Microlocal analysis of the generalized Radon transform arising in a model of pure industry

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Microlocal analysis of the generalized Radon transform arising in a model of pure industry

Alexey Agaltsov

To cite this version:

Alexey Agaltsov. Microlocal analysis of the generalized Radon transform arising in a model of pure

industry. 2015. �hal-01133240v2�

Microlocal analysis of the generalized Radon transform arising in a model of pure industry

A. D. Agaltsov

^1,2

We show that the generalized Radon transform arising in a model of pure industry taking into account the substitution of production factors at the micro-level is an integral Fourier operator satisfying the condition of microlocal regularity. We describe a method for reconstruction of singularities of a function from the singularities of its generalized Radon transform.

Keywords: generalized Radon transform, microlocal analysis, in- tegral Fourier operators, model of pure industry, constant elasticity of substitution

AMS classication: 44A12 (Radon transform), 35S30 (Fourier in- tegral operators), 46N10 (applications of functional analysis in eco- nomics)

1 Introduction

Modern production systems are experiencing such eects as standartization and globalization which, in particular, lead to the situation when dierent produc- tion factors (resources) can substitute each other in the production process.

One of the actual problems of the production theory is to give a mathematical description of the production process in a pure industry taking into account this eect of substitution of production factors and allowing its quantitative analysis.

One of the models of pure industry taking into account the aforementioned eect was proposed by A. A. Shananin in [Sha2], [Sha3]. This model generalizes the HouthakkerJohansen model of pure industry (see [Sha1]) allowing an arbi- trary continuous positively homogeneous monotone function as the production function at the micro-level whereas the HouthakkerJohansen model allowed only the Leontief production functions.

In the present paper we consider Shananin's model of pure industry in one of the most important particular cases, when the elasticity of substitution of production factors is constant. We also restrict our attention to the case of

1Centre de Mathematiques Appliquees, Ecole Polytechnique Route de Saclay

91128 PALAISEAU Cedex, France

2Lomonosov Moscow State University, GSP-1, Leninskie Gory

119991 MOSCOW, Russia email: [email protected]

two production factors in order to simplify the mathematical formulas but the multidimensional case is completely similar.

In Shananin's model the main tool of quantitative description of a pure industry at the macro-level is the prot function which, under the assumption of constant elasticity of substitution of production factors, is given by the following formula:

Π

α

(p

0

, p

1

, p

2

) = Z

p₀

0

Z

t 0

1 τ R

α

p

1

τ , p

2

τ

dτ dt α ≤ 1, (1.1) where p

0

is the unit price of the nal product, p

0

, p

1

are the unit prices of the production factors, (1 − α) is the elasticity of substitution of production factors and R

α

is the generalized Radon transform on the positive quadrant X = {(x

1

, x

2

) : x

1

> 0, x

2

> 0} which maps a suciently regular real-valued function u on X to the function R

α

u on P = {(p

1

, p

2

) : p

1

> 0, p

2

> 0} dened as follows:

R

α

u(p) = Z

X_α,p

u(x) ω

α,p

, (1.2)

where p = (p

1

, p

2

) ,

X

α,p

=

(x

1

, x

2

) ∈ X : q

α

(p

1

x

1

, p

2

x

2

) = 1 , (1.3) q

α

(x

1

, x

2

) = (x

^α₁

+ x

^α₂

)

^1/α

, x

1

, x

2

> 0, (1.4) ω

α,x

is the GelfandLeray 1 -form given by the interior product:

ω

α,p

= d

x

q

α

(p

1

x

1

, p

2

x

2

) y (dx

1

∧ dx

2

), (1.5) and orientation on X

α,p

is given by the volume form |∇

x

q

α

(p

1

x

1

, p

2

x

2

)|ω

α,p

.

Recall that by denition ω

α,p

is the restriction to X

α,p

of any 1 -form e ω

α,p

on X satisfying

d

_x

q

_α

(p

₁

x

₁

, p

₂

x

₂

) ∧ ω e

_α,p

= dx

₁

∧ dx

₂

. The form ω

_α,p

doesn't depend on the choice of ω e

_α,p

.

In the setting of Shananin's model q

_α

is the unit cost function of the nal product, u has the meaning of density of distribution of production powers over technologies and Π

_α

(p

₀

, p

₁

, p

₂

) is the total prot of the industry for a production cycle. In this context function q

α

dened in (1.4) is called the constant elasticity of substitution function (CES-function), corresponding to elasticity (1 − α) .

The case α = 1 corresponds to production systems with no substitution of production factors at the microlevel. This assumption is not fullled in the majority of production systems experiencing the eects of standartization and globalization. In this case transform (1.2) is the classical Radon transform in R

²

with limited data: only those straight lines that intersect both coordinate axes in the rst (nonnegative) quadrant are considered. The properties of this transform were studied in [HS].

Passing α → 0 in (1.2) we obtain the transform R

₀

that integrates function

u over the sets x

₁

x

₂

= const . This case corresponds to production systems

with the CobbDouglas production functions. Note that transform R

₀

maps

a function of two variables to a function of one variable (after an appropriate change of variables). We exclude this case from consideration.

As it was mentioned before, the prot function Π

α

is the main tool of quan- titative description of the production system at the macro-level (as well as the production function, but they are dual, see [Sha2]). Note that we consider Π

α

(resp. R

_α

) as an integral transform acting on functions u dened in X . From this point of view it follows from formula (1.1) that transforms Π

_α

and R

_α

are essentially the same (they are related by dierentiation and integration, respec- tively). This means that the problem of quantitative analysis of production process in Shananin's model reduces to the study of the transform R

_α

.

We are mainly interested in the following problems for transform R

α

. Problem 1 (Uniqueness). Find sucient conditions for injectivity of transform R

α

on a suitable space. Find examples of non-injectivity of transform R

α

. Problem 2 (Characterization). Decribe the range of transform R

α

on a suitable space.

Problem 3 (Inversion). Find a stable algorithm (or formula) for inversion of R

α

on a suitable space. Obtain the stability estimates.

Problem 4 (Microlocal properties). Describe the propagation of singularities by transform R

_α

.

By ¾suitable spaces¿ we understand the spaces which naturally arise in the above setting of the generalized model of pure industry, for example, the space of non-negative compactly supported continuous functions or more general spaces.

Transform R

α

was introduced in Ref. [HS] in the case when α = 1 . It was studied for the rst time under the assumption that α > 0 in Ref. [Ag1]. In Ref.

[Ag2] the more general transform was considered, where integration in (1.2) is over the level hypersurfaces of an arbitrary positively homogeneous function in positive orthant satisfying some regularity assumptions.

Problem 1 was considered in Ref. [Ag2]. In this work it was shown that transform R

α

is injective on the subspace of non-negative functions (and even measures) growing not faster than exponents at innity. Besides, it was shown that the ranges of transforms R

α

and R

β

for α > 0 , β > 0 , α 6= β , on a subspace of functions (and even measures) with sucient decay at innity intersect only by zero function.

Problem 2 was studied in Ref. [Ag1]. In a subsequent work we will present results on Problem 2 in the case of more general integral transform over the level hypersurfaces of an arbitrary suciently regular positively homogeneous function.

Problem 3 was addressed in Ref. [Ag1] in the case of transform R

α

and in Ref. [Ag2] in the case of more general integral transform.

