 (use Remark 2.1-(iii)) we have

ξ−ξ⁰+ ∆∂xb1(z, x⁰, ξ) = 0, (2.11)

x⁰−x+ ∆∂ξb1(z, x⁰, ξ) = 0, (2.12)

ξ⁰−ξ⁰⁰+ ∆⁰∂xb1(z⁰, x⁰⁰, ξ⁰) = 0, (2.13)

x⁰⁰−x⁰+ ∆⁰∂ξb1(z⁰, x⁰⁰, ξ⁰) = 0, (2.14)

where ∆ :=z⁰−z and ∆⁰:=z⁰⁰−z⁰. DefineF(ξ⁰) :=ξ⁰⁰−∆⁰∂xb1(z⁰, x⁰⁰, ξ⁰). It follows that

|F(ξ⁰)−F( ˜ξ⁰)| ≤∆⁰sup(|∂ξi∂xjb1(z⁰, x⁰⁰, ξ⁰)|)|ξ⁰−ξ˜⁰|,

where the supremum is taken overz∈[0, Z],x⁰⁰∈Rⁿ,ξ⁰∈Rⁿand 1≤i, j≤n.

As b1 ∈ C⁰([0, Z], S¹(X ×Rⁿ)) it follows that ∂ξi∂xjb1(z⁰, x⁰⁰, ξ⁰) is globally bounded. Thus, for ∆⁰ sufficiently small the map F is a contraction and ξ⁰ in (2.13) is uniquely defined by the fixed point theorem. Equation (2.14) then shows that x⁰ is uniquely defined by the above identities if ∆⁰ is sufficiently small. Hence the map π is injective. (Notice that we only need either ∆ or ∆⁰ to be sufficiently small to reach the conclusion; in fact we could form G(x⁰) = x−∆∂ξb1(z, x⁰, ξ) and prove that it is contracting for sufficiently small ∆.) Let now K ⊂ T^∗(X⁰⁰×X)\0 be a compact set. As π⁻¹(K) is closed we just have to prove that it is bounded. Note that the equations above give x⁰ =x+ ∆∂ξb1(z, x⁰, ξ) and since ∂ξb1 ∈C⁰([0, Z], S⁰(X×Rⁿ)), it is globally bounded. Assume then thatγ∈K. Thenxstays in a bounded set and so does x⁰. We also haveξ⁰=ξ+ ∆∂xb1(z, x⁰, ξ). Asx⁰andξ stay in a bounded domain so doesξ⁰ by (2.11). Therefore,π is a proper map.

Lemma 2.5. There exists ∆6 > 0 such that if z⁰⁰ ≥ z⁰ ≥ z ∈ [0, Z] with z⁰⁰−z⁰ ≤∆6 andz⁰−z≤∆6 then

φ(z⁰⁰,z⁰,z)(x⁰⁰, x, ξ⁰, x⁰, ξ) :=φ(z⁰⁰,z⁰)(x⁰⁰, x⁰, ξ⁰) +φ(z⁰,z)(x⁰, x, ξ) is a non-degenerate phase function of positive type in X⁰⁰×X×(R³ⁿ\0).

This follows from a more general result which will be of use in the sequel as well Lemma 2.6. There exists ∆6 >0 such that if z^(N) ≥z^(N−1) ≥ · · · ≥ z⁽⁰⁾ ∈ [0, Z]with z⁽ⁱ⁾−z⁽ⁱ⁻¹⁾≤∆6, i= 1, . . . , N, then

(2.15) φ_(z^(N)_,...,z⁽⁰⁾₎(x^(N), x⁽⁰⁾, ξ^(N−1), x^(N−1), . . . , ξ⁽¹⁾, x⁽¹⁾, ξ⁽⁰⁾) :=

XN i=1

φ_(z⁽ⁱ⁾_,z⁽ⁱ⁻¹⁾₎(x⁽ⁱ⁾, x⁽ⁱ⁻¹⁾, ξ⁽ⁱ⁻¹⁾)

= XN i=1

hx⁽ⁱ⁾−x⁽ⁱ⁻¹⁾|ξ⁽ⁱ⁻¹⁾i+ (z⁽ⁱ⁾−z⁽ⁱ⁻¹⁾)a1(z⁽ⁱ⁻¹⁾, x⁽ⁱ⁾, ξ⁽ⁱ⁻¹⁾)

is a phase function of positive type in X^(N)×X⁽⁰⁾×(R^n(2N−1)\0) which is non-degenerate.