Besides, Problems 1, 2, 3 for transform R

α

were studied in Ref. [HS] in the

case when α = 1 .

In the present paper we study Problem 4. More precisely, we show that R

α

, α ∈ R \ {0, 1} , is a Fourier integral operator satisfying the property of microlo- cal regularity and we propose a method for reconstruction of singularities, see Theorem 2.1 and Proposition 2.1 of Section 2.

We don't consider the case α = 1 for the following reason. On the one hand, our method of proof of microlocal regularity is based on a version of method of stationary phase presented in Section 4 which uses the fact that the linear function S (x, ξ) = ξ

₁

x

₁

+ ξ

₂

x

₂

at xed ξ

₁

, ξ

₂

with ξ

₁

ξ

₂

> 0 attains the unique extremum x

^∗

(p, ξ) on X

_α,p

for any p ∈ P when α ∈ R \ {0, 1} and this is not true when α = 1 . On the other hand, restrictions in space prevent us from studying the microlocal properties of R

1

as an exceptional case in the present paper. Finally, our study of transform R

α

is mainly motivated by the study of production systems with substitution of production factors at the microlevel and this case corresponds to α 6= 1 .

It is important to note that in many applications related with generalized Radon transforms we are interested in anomalies in distribution of a quantity described by function u rather than in exact values of u . In this case the property of microlocal regularity allows us to estimate these anomalies via the singularities of R

α

u without constructing the inverse operator R

⁻¹_α

.

Besides, the knowledge of microlocal properties of operator R

_α

is very im- portant if we are interested in reconstruction problem for this operator from incomplete data (or in presence of noise). In particular, even if the reconstruc- tion is possible it may be not convenient for numerical computations if the generalized Radon transform has bad microlocal properties.

On the other hand, even if the generalized Radon transform is not injective, it is sometimes possible to construct a convenient pseudo-inverse operator if the generalized Radon transform has good microlocal properties. For related discussion, see, e.g., Ref. [Fa].

In addition, the property of microlocal regularity (in the case of analytic singularities) was used by some authors to estimate the support of function u given the support of its generalized Radon transform, see, e.g., Refs. [BQ1], [BQ2], [Bo], [Qu2], [Qu3], [Kr].

Note that the history of microlocal approach to generalized Radon trans- forms goes back to Refs. [Gu], [GS].

Concerning the results given in literature on microlocal properties of gener- alized Radon transforms, see, e.g., Refs. [ABKQ], [Bo], [BQ1], [BQ2], [FLU], [FQ], [GU], [KLQ], [Qu2], [Qu3], [SU] and references therein.

2 Main results

To formulate main results we need to introduce some notations. For xed α ∈ R \ 0 , x ∈ X we dene

P

_α,x

=

(p

₁

, p

₂

) ∈ P : q

_α

(p

₁

x

₁

, p

₂

x

₂

) = 1 ,

with orientation given by the volume form |∇

p

q

α

(p

1

x

1

, p

2

x

2

)|ω

α,x

, where ω

α,x

= d

p

q(p

1

x

1

, p

2

x

2

) y (dp

1

∧ dp

2

) .

The dual transform R

^∗_α

to R

α

maps a suciently regular real-valued function v on P to the function R

^∗_α

v on X given by the formula

R

^∗_α

v(x) = Z

Pα,x

v(p) ω

_α,x

. (2.1)

Note that since P ∼ = X , in fact, transform R

α

is self-dual.

Let Z

α

be the following incidence relation between the points in X and P : Z

_α

==

^def

(p

₁

, p

₂

; x

₁

, x

₂

) ∈ P × X : q

_α

(p

₁

x

₁

, p

₂

x

₂

) = 1 . We consider Z

_α

as a submanifold of P × X .

The total space of conormal bundle N

^∗

Z

_α

of Z

_α

in P × X is given by the following formula:

N

^∗

Z

_α

=

p

₁

, p

₂

; λ X

²

j=1

x

^α_j

p

^α−1_j

dp

_j

; x

₁

, x

₂

; λ X

²

j=1

p

^α_j

x

^α−1_j

dx

_j

∈

∈ T

^∗

P × T

^∗

X ∼ = T

^∗

(P × X) : q

_α

(p

₁

x

₁

, p

₂

x

₂

) = 1, λ ∈ R .

(2.2) We denote by N ˙

^∗

Z

α

the manifold N

^∗

Z

α

with zero section removed.

In a similar way, the total space of conormal bundle N

^∗

X

α,p

of X

α,p

in X is given by formula

N

^∗

X

α,p

=

(x

1

, x

2

; λ X

²

j=1

p

^α_j

x

^α−1_j

dx

j

) : (x

1

, x

2

) ∈ X, q

α

(p

1

x

1

, p

2

x

2

) = 1, λ ∈ R , p ∈ P.

(2.3) We dene the canonical relation C

α

to be the total space of the twisted conormal bundle of Z

α

in P × X with zero section removed:

C

α

== ( ˙

def

N

^∗

Z

α

)

⁰

= {(p, ηdp; x, −ξdx) ∈ T

^∗

P × T

^∗

X :

(p, ηdp, x, ξdx) ∈ N ˙

^∗

Z

α

}, (2.4) where

p = (p

₁

, p

₂

), η = (η

₁

, η

₂

), x = (x

₁

, x

₂

), ξ = (ξ

₁

, ξ

₂

), ηdp = η

1

dp

1

+ η

2

dp

2

, ξdx = ξ

1

dx

1

+ ξ

2

dx

2

. Theorem 2.1. Let α ∈ R \ {0, 1} be xed. Then:

1. R

α

is a Fourier integral operator associated with canonical relation C

α

. Hence R

α

is the linear continuous operator from C

_c^∞

(X ) to C

^∞

(P ) and from E

⁰

(X ) to D

⁰

(P) , where E

⁰

(X ) denotes the space of compactly sup- ported distributions from D

⁰

(X ) ;

2. (microlocal regularity) for any xed u ∈ E

⁰

(X ) the condition p 6∈ sing supp R

α

u is satised if and only if

W F (u) ∩ N

^∗

X

α,p

= ∅ , (2.5)

where W F denotes the wave front set and N

^∗

X

α,p

is dened in (2.3);

3. for any u ∈ E

⁰

(X) if (x, ξdx) ∈ W F (R

α

u) , ξ = (ξ

1

, ξ

2

) , then ξ

1

ξ

2

> 0 . Statement 2 of Theorem 2.1 tells us that if distribution u ∈ E

⁰

(X ) has a singularity at point x ∈ X

α,p

which is conormal to curve X

α,p

, then distribution R

α

u has a singularity at point p . Furthermore, all singularities of R

α

u arise in this way.

Note that since the dual transform R

^∗_α

dened in (2.1) is given by the same formula as transform R

α

(after changing x to p and vice versa), Theorem 2.1 also holds for R

_α^∗

if we replace X by P , X

α,p

by P

α,x

and vice versa in its statement.

Theorem 2.1 is proved in Section 3. The proof of the only if part of statement 2 of Theorem 2.1 is based on Lemma 3.1 of Section 3 which is proved in Section 5.

In order to formulate the following result we need to introduce two notations.

For xed 0 < δ <

¹₂

chose any χ

_δ

∈ C

^∞

(0, +∞) such that χ

δ

(t) =

( 0, if t < δ or t > δ

⁻¹

,

1, if t ≥ 2δ and t ≤

¹₂

δ

⁻¹

. (2.6) For xed α ∈ R \ {0, 1} dene γ(α) by formula

γ(α) =



 

 

1 + α

⁻¹

, α > 1,

−1 + 3α

⁻¹

, 0 < α < 1, 1 − 3α

⁻¹

, α < 0.

(2.7)

Proposition 2.1 (reconstruction of singularities). Let α ∈ R \ {0, 1} and 0 <

ε <

¹₂

be xed. Dene γ(α) by formula (2.7) and put δ = 2

^−1α−1

ε

^γ(α)

. Then for any u ∈ E

⁰

(X ) such that

supp u ⊆ {(x

1

, x

2

) ∈ X : ε < x

j

< ε

⁻¹

, j = 1, 2}, (2.8) and for all (x, ξdx) ∈ T

^∗

X , x = (x

1

, x

2

) , ξ = (ξ

1

, ξ

2

) , satisfying ε < x

j

< ε

⁻¹

, j = 1 , 2 , ε <

^ξ_ξ¹

2

< ε

⁻¹

, the following formula holds:

(x, ξdx) ∈ W F (u) if and only if (x, ξdx) ∈ W F R

^∗_α

χ

¹_δ

χ

²_δ

R

α

u

, (2.9) where χ

^j_δ

denotes the operator of multiplication by χ

δ

(p

j

) , j = 1 , 2 , and χ

δ

is dened in formula (2.6).

Proposition 2.1 is proved in Section 3. The proof is based on Lemma 3.1 of Section 3.

Proposition 2.1 allows us to reconstruct singularities of u given R

α

u and a priori information about bounds on support of u , by choosing an appropriate δ and applying operator R

^∗_α

χ

¹_δ

χ

²_δ

to R

α

u . For numerical examples, see Section 6.

Remark 2.1. Note that replacing function q

α

in formula (1.4) by the function of the same form but with n ≥ 3 arguments, we could consider the generalized Radon transform over the level hypersurfaces of CES-function in dimension n .

The generalization of Theorem 2.1 and Proposition 2.1 to this case is straight-

forward. We consider the two-dimensional case in order to simplify the proofs.

Remark 2.2. In fact, statements 1, 3 and the if part of statement 2 of Theorem 2.1 remain valid if we replace function q

α

in denition (1.2) by much more general smooth function q . In particular, we could consider functions q ∈ C

^∞

(X ) satisfying the following conditions:

Q1. q(λx) = λq(x) , x ∈ X , λ > 0 ,

Q2. q(x) , ∂

₁

q(x) , ∂

₂

q(x) are positive for x ∈ X , where ∂

_j

q = ∂q/∂x

_j

, j = 1 , 2 ; Q3. for any xed p ∈ P the map Φ

q,p

: X

q,p

→ (0, +∞) dened by

Φ

_q,p

(x) = x

₁

∂

₁

q(p

₁

x

₁

, p

₂

x

₂

) x

2

∂

2

q(p

1

x

1

, p

2

x

2

) , X

q,p

=

(x

1

, x

2

) ∈ X : q(p

1

x

1

, p

2

x

2

) = 1 , (2.10) is bijective.

These conditions are satised, in particular, by functions q

α

, α ∈ R \ 0 . In the above interpretation of transform (1.2) in the setting of the generalized model of pure industry functions q

_α

, α ≤ 1 , correspond to unit cost functions for production systems with constant coecient of elasticity of substitution of production factors whereas concave functions q dierent from q

_α

, α ≤ 1 , and satisfying assumptions Q1, Q2 correspond to production systems with variable coecient of elasticity of substitution.

Let us sketch the proof of statements 1, 2 (if part) and 3 of Theorem 2.1 in the case of smooth functions q satisfying Q1, Q2, Q3, see also the proof of Theorem 2.1.

Denote by R

q

the generalized Radon transform dened by formula (1.2) using function q instead of q

α

. Denote by C

q

the corresponding canonical relation.

The proof of statement 1 of Theorem 2.1 for this class of functions q remains the same as for functions q

α

, see Section 3.

It follows from Q2 that for any set M ⊆ T

^∗

X , if (p, ηdp) ∈ C

q

◦ M , η = (η

₁

, η

₂

) , then η

₁

η

₂

> 0 , where

C

_q

◦ M ==

^def

(p, ηdp) ∈ T

^∗

P : ∃(x, ξdx) ∈ M : (p, ηdp; x, ξdx) ∈ C

_q

. (2.11) This implies statement 3 of Theorem 2.1 if we take into account the inclusion W F (R

_q

u) ⊆ C

_q

◦ W F (u) , u ∈ E

⁰

(X ) .

It follows from assumptions Q1, Q2, Q3 that the canonical projection from C

q

to T

^∗

X is injective and that the following inclusion holds:

C

q

◦ W F (u) ∩ T

_p^∗

P ⊆ C

q

◦ N ˙

^∗

X

q,p

. (2.12) Using (2.12) we can obtain the if part of statement 2 of Theorem 2.1.

The only if part of statement 2 of Theorem 2.1 is the most dicult to

prove. To generalize it to the case of smooth functions q satisfying Q1, Q2, Q3

we need to obtain an analogue of Lemma 3.1 below. Restrictions in space and

time prevent us from obtaining this analogue in the present paper. Nevertheless,

we prove the main part of Lemma 3.1 (see also Lemma 4.1) for the general case of smooth functions q satisfying Q1, Q2 to be able to use it in a subsequent paper.

Remark 2.3. It follows from denitions (1.2), (2.1) that:

1. even if u is compactly supported in X , v = R

_α

u doesn't have compact support in general;

2. if v is not compactly supported in P , the value of R

^∗_α

v is not always dened.

Hence R

_α^∗

R

α

u is not always dened for u ∈ C

_c^∞

(X ) .

In the Gelfand's approach to generalized Radon transforms via double - brations (see, e.g., Ref. [GGS]) this corresponds to the fact that the canonical projection from Z

α

to X is not a proper map. Furthermore, this problem can't be resolved even by considering transform R

α

on functions which have supports in some xed compact subset of X .

Therefore, we are not able to deduce the only if part of statement 2 of Theorem 2.1 from the microlocal regularity properties of operator R

^∗_α

R

α

as it can be done when the canonical projection from the incidence relation to the source space is a proper map and the double bration corresponding to the generalized Radon transform satises the so-called Bolker assumption, see, e.g., Refs. [Gu], [GS], [Qu1] for more details.

3 Proof of Theorem 2.1 and Proposition 2.1

3.1 Approximation of R

_α

by proper Fourier integral oper- ators

In this subsection we show that R

α

is the Fourier integral operator associated with C

α

and we formulate an auxilary lemma, which is crucial in the proof of the only if part of statement 2 of Theorem 2.1.