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We collect the phase variables of φ(z^(N),...,z⁽⁰⁾)as

θN−1:= (ξ^(N−1), x^(N−1), . . . , ξ⁽¹⁾, x⁽¹⁾, ξ⁽⁰⁾)∈R^n(2N−1). (2.16)

The functionφis homogeneous of degree 1 in

θ˜N−1:= (ξ^(N−1), λx^(N−1), . . . , ξ⁽¹⁾, λx⁽¹⁾, ξ⁽⁰⁾)∈R^n(2N−1)\0,

whereλ:=|(ξ^(N−1), . . . , ξ⁽⁰⁾)|. Apart from the reasoning immediately below we shall, as usual, omit this scaling. Yet one should keep in mind that ˜θN−1is the actual phase variable for φ_(z^(N)_,...,z⁽⁰⁾₎. When we write that the phase variable belongs to (R^n(2N−1))\0 in the statement of the lemma, it is meant in the sense that ˜θN−1∈(R^n(2N−1))\0.

Proof. For simplicity, we writeφinstead ofφ_(z^(N)_,...,z⁽⁰⁾₎. Suppose dφ= 0 then

∂x⁽⁰⁾φ=· · ·=∂x^(N−1)φ= 0

yieldξ⁽⁰⁾=· · ·=ξ^(N−1)= 0 and with the scaling byλwe have ˜θN−1= 0. Thus dφ6= 0 in X^(N)×X⁽⁰⁾×(R^n(2N−1)\0). Clearly Imφ≥0. It remains to show that the differentialsd(∂_x⁽ⁱ⁾φ),i= 1, . . . , N−1, andd(∂_ξ^(j)φ),j= 0, . . . , N−1 are linearly independent. We observe that

∂_x⁽ⁱ⁾φ=ξ⁽ⁱ⁻¹⁾−ξ⁽ⁱ⁾+i∆⁽ⁱ⁻¹⁾∂xa_z⁽ⁱ⁻¹⁾(x⁽ⁱ⁾, ξ⁽ⁱ⁻¹⁾), i= 1, . . . , N −1,

∂_ξ^(j)φ=x^(j+1)−x^(j)+i∆^(j)∂ξa_z^(j)(x^(j+1), ξ^(j)), j= 0, . . . , N −1, where ∆⁽ⁱ⁾:=z⁽ⁱ⁺¹⁾−z⁽ⁱ⁾. The structure of the partial differentials∂(∂_x(i)φ), i = 1, . . . , N −1, and ∂(∂_ξ^(j)φ), j = 0, . . . , N −1, w.r.t. x⁽⁰⁾, . . . , x^(N) and ξ⁽⁰⁾, . . . , ξ^(N−1)can be summarized as follows

x⁽⁰⁾ ξ⁽⁰⁾ x⁽¹⁾ ξ⁽¹⁾ x⁽²⁾ ξ⁽²⁾ ... ξ^(N−2) x^(N−1) ξ^(N−1) x^(N)

∂(φ⁰_ξ(0)) ∂(φ⁰_x(1)) ∂(φ⁰_ξ(1)) ∂(φ⁰_x(2)) . . . ∂(φ⁰_x(N−1)) ∂(φ⁰_ξ(N−1))

−I 0 0 0 . . . 0 0

0 0 . . . 0 0

−I 0 . . . 0 0

0 −I . . . 0 0

0 0 . . . 0 0

0 0 0 −I . . . 0 0

... . .. ...

0 0 0 0 . . . 0

0 0 0 0 . . . −I

0 0 0 0 . . . 0

where is some n×n matrix and is a n×n matrix of the form I + i∆^(j)∂x∂ξa_z^(j) for some 0 ≤ j ≤ N −1. As ∂xk∂ξla_z^(j) ∈ S⁰(X ×Rⁿ) continuously w.r.t. z^(j), it is globally bounded. Thus for ∆^(j) sufficiently small every matrix is invertible. The partial differentials of interest are thus of maximal rank.

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Definition 2.7. For z⁰⁰ ≥ z⁰ ≥ z ∈ [0, Z] we write G_(z⁰⁰_,z⁰_,z) := G_(z⁰⁰_,z⁰₎◦ G_(z⁰_,z) and more generally for z^(N) ≥ z^(N−1) ≥ · · · ≥ z⁽⁰⁾ ∈ [0, Z] we write G_(z(N),...,z⁽⁰⁾):=G_(z(N),z^(N−1))◦ · · · ◦ G_(z(1),z⁽⁰⁾).

Proposition 2.8. Letz⁰⁰≥z⁰ ≥z ∈[0, Z]. The operator G(z⁰⁰,z⁰,z) is a global Fourier integral operator if z⁰⁰−z⁰ ≤ min(∆4,∆5,∆6) and z⁰−z ≤ ∆5. Its kernelG(z⁰⁰,z⁰,z) is inI⁰(X⁰⁰×X,(J(z⁰⁰,z⁰,z))⁰,Ω^1/2_X00×X)where the canonical ideal is given by J(z⁰⁰,z⁰,z):=J(z⁰⁰,z⁰)◦J(z⁰,z) with transversal composition. J(z⁰⁰,z⁰,z)

is globally parameterized by the non-degenerate phase function of positive type φ(z⁰⁰,z⁰,z).

I⁰(X⁰⁰×X,(J(z⁰⁰,z⁰,z))⁰,Ω^1/2_X00×X) is the set of Lagrangian-distribution half-densi- ties on X⁰⁰×X of order 0 associated to the Lagrangian ideal (J(z⁰⁰,z⁰,z))⁰ (see [10, Definition 25.4.9]).

Proof. We apply Theorem 25.5.5 in [10] and we use Lemmas 2.2, and 2.4.

Lemma 2.5 and Proposition A.4 yield transversal composition for the two canonical ideals J(z⁰⁰,z⁰) and J(z⁰,z). Observe that J(z⁰⁰,z⁰,z)R =J(z⁰⁰,z⁰)R◦J(z⁰,z)R by proposition A.3. At every point of J(z⁰⁰,z⁰)R◦J(z⁰,z)Rthe non-degenerate phase function φ_(z⁰⁰_,z⁰_,z) locally defines J_(z⁰⁰_,z⁰₎◦J_(z⁰_,z) by Proposition 25.5.4 in [10]

hence we obtain thatφ_(z⁰⁰_,z⁰_,z)is a global phase function forJ_(z⁰⁰_,z⁰_,z)and con- sequently forG(z⁰⁰,z⁰,z). The order ofG(z⁰⁰,z⁰,z)follows since both kernelsG(z⁰⁰,z⁰)

andG(z⁰,z)are inI⁰.

Theorem 2.9. Letz^(N) ≥z^(N−1)≥ · · · ≥z⁽⁰⁾ ∈ [0, Z] with ∆⁽ⁱ⁾ :=z⁽ⁱ⁺¹⁾− z⁽ⁱ⁾≤min(∆4,∆5,∆6), for all i= 0, . . . , N−1. ThenG_(z^(N)_,...,z⁽⁰⁾₎ is a global Fourier integral operator with complex phase and with distribution kernel

G_(z^(N)_,...,z⁽⁰⁾₎∈I⁰(X^(N)×X⁽⁰⁾,(J_(z^(N)_,...,z⁽⁰⁾₎)⁰,Ω^1/2_X(N)×X⁽⁰⁾),

where J(z^(N),...,z⁽⁰⁾) :=J(z^(N),z^(N−1))◦ · · · ◦J(z⁽¹⁾,z⁽⁰⁾) with transversal composi- tions. J_(z(N),...,z⁽⁰⁾) is globally parameterized by the non-degenerate phase function of positive typeφ_(z^(N)_,...,z⁽⁰⁾₎. We haveJ_(z^(N)_,...,z⁽⁰⁾₎_R=J_(z^(N)_,z^(N−1)₎_R◦· · ·◦

J_(z⁽¹⁾_,z⁽⁰⁾₎_R.

Proof. We proceed by induction assuming the result is true for G_(z^(N)_,...,z⁽⁰⁾₎ andJ_(z^(N)_,...,z⁽⁰⁾₎. By Lemma 2.6 and Proposition A.4 we see thatJ_(z^(N+1)_,z^(N)₎ andJ_(z^(N)_,...,z⁽⁰⁾₎compose transversally. Lemma 2.3 shows thatJ_(z^(N+1)_,z^(N)₎_R⊂ T^∗(X^(N+1))\0×T^∗(X^(N))\0. In the induction we assume thatJ_(z^(N)_,...,z⁽⁰⁾_)R= Je_(z^(N)_,...,z⁽⁰⁾_)R(see (2.10)) thusJ_(z^(N)_,...,z⁽⁰⁾_)R⊂T^∗(X^(N))\0×T^∗(X⁽⁰⁾)\0. At this point we claim

Lemma 2.10. The map

πN:J_(z^(N+1)_,z^(N)₎_R×J_(z^(N)_,...,z⁽⁰⁾₎_R∩T^∗(X^(N+1))×∆T^∗(X^(N))×T^∗(X⁽⁰⁾)

→T^∗(X^(N+1)×X⁽⁰⁾)\0

(x^(N+1), ξ^(N+1), x^(N), ξ^(N), x^(N), ξ^(N), x⁽⁰⁾, ξ⁽⁰⁾)7→(x^(N+1), ξ^(N+1), x⁽⁰⁾, ξ⁽⁰⁾) is injective and proper if ∆⁽ⁱ⁾=z⁽ⁱ⁺¹⁾−z⁽ⁱ⁾≤∆5, i= 0, . . . , N−1.