Let α ∈ R \ {0, 1} be xed. For any u ∈ C

_c^∞

(X ) the following chain of equalities holds:

R

_α

u(p) = Z

X_α,p

u(x)ω

_α,p

= Z

R

δ(s − 1) Z

q_α(p₁x₁,p₂x₂)=s

u(x)ω

_α,p

ds

= 1 2π

Z

R

Z

R

e

^iθ(s−1)

Z

q_α(p₁x₁,p₂x₂)=s

u(x)ω

_α,p

dθds

= 1 2π

Z

R

Z

X

e

^{iφα(p,x,θ)}

u(x) dxdθ,

(3.1)

where p ∈ P and φ

α

(p, x, θ) is the real phase function:

φ

α

(p, x, θ) = θ(q

α

(p

1

x

1

, p

2

x

2

) − 1), p ∈ P, x ∈ X, θ ∈ R \ 0. (3.2)

It follows from (3.1), (3.2) that R

α

is a Fourier integral operator associated with C

α

. Hence R

α

is a linear continuous operator from C

_c^∞

(X ) to C

^∞

(P) and from E

⁰

(X) to D

⁰

(P ) . Statement 1 of Theorem 2.1 is proved.

Formula (3.1) implies that R

α

is not a proper operator. We are going to approximate R

α

by proper Fourier integral operators.

Let 0 < δ <

¹₂

be xed. Chose χ

_δ

∈ C

^∞

(0, +∞) as in (2.6) and χ e

_δ

∈ C

^∞

( R ) such that

χ e

δ

(t) =

( 1, |t| <

¹₂

δ, 0, |t| > δ.

Put

æ

α,δ

(p, x) = χ

δ

(p

1

)χ

δ

(p

2

)χ

δ

(x

1

)χ

δ

(x

2

) χ e

δ

(q

α

(p

1

x

1

, p

2

x

2

) − 1), (3.3) where p = (p

1

, p

2

) ∈ P , x = (x

1

, x

2

) ∈ X .

We dene the Fourier integral operator R

α,δ

by the following formula:

R

α,δ

u(p) = 1 2π

Z

R

Z

X

e

^{iφα(p,x,θ)}

æ

α,δ

(p, x)u(x) dxdθ, p ∈ P, (3.4) where u ∈ C

_c^∞

(X) and φ

_α

is dened in (3.2).

In a similar way with formula (3.1) we can obtain that R

α,δ

u(p) =

Z

X_α,p

æ

α,δ

(p, x)u(x)ω

α,p

, p ∈ P. (3.5) In particular, the denition of R

_α,δ

doesn't depend on the choice of function χ e

_δ

. It follows from formula (3.4) that the Schwartz kernel K

_α,δ

of operator R

_α,δ

is given by formula

K

α,δ

(p, x) = 1 2π

Z

R

e

^{iφα(p,x,θ)}

æ

α,δ

(p, x) dθ. (3.6) Formulas (3.3), (3.6) imply that

supp K

α,δ

⊆

(p

1

, p

2

; x

1

, x

2

) ∈ P × X :

q

_α

(p

₁

x

₁

, p

₂

x

₂

) ≤ δ, δ ≤ x

_j

≤ δ

⁻¹

, δ ≤ p

_j

≤ δ

⁻¹

, j = 1, 2 . (3.7) Hence, K

α,δ

is compact and R

α,δ

is a proper operator.

As a proper Fourier integral operator, R

_α,δ

continuously maps C

_c^∞

(X ) to C

_c^∞

(P ) and E

⁰

(X ) to E

⁰

(P ) . Note that the dual operator R

^∗_α,δ

is given by the same formula (3.4) (after changing x to p and vice versa). Hence, the composi- tion R

^∗_α,δ

R

_α,δ

is a well-dened proper linear continuous operator on C

_c^∞

(X) and on E

⁰

(X ) . We will show that, in fact, R

^∗_α,δ

R

_α,δ

is a pseudo-dierential operator and we will compute its principal symbol.

In order to formulate the following result we need to introduce the set Σ , 0 < < 1 :

Σ =

(ξ

1

, ξ

2

) ∈ R

²

: < ξ

₁

ξ

2

<

⁻¹

. (3.8)

Lemma 3.1. Let α ∈ R \ {0, 1} and 0 < δ <

¹₂

be xed. Then R

^∗_α,δ

R

α,δ

is a classical proper pseudo-dierential operator of order 1 on X with principal symbol σ

^P

(x, ξdx) , (x, ξdx) ∈ T

^∗

X , given by formula

σ

^P_α,δ

(x, ξdx) =



 

 

2π

|α|

exp i(π/4) · sgn(1 − α) · (1 − sgn ξ

₁

)

(x

₁

x

₂

)

^−1β

×

×|ξ

1

ξ

2

|

^−1β

|ξx|

^1−α2

æ

²_α,δ

(p

^∗

(x, ξ), x), if ξ

1

ξ

2

> 0, 0, otherwise ,

(3.9)

where x = (x

₁

, x

₂

) , ξ = (ξ

₁

, ξ

₂

) , p

^∗

(x, ξ) = (p

^∗₁

(x, ξ), p

^∗₂

(x, ξ)) , ξx = ξ

₁

x

₁

+ ξ

₂

x

₂

, 1/α + 1/β = 1 ,

p

^∗_j

(x, ξ) = |ξx|

^−α1

|ξ

j

|

^α1

x

⁻

1 β

j

, j = 1, 2. (3.10)

Besides, there exists = (α, δ) > 0 such that the full symbol σ

α,δ

(x, ξdx) of R

^∗_α,δ

R

α,δ

is zero for x ∈ X , ξ ∈ R

²

\ (0 ∪ Σ ∪ −Σ ) .

For the denition of classical pseudo-dierential operator see, e.g., Ref.

[Shu], Denition 3.5.

The biggest part of the present paper is devoted to the proof of Lemma 3.1.

In Section 4 we will present and prove a version of method of stationary phase and in Section 5 we will apply this method to prove Lemma 3.1.

3.2 Final part of the proof of Theorem 2.1

We begin this subsection by proving a lemma which follows from Lemma 3.1 and which will be used in the proof of the only if part of statement 2 of Theorem 2.1. In the end of this subsection we prove statements 2, 3 of Theorem 2.1.

We need to introduce some notations. For canonical relation C

_α

dened in (2.4) the transposed canonical relation C

_α^t

is dened by the following formula:

C

_α^t

==

^def

{(x, ξdx, p, ηdp) ∈ T

^∗

X × T

^∗

P : (p, ηdp, x, ξdx) ∈ C

α

} =

=

(x

1

, x

2

; λ X

²

j=1

p

^α_j

x

^α−1_j

dx

j

; p

1

, p

2

; −λ X

²

j=1

x

^α_j

p

^α−1_j

dp

j

) ∈

∈ T

^∗

X × T

^∗

P : q

α

(p

1

x

1

, p

2

x

2

) = 1, λ ∈ R \ 0 .

(3.11)

In a similar way with (2.11), for a subset M

⁰

⊆ T

^∗

P we denote C

_α^t

◦ M

⁰

==

^def

(x, ξdx) ∈ T

^∗

X : ∃(p, ηdp) ∈ M

⁰

: (x, ξdx; p, ηdp) ∈ C

_α^t

. (3.12) We also need to introduce the following sets:

Σ =

ξ = (ξ

1

, ξ

2

) ∈ R

²

: ξ

1

> 0, ξ

2

> 0 , T

_±^∗

X =

(x, ξdx) ∈ T

^∗

X : ξ ∈ ±Σ , T

_±^∗

P =

(p, ηdp) ∈ T

^∗

P : η ∈ ±Σ}, (3.13) X

δ

=

(x

1

, x

2

) ∈ X : δ < x

j

< δ

⁻¹

, j = 1, 2 , 0 < δ < 1, P

_δ

=

(p

₁

, p

₂

) ∈ P : δ < p

_j

< δ

⁻¹

, j = 1, 2 , 0 < δ < 1. (3.14)

Lemma 3.2. Let α ∈ R \ {0, 1} and u ∈ E

⁰

(X ) be xed. Then W F (u) ∩ T

₊^∗

X ∪ T

₋^∗

X

⊆ C

_α^t

◦ W F (R

α

u). (3.15) Proof. Let u ∈ E

⁰

(X ) and (x

⁰

, ξ

⁰

dx) ∈ W F (u) with ξ

⁰

∈ Σ ∪ (−Σ) be xed.

Lemma 3.2 will be proved if we show that

(x

⁰

, ξ

⁰

dx) ∈ C

_α^t

◦ W F (R

_α

u). (3.16) Chose 0 < δ <

¹₂

such that

supp u ⊆ X

_2δ

and p

^∗

(x

⁰

, ξ

⁰

) ∈ P

_2δ

, (3.17) where p

^∗

(x

⁰

, ξ

⁰

) = (p

^∗₁

(x

⁰

, ξ

⁰

), p

^∗₂

(x

⁰

, ξ

⁰

)) is dened in (3.10).

If follows from (3.3), (3.9), (3.17) that æ

_α,δ

(p

^∗

(x

⁰

, ξ

⁰

), x

⁰

) = 1 and

σ

_α,δ^P

(x

⁰

, ξ

⁰

dx) 6= 0. (3.18) Formula (3.18) implies that

(x

⁰

, ξ

⁰

dx) ∈ W F R

^∗_α,δ

R

α,δ

u

. (3.19)

Operator R

^∗_α,δ

is a Fourier integral operator associated with canonical rela- tion C

_α^t

and the following inclusion holds:

W F R

^∗_α,δ

R

α,δ

u

⊆ C

_α^t

◦ W F (R

α,δ

u). (3.20) It follows from formulas (2.6), (3.3), (3.5), (3.17) that

(R

α,δ

u)(p) = χ

δ

(p

1

)χ

δ

(p

2

)(R

α

u)(p), p ∈ P.

Hence

W F (R

_α,δ

u) ⊆ W F (R

_α

u). (3.21) Formula (3.16) follows from (3.19), (3.20), (3.21). Lemma 3.2 is proved.

Proof of statements 2, 3 of Theorem 2.1. We denote by π

_P

, π

_X

, e π

_P

, π e

_X

the canonical projections:

C

_α

π^P

←− − −−→

^πX

T ˙

^∗

P T ˙

^∗

X

C

_α^t

eπ^X

←− − −−→

^eπP

T ˙

^∗

X T ˙

^∗

P As above, the dot means that we exclude the zero section.

Note that the maps π

P

, π

X

, e π

P

, e π

X

are injective. Consider, for example, the map π

_P

. By denition

π

P

p

1

, p

2

; λ X

²

j=1

x

^α_j

p

^α−1_j

dp

j

; x

1

, x

2

; −λ X

²

j=1

p

^α_j

x

^α−1_j

dx

j

= p

₁

, p

₂

; λ X

²

j=1

x

^α_j

p

^α−1_j

dp

_j

.

(3.22)

It follows from (3.22) that the value of π

P

(p, ηdp, x, ξdx) determines uniquely the ratio x

1

/x

2

. Since x

^α₁

p

^α₁

+ x

^α₂

p

^α₂

= 1 , it also determines x

1

, x

2

. Hence, it also determines λ and the map π

P

is injective.

The following formulas follow from (2.2), (2.3), (2.4), (3.11):

π

P

◦ π

_X⁻¹

(W F (u)) ⊆ T

₊^∗

P ∪ T

₋^∗

P, (3.23) π

P

◦ π

_X⁻¹

( ˙ N

^∗

X

α,p

) = T

_+,p^∗

P ∪ T

_−,p^∗

P, (3.24) e π

_X

◦ e π

⁻¹_P

(T

_p^∗

X \ 0) = ˙ N

^∗

X

_α,p

. (3.25) where p ∈ P , T

_±^∗

P is dened in (3.13) and T

_±,p^∗

= T

_±^∗

P ∩ T

_p^∗

P .

Besides, using denitions (2.4), (3.11), (2.11), (3.12) we obtain that

π

_P

◦ π

⁻¹_X

(W F (u)) = C

_α

◦ W F (u), (3.26) π e

X

◦ π e

_P⁻¹

(W F (R

α

u)) = C

_α^t

◦ W F (R

α

u). (3.27) Statement 3. Using (3.23), (3.26) and inclusion W F(R

α

u) ⊆ C

α

◦ W F (u) we obtain that if (x, ξdx) ∈ W F (R

α

u) , ξ = (ξ

1

, ξ

2

) , then ξ

1

ξ

2

> 0 . Statement 3 of Theorem 2.1 is proved.

Statement 2: if part. Suppose that (2.5) holds. Using injectivity of π

P

we obtain that

π

P

◦ π

_X⁻¹

(W F (u)) ∩ π

P

◦ π

_X⁻¹

( ˙ N

^∗

X

α,p

) = ∅ .

This formula together with formulas (3.23), (3.24), (3.26) imply that C

α

◦ W F (u) ∩ T

_p^∗

P = ∅ . (3.28) Using formula (3.28) and inclusion W F (R

α

u) ⊆ C

α

◦ W F (u) we obtain that p 6∈ sing supp R

α

u .

Statement 2: only if part. Suppose that p 6∈ sing supp R

α

u . Equivalently, W F (R

α

u) ∩ (T

_p^∗

P \ 0) = ∅ . (3.29) Using injectivity of π e

X

and formulas (3.25), (3.27) we obtain from (3.29) the following formula:

C

_α^t

◦ W F (R

α

u) ∩ N ˙

^∗

X

α,p

= ∅ . (3.30) Note that formula (2.3) implies that

N ˙

^∗

X

α,p

⊆ T

₊^∗

X ∪ T

₋^∗

X. (3.31) Using inclusion (3.15) of Lemma 3.2 and formulas (3.30), (3.31), we obtain formula (2.5). Theorem 2.1 is proved.

3.3 Proof of Proposition 2.1

Let u ∈ E

⁰

(X ) and (x, ξdx) ∈ T

^∗

X satisfying the conditions of Proposition 2.1 be xed. It follows from the denition of δ and from (3.10) that

σ

^P_α,δ

(x, ξdx) 6= 0, (3.32)

where σ

_α,δ^P

is the symbol of operator R

^∗_α,δ

R

α,δ

dened in (3.9).

Only if part. Suppose that (x, ξdx) ∈ W F (u) . Then it follows from (3.32) that

(x, ξdx) ∈ W F R

_α,δ^∗

R

α,δ

u

. (3.33)

Since u satises (2.8) and 2δ < ε , the following formula is valid:

R

_α,δ

u = χ

¹_δ

χ

²_δ

R

_α

u. (3.34) Also note that

R

^∗_α,δ

= ˜ χ

¹_δ

χ ˜

²_δ

R

^∗_α

χ

¹_δ

χ

²_δ

, (3.35) where χ ˜

^j_δ

is the operator of multiplication by χ

_δ

(x

_j

) , j = 1 , 2 .

Formulas (3.34), (3.35) imply that W F R

_α,δ^∗

R

_α,δ

u

⊆ W F (R

^∗_α

(χ

¹_δ

)

²

(χ

²_δ

)

²

R

_α

u .