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The proof of this lemma can be copied to a large extent from that of Lemma 2.4 (with an induction). (This lemma also follows directly from (2.20) and Re- mark 2.13-(i) below.)

With the above observations we can apply Theorem 25.5.5 in [10], which yields the first part of the result. Now, Lemma A.3 yields

J_(z^(N+1)_,...,z⁽⁰⁾_)R=J_(z^(N+1)_,z^(N)_)R◦J_(z^(N)_,...,z⁽⁰⁾_)R=Je_(z^(N+1)_,...,z⁽⁰⁾_)R. which completes the induction.

Corollary 2.11. Let P = {z⁽⁰⁾, z⁽¹⁾, . . . , z^(N)}be a subdivision of [0, Z] with 0 = z⁽⁰⁾ < z⁽¹⁾ < · · · < z^(N) = Z such that z⁽ⁱ⁺¹⁾−z⁽ⁱ⁾ ≤ ∆P. Let z ∈ [0, Z]. Then the operatorWP,zgiven in Definition 1.6 is a global Fourier integral operator of order 0if∆P<min(∆4,∆5,∆6).

Letz^(N)≥z^(N−1)· · · ≥z⁽⁰⁾∈[0, Z]. Note that

(x^(N), ξ^(N), x⁽⁰⁾, ξ⁽⁰⁾)∈J_(z^(N)_,...,z⁽⁰⁾_)R

if and only if there existsθN−1∈R^n(2N−1)\0 as defined in (2.16) such that ξ^(j)−ξ^(j+1)+ ∆^(j)∂xb1(z^(j), x^(j+1), ξ^(j)) = 0,

(2.17)

x^(j+1)−x^(j)+ ∆^(j)∂ξb1(z^(j), x^(j+1), ξ^(j)) = 0, (2.18)

forj= 0, . . . , N, and

c1(z^(j), x^(j+1), ξ^(j)) = 0, j= 0, . . . , N−1 (2.19)

(see Remark 2.1-(iii)).

Letz^(N)≥z^(N−1)· · · ≥z⁽⁰⁾ ∈[0, Z] we define (2.20) J_(z^(N)_,...,z⁽⁰⁾₎=n

(x^(N), ξ^(N), x⁽⁰⁾, ξ⁽⁰⁾)| ∃ θN−1∈R^n(2N−1)\0 as defined in (2.16) such that (2.17)–(2.18) are satisfiedo

. Note that J(z⁰,z)R=J(z⁰,z)∩ {(x⁰, ξ⁰, x, ξ)|c1(z, x⁰, ξ) = 0}and

(2.21) J_(z^(N)_,...,z⁽⁰⁾₎_R=n

(x^(N), ξ^(N), x⁽⁰⁾, ξ⁽⁰⁾)∈J(z^(N),...,z⁽⁰⁾)| forθN−1∈R^n(2N−1)\0 defined above

c1(z^(j), x^(j+1), ξ^(j)) = 0, forj= 0, . . . , N−1o . Note also thatJ_(z^(N)_,...,z⁽⁰⁾₎is locally a canonical relation fromT^∗(X⁽⁰⁾)\0 into T^∗(X^(N))\0: simply apply the classical results for real phase functions [4, 10]

to the non-degenerate phase functionϕ_(z(N),...,z⁽⁰⁾)= Reφ_(z(N),...,z⁽⁰⁾), that is, (2.22) ϕ_(z^(N)_,...,z⁽⁰⁾₎(x^(N), x⁽⁰⁾, ξ^(N−1), x^(N−1), . . . , ξ⁽¹⁾, x⁽¹⁾, ξ⁽⁰⁾)

:= Re XN i=1

φ_(z⁽ⁱ⁾_,z⁽ⁱ⁻¹⁾₎(x⁽ⁱ⁾, x⁽ⁱ⁻¹⁾, ξ⁽ⁱ⁻¹⁾)

= XN i=1

hx⁽ⁱ⁾−x⁽ⁱ⁻¹⁾|ξ⁽ⁱ⁻¹⁾i+ (z⁽ⁱ⁾−z⁽ⁱ⁻¹⁾)b1(z⁽ⁱ⁻¹⁾, x⁽ⁱ⁾, ξ⁽ⁱ⁻¹⁾).