From this formula, taking into account that we can replace (χ

^j_δ

)

²

by χ

^j_δ

, j = 1 , 2 , and from formula (3.33) it follows that

(x, ξdx) ∈ W F R

^∗_α

χ

¹_δ

χ

²_δ

R

_α

u

. (3.36)

If part. Suppose now that (3.36) holds. Then we have the following se- quence of inclusions:

(x, ξdx) ∈ W F R

^∗_α

χ

¹_δ

χ

²_δ

R

_α

u

⊆ C

_α^t

◦ W F (χ

¹_δ

χ

²_δ

R

α

u) ⊆ C

_α^t

◦ W F (R

α

u)

⊆ C

_α^t

◦ C

α

◦ W F (u) ⊆ W F (u),

since R

^∗_α

is the Fourier integral operator associated with canonical relation C

_α^t

and the inclusion C

_α^t

◦ C

_α

⊆ diag(T

^∗

X × T

^∗

X) holds, where

diag(T

^∗

X × T

^∗

X ) ==

^def

(x, ξdx, x, ξdx) ∈ T

^∗

X × T

^∗

X , (3.37) C

_α^t

◦ C

_α

==

^def

(y, ζdy, x, ξdx) ∈ T

^∗

X × T

^∗

X :

∃(p, ηdp) ∈ T

^∗

P : (y, ζdy, p, ηdp) ∈ C

_α^t

, (p, ηdp, x, ξdx) ∈ C

_α

. (3.38) Proposition 2.1 is proved.

4 Method of stationary phase

In this section we formulate and prove a version of method of stationary phase which will be used in Section 5 in the proof of Lemma 3.1.

We say that a subset Σ b ⊆ R

²

\ 0 is conic if

tξ ∈ R

²

: t > 0, ξ ∈ Σ b = Σ. b

Let Σ b ⊆ R

²

\ 0 be an open conic set. We denote by S

^γ

(X × Σ) b , γ ∈ R, the set of functions a ∈ C

^∞

(X × Σ) b satisfying the following property: for any compact K ⊂ X × Σ b and for any α = (α

₁

, α

₂

) , β = (β

₁

, β

₂

) ∈ Z

²+

there exists such constant C

_K^α,β

> 0 that

D

_x^α

D

^β_ξ

a(x, ξ)

≤ C

_K^α,β

|ξ|

^γ−|β|

, |ξ| ≥ 1, ξ = t ξ, t > e 0, (x, ξ) e ∈ K, where

D

_x^α

= ∂

^|α|

∂x

^α₁¹

∂x

^α₂²

, D

^β_ξ

= ∂

^|β|

∂ξ

^β₁¹

∂ξ

₂^β2

, and |α| = α

₁

+ α

₂

, |β| = β

₁

+ β

₂

.

The space S

^γ

(P × Σ) b , γ ∈ R, is dened in a similar way. We also put S

^−∞

(X × Σ) = b ∩

_γ∈R

S

^γ

(X × Σ) b and S

^−∞

(P × Σ) = b ∩

_γ∈R

S

^γ

(P × Σ) b .

We will formulate and prove a version of method of stationary phase for the integrals of the following form:

I(p, ξ) = Z

X_q,p

e

^iS(x,ξ)

f (x, ξ) ω

_q,p

, (4.1) where q ∈ C

^∞

(X ) satises Q1, Q2 of Remark 2.2, X

q,p

is dened in (2.10) with orientation given by the volume form |∇

x

q(p

1

x

1

, p

2

x

2

)|ω

q,p

, where ω

q,p

is the GelfandLeray form:

ω

q,p

= d

x

q(p

1

x

1

, p

2

x

2

) y (dx

1

∧ dx

2

), and S ∈ C

^∞

(X × Σ) b satises the following assumption:

S1. S(x, λξ) = λS(x, ξ) , x ∈ X , ξ ∈ Σ b , λ > 0 .

Assumtion Q2 of Remark 2.2 implies that at xed p ∈ P the curve X

q,p

is the graph of C

^∞

function

x

2

= x

2

(x

1

, p), x

⁻₁

(p) < x

1

< x

⁺₁

(p) ≤ +∞. (4.2) Put S(t, p, ξ) = e S((t, x

2

(t, p)), ξ) .

We suppose that q and S satisfy the following additional assumptions:

QS1. at xed p ∈ P , ξ ∈ Σ b function S(·, p, ξ) e has the unique local extremum x

^∗₁

(p, ξ) on the interval (x

⁻₁

(p), x

⁺₁

(p)) which is the unique zero of function S e

t

(·, p, ξ) ; the map x

^∗₁

: X × Σ b → (0, +∞) is C

^∞

and open;

QS2. the second derivative S e

_tt

(x

^∗₁

(p, ξ), p, ξ) is nonzero and has the same sign for all p ∈ P , ξ ∈ Σ b .

Lemma 4.1. Let Σ b be Σ or −Σ . Let q ∈ C

^∞

(X) satisfy assumptions Q1, Q2 of

Remark 2.2 and S ∈ C

^∞

(X × Σ) b satisfy assumption S1 above. Suppose also that

q , S satisfy assumptions QS1, QS2 dened above. Suppose that f ∈ S

^m

(X × Σ) b ,

m ∈ R, and f (·, ξ) has compact support in X for any xed ξ ∈ Σ b . Then:

1. the following inclusion holds:

e

^{−iS∗(p,ξ)}

I(p, ξ) ∈ S

^m−12

(P × Σ), b (4.3) where I(p, ξ) is dened in (4.1), S

^∗

(p, ξ) = S(x

^∗

(p, ξ), ξ) , x

^∗

(p, ξ) = (x

^∗₁

(p, ξ), x

^∗₂

(p, ξ)) , x

^∗₁

is dened in assumption QS1 above, x

^∗₂

(p, ξ) = x

2

(x

^∗₁

(p, ξ), p) ;

2. if, in addition, function f satises

f (x, λξ) = λ

^m

f (x, ξ), x ∈ X, ξ ∈ Σ, λ > b 0, (4.4) then there exist the unique functions a

j

∈ C

^∞

(P × Σ) b , j ∈ N ∪0 , satisfying a

j

(p, λξ) = λ

^m−12−j

a

j

(p, ξ), p ∈ P, ξ ∈ Σ, λ > b 0, j ∈ N ∪ 0, (4.5) and such that for any N ∈ N ∪ 0 the following formula is valid:

e

^{−iS∗(p,ξ)}

I(p, ξ) − X

^N

j=0

a

j

(p, ξ) = r

N

(p, ξ) ∈ S

^m−N−32

(P × Σ); b (4.6) furthermore, the following formula is holds:

a

0

(p, ξ) = (2π)

^1/2

e

^iπ/4

f (x

^∗

(p, ξ), ξ) S e

tt

(x

^∗₁

(p, ξ), p, ξ)

^−1/2

×

×p

⁻¹₂

∂

2

q p

1

x

^∗₁

(p, ξ), p

2

x

^∗₂

(p, ξ)

−1

. (4.7)

Proof. We suppose that Σ = Σ b and

S e

tt

(x

^∗₁

(p, ξ), p, ξ) > 0, p ∈ P, ξ ∈ Σ. (4.8) The other three cases ( Σ = b −Σ , S e

tt

> 0 ; Σ = Σ b , S e

tt

< 0 ; Σ = b −Σ , S e

tt

< 0 ) can be considered in a similar way.

The proof of Lemma 4.1 consists of six steps.

Step 1: uniform Morse coordinate. Using integration by parts and taking into account that S e

t

(x

^∗₁

(p, ξ), p, ξ) = 0 we obtain that at xed p ∈ P , ξ ∈ Σ for x

1

suciently close to x

^∗₁

(p, ξ) the following chain of equalities holds:

S(x e

₁

, p, ξ) − S

^∗

(p, ξ)

= Z

1

0

(1 − τ ) d

²

dτ

²

S x e

^∗₁

(p, ξ) + τ(x

1

− x

^∗₁

(p, ξ)), p, ξ dτ

= (x

1

− x

^∗₁

(p, ξ))

²

Z

1

0

(1 − τ) S e

tt

x

^∗₁

(p, ξ) + τ(x

1

− x

^∗₁

(p, ξ), p, ξ dτ

= 1

2 z

₁²

(x

₁

, p, ξ),

(4.9)

where z

1

(x

1

, p, ξ) is dened for x

1

suciently close to x

^∗₁

(p, ξ) : z

1

(x

1

, p, ξ) = √

2(x

1

− x

^∗₁

(p, ξ))×

× Z

1

0

(1 − τ) S e

tt

x

^∗₁

(p, ξ) + τ (x

1

− x

^∗₁

(p, ξ)), p, ξ dτ

^1/2

. (4.10)

Let 0 < < 1 be xed. Dene s = x

1

− x

^∗₁

(p, ξ) for x

1

∈ (x

⁻₁

(p), x

⁺₁

(p)) . Recall that the sets Σ , P , > 0 , are dened in formulas (3.8), (3.14). Since x

^∗₁

is open and continuous, the set x

^∗₁

(P × Σ ) in an interval and x

^∗₁

(P × Σ ) b (x

⁻₁

(p), x

⁺₁

(p)) .

Hence, for all p ∈ P and ξ ∈ Σ the range of variable s contains an interval (− ν e

, e ν

) , e ν

> 0 , not depending on p , ξ .

Fix p e ∈ P and ξ e ∈ Σ . It follows from (4.8), (4.10) that

∂z

1

∂x

₁

x

^∗₁

( p, e ξ), e p, e ξ e

= S e

_tt

(x

^∗₁

( p, e ξ), e p, e ξ e

^1/2

> 0. (4.11) Since the derivatives S

_tt

(x

₁

, p, ξ) and ∂z

₁

/∂x

₁

(x

₁

, p, ξ) are continuous w.r.t.

(x

₁

, p, ξ) , there exist open neighborhood U

₁

( p, e ξ) e of point 0 ∈ R, open neigh- borhood U

₂

( p, e ξ) e of point p e ∈ P and open conic neighborhood U

₃

( p, e ξ) e of point ξ e ∈ Σ such that for any (s, p, ξ) ∈ Π

³_j=1

U

_j

( e p, ξ) e function z

₁

(x

₁

, p, ξ) is well- dened and the following inequality holds:

∂z

1

∂x

₁

s + x

^∗₁

(p, ξ), p, ξ

> 0.

The set P

is compact and the set Σ

is conically compact (i.e., it is conic and its intersection with the unit circle S

¹

in R

²

is compact). The system of sets Π

³_j=2

U

_j

( e p, ξ) e , p e ∈ P , ξ e ∈ Σ , is an open cover of P

× Σ

. Let Π

³_j=2

U

_j

( p e

^k

, ξ e

^k

) , k = 1 , . . . , N

₁

, be its nite subcover. Denote by (−ν

, ν

) any symmetric with respect to zero interval contained in ∩

^N_k=1¹

U

₁

( p e

^k

, ξ e

^k

) . Then z

₁

(x

₁

, p, ξ) is well-dened for x

₁

− x

^∗₁

(p, ξ) ∈ (−ν

, ν

) , p ∈ P

, ξ ∈ Σ

and

∂z

1

∂x

₁

(x

1

, p, ξ) > 0 for x

1

− x

^∗₁

(p, ξ) ∈ (−ν , ν ) , p ∈ P , ξ ∈ Σ . (4.12) Step 2: partition of unity. Fix p e ∈ P , ξ e ∈ Σ . Since x

^∗₁

is continuous, there exist an open neighborhood P e ( p) e of p e in P

and an open conic neighborhood Σ(e e ξ) of ξ e in Σ

such that |x

^∗₁

( P e ( p) e × Σ(e e ξ))| <

¹₂

ν

.

The set P

₂

is compact and the set Σ

₂

is conically compact. The sets P( e p)× e Σ(e e ξ) , p e ∈ P

, ξ e ∈ Σ

, form an open cover of P

₂

×Σ

₂

. Let P( e p e

^k

)× Σ(e e ξ

^k

) , k = 1 , . . . , N

₂

be its nite subcover, so that

P

2

× Σ

2

⊆ [

^N2

k=1

P e ( e p

^k

) × Σ(e e ξ

^k

)

⊆ P × Σ . (4.13) Denote (x

⁻_1,k

, x

⁺_1,k

) = x

^∗₁

( P e ( e p

^k

) × Σ(e e ξ

^k

)) , k = 1 , . . . , N

2

. Then

x

⁺_1,k

− x

⁻_1,k

<

¹₂

ν , k = 1, . . . , N

2

. (4.14) It follows from (4.12), (4.14) that

∂z

1

∂x

1

(x

1

, p, ξ) > 0 on (x

⁻_1,k

−

¹₂

ν , x

⁺_1,k

+

¹₂

ν ) × P( e p e

^k

) × Σ(e e ξ

^k

) . (4.15)

Fix k ∈ {1, . . . , N

2

} . Let ϕ

^k₁

∈ C

^∞

(0, +∞) satisfy 0 ≤ ϕ

^k₁

(x

1

) ≤ 1 , x

1

> 0 , ϕ

^k₁

(x

₁

) =

( 1, if x

⁻_1,k

−

¹₈

ν < x

1

< x

⁺_1,k

+

¹₈

ν ,

0, if x

1

< x

⁻_1,k

−

¹₄

ν or x

1

> x

⁺_1,k

+

¹₄

ν . (4.16) We dene ϕ

^k₂

= 1 − ϕ

^k₁

. Formula (4.16) implies that

dist supp ϕ

^k₂

, x

^∗₁

( P( e p e

^k

) × Σ(e e ξ

^k

)

≥

¹₈

ν . (4.17) At xed k ∈ {1, . . . , N

2

} , p ∈ P , ξ ∈ Σ we rewrite integral (4.1) in the following form:

I(p, ξ) = I

₁^k

(p, ξ) + I

₂^k

(p, ξ), (4.18) I

_j^k

(p, ξ) =

Z

x⁺₁(p) x⁻₁(p)

e

^{iS(xe 1,p,ξ)}

f (x

1

, x

2

(x

1

, p), ξ)ϕ

^k_j

(x

1

) dx

1

p

2

∂

2

q(p

1

x

1

, p

2

x

2

(x

1

, p)) , j = 1, 2, (4.19) where ∂

2

q(y

1

, y

2

) =

_∂y^∂q

2

(y

1

, y

2

) .