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Proposition 1.3, in the case of a real phase function, yields thatJ(z^(N),...,z⁽⁰⁾)is a canonical relation globally defined byϕ(z^(N),...,z⁽⁰⁾). We can actually say more aboutJ(z^(N),...,z⁽⁰⁾):

Lemma 2.12. There exists∆7>0such that ifz^(N)≥z^(N−1)· · · ≥z⁽⁰⁾∈[0, Z]

with z⁽ⁱ⁾−z⁽ⁱ⁻¹⁾≤∆7 then J_(z^(N)_,...,z⁽⁰⁾₎ is a one-to-one canonical transformation fromT^∗(X⁽⁰⁾)\0ontoT^∗(X^(N))\0.

Proof. It suffices to prove the result forJ(z⁰,z), with z⁰−z sufficiently small, as

J(z^(N),...,z⁽⁰⁾)=J(z^(N),z^(N−1))◦ · · · ◦J(z⁽¹⁾,z⁽⁰⁾).

The canonical relationJ(z⁰,z)is globally generated by the non-degenerate real phase functionϕ(z⁰,z)=hx⁰−x|ξi+ (z⁰−z)b1(z, x⁰, ξ). For ∆ sufficiently small we see that ϕ(z⁰,z)− hx|ξi satisfies Definition 1.2 in [13, Chapter 10]. Then Proposition 3.13 in [13, Chapter 10] applies.

Remark 2.13. (i). With the results obtained so far we immediately deduce that the projection

e

π:J(z⁰⁰,z⁰)×J(z⁰,z)∩T^∗(X⁰⁰)×diag(T^∗(X⁰))×T^∗(X)

→T^∗(X⁰⁰×X)\0 (x⁰⁰, ξ⁰⁰, x⁰, ξ⁰, x⁰, ξ⁰, x, ξ)7→(x⁰⁰, ξ⁰⁰, x, ξ) and also

e

πN:J(z^(N+1),z^(N))×J(z^(N),...,z⁽⁰⁾)∩T^∗(X^(N+1))×∆T^∗(X^(N))×T^∗(X⁽⁰⁾)

→T^∗(X^(N+1)×X⁽⁰⁾)\0 (x^(N+1), ξ^(N+1), x^(N), ξ^(N), x^(N), ξ^(N),x⁽⁰⁾, ξ⁽⁰⁾)

7→(x^(N+1), ξ^(N+1), x⁽⁰⁾, ξ⁽⁰⁾) are injective and proper (see [9, pages 174-175]). This alternatively yields the results of Lemma 2.4 and Lemma 2.10 as J_(z^(N)_,...,z⁽⁰⁾_)R is closed in J_(z^(N)_,...,z⁽⁰⁾₎.

(ii). Since J(z⁰⁰,z⁰) and J(z⁰,z), z⁰⁰ ≥ z⁰ ≥ z are canonical transformations, they compose transversally (see [9, pages 174-175]). However, this does not apply toJ(z⁰⁰,z⁰)andJ(z⁰,z)since their tangent spaces, in the complex- ification of the tangent space of T^∗(X⁰⁰×X⁰\0) and T^∗(X⁰×X \0) at γ⁰ = (x⁰⁰, ξ⁰⁰, x⁰, ξ⁰) and γ = (x⁰, ξ⁰, x, ξ), may differ from those of J(z⁰⁰,z⁰) andJ(z⁰,z). In factTγ(J(z⁰,z)) is defined bydvξj = 0,dvxj = 0, j= 1, . . . , nby Proposition 1.3, whileTγ(J(z⁰,z)) is defined by dveξj = 0, dvexj = 0,j= 1, . . . , nwith

e

vξj(x⁰, x, ξ⁰, ξ) =ξj−ξ_j⁰ +i∆∂xjb1(z, x⁰, ξ), j= 1, . . . , n, e

vxj(x⁰, x, ξ⁰, ξ) =x⁰_j−xj+i∆∂ξjb1(z, x⁰, ξ), j= 1, . . . , n.

Note that J(z^(N),...,z⁽⁰⁾)(x⁽⁰⁾, ξ⁽⁰⁾) now means the image of (x⁽⁰⁾, ξ⁽⁰⁾) under the map defined according to Lemma 2.12. In a similar fashion, we shall write J_(z^(N)_,...,z⁽⁰⁾₎_R(x⁽⁰⁾, ξ⁽⁰⁾) as the image,if it exists, of (x⁽⁰⁾, ξ⁽⁰⁾) under the relation J_(z^(N)_,...,z⁽⁰⁾₎_R.