Step 3: integral I

₁^k

(p, ξ) . It follows from (4.15) that at xed p ∈ P e ( p e

^k

) , ξ ∈ Σ(e e ξ

^k

) function z

₁

(·, p, ξ) on (x

⁻_1,k

−

¹₂

ν

, x

⁺_1,k

+

¹₂

ν

) has the inverse x

₁

(·, p, ξ) .

It follows from (4.9), (4.16), (4.19) that I

₁^k

(p, ξ) = e

^iS∗(p,ξ)

Z

+∞

−∞

e

^2iz21

g

^k

(z

1

, p, ξ) dz

1

, (4.20) where

g

^k

(z

1

, p, ξ) =



 

 

 

 

f (x

1

(z

1

, p, ξ), x

2

(z

1

, p, ξ), ξ)

^∂x_∂z¹

1

(z

1

, p, ξ )×

×p

⁻¹₂

∂

₂

q(p

₁

x

₁

(z

₁

, p, ξ), p

₂

x

₂

(z

₁

, p, ξ))

−1

×

×ϕ

^k₁

(x

1

(z

1

, p, ξ)), if x

1

(z

1

, p, ξ) ∈ supp ϕ

^k₁

, 0, otherwise ,

(4.21)

and x

₂

(z

₁

, p, ξ) = x

₂

(x

₁

(z

₁

, p, ξ), p) .

Dene e z

1

(x

1

, p, ξ) = |ξ|

⁻¹²

z

1

(x

1

, p, ξ) . It follows from assumption S1 and from formula (4.10) that z e

1

(x

1

, p, λξ) = z e

1

(x

1

, p, ξ) , x

1

∈ (x

⁻_1,k

−

¹₂

ν , x

⁺_1,k

+

¹₂

ν ) , p ∈ P e ( p e

^k

) , ξ ∈ Σ(e e ξ

^k

) , λ > 0 . Note that at xed p ∈ P e ( p e

^k

) , ξ ∈ Σ(e e ξ

^k

) the inverse e x

1

(·, p, ξ) of z e

1

(·, x, ξ) on (x

⁻_1,k

−

¹₂

ν , x

⁺_1,k

+

¹₂

ν ) exists and the following formulas hold:

x

1

(z

1

, p, ξ) = e x

1

(|ξ|

⁻¹²

z

1

, p, ξ), (4.22) e x

1

( e z

1

, p, λξ) = e x

1

( e z

1

, p, ξ), λ > 0, (4.23) for all z

1

(resp. e z

1

) in the domain of denition of x

1

(·, p, ξ) (resp. x e

1

(·, p, ξ) ), p ∈ P e ( p e

^k

) , ξ ∈ Σ(e e ξ

^k

) .

It follows from assumption S1 and from formula (4.10) that the quantity

|ξ|

⁻¹²

z

1

(x

1

, p, ξ) is uniformly bounded for x

1

∈ supp ϕ

^k₁

, p ∈ P e ( p e

^k

) , ξ ∈ Σ(e e ξ

^k

) .

Taking this into account and using formulas (4.22), (4.23) we obtain that at xed k ∈ {1, . . . , N

2

} for any α , β ∈ Z

²+

there exist constants C

_1,k^α,β

> 0 such that the following inequalities hold:

D

^α_p

D

^β_ξ

x

₁

(z

₁

, p, ξ) ,

D

_p^α

D

^β_ξ

x

₂

(z

₁

, p, ξ) ,

D

^α_p

D

^β_ξ

ϕ

^k₁

(x

₁

(z

₁

, p, ξ)) , D

^α_p

D

^β_ξ

q(p

₁

x

₁

(z

₁

, p, ξ), p

₂

x

₂

(z

₁

, p, ξ))

−1

≤ C

_1,k^α,β

|ξ|

^−|β|

, D

^α_p

D

^β_ξ

∂x

₁

∂z

1

(z

₁

, x, ξ)

≤ C

_1,k^α,β

|ξ|

^−|β|−12

, D

^α_p

D

^β_ξ

f(x

₁

(z

₁

, p, ξ), x

₂

(z

₁

, p, ξ), ξ)

≤ C

_1,k^α,β

|ξ|

^m−|β|

,

(4.24)

where p ∈ P( e p e

^k

) , ξ ∈ Σ(e e ξ

^k

) , |ξ| ≥ 1 , z

1

∈ z

1

(supp ϕ

^k₁

, p, ξ) .

Formulas (4.20), (4.21) and inequalities (4.24) imply that at xed k ∈ {1, . . . , N

2

} for any α , β ∈ Z

²+

there exist constants C

_2,k^α,β

> 0 , C

_3,k^α,β

> 0 such that

D

^α_p

D

^β_ξ

g

^k

(z

₁

, p, ξ)

≤ C

_2,k^α,β

|ξ|

^{m−12−|β|}

, (4.25) D

^α_p

D

^β_ξ

e

^{−iS∗(p,ξ)}

I

₁^k

(p, ξ)

≤ C

_3,k^α,β

|ξ|

^{m−12−|β|}

. (4.26) where z

1

∈ R, p ∈ P e ( p e

^k

) , ξ ∈ Σ(e e ξ

^k

) , |ξ| ≥ 1 .

We need to introduce some notations. Let P e ⊆ P be an open set and Σ e ⊆ Σ be an open conic set. Let γ ∈ R be xed. We denote by S e

^γ

( P e × Σ) e the set of functions a ∈ C

^∞

( P e × Σ) e such that for any α , β ∈ Z

²+

there exists a constant C e

^α,β

> 0 such that

D

_p^α

D

^β_ξ

a(p, ξ)

≤ C e

^α,β

|ξ|

^γ−|β|

, p ∈ P , ξ e ∈ Σ, e |ξ| ≥ 1. (4.27) Besides, we denote S e

^−∞

( P e × Σ) = e ∩

γ∈R

S e

^γ

( P e × Σ) e .

Note that we can rewrite formula (4.26) in the following form:

e

^{−iS∗(p,ξ)}

I

₁^k

(p, ξ) ∈ S e

^m−12

( P e ( p e

^k

) × Σ(e e ξ

^k

)), k = 1, . . . , N

2

. (4.28) Step 4: functions a

j

(p, ξ) . Throughout the Steps 4, 5 we suppose that (4.4) holds.

Using assumptions Q1, S1 and formulas (4.4), (4.19), (4.20) we obtain the identity

Z

+∞

−∞

e

^i2z12

g

^k

(z

1

, p, λξ) dz

1

= λ

^m

Z

+∞

−∞

e

^i2λz12

g

^k

(z

1

, p, ξ) dz

1

, (4.29) where p ∈ P( e p e

^k

) , ξ ∈ Σ(e e ξ

^k

) , λ > 0 .

Using the Plancherel identity we get Z

+∞

−∞

e

^i2λz21

g

^k

(z

₁

, p, ξ) dz

₁

= (2πλ)

⁻¹²

e

^iπ4

Z

+∞

−∞

e

^it

2 2λ

g b

^k

(t, p, ξ) dt, (4.30)

where p ∈ P( e p e

^k

) , ξ ∈ Σ(e e ξ

^k

) , λ > 0 and

b g

^k

(t, p, ξ) = Z

+∞

−∞

e

^−itz1

g

^k

(z

1

, p, ξ) dz

1

.

Note that the following formula holds for t ∈ R, λ > 0 , N ∈ N ∪ 0 : e

^it

2 2λ

=

N

X

j=0

1 j!

it

²

2λ

^j

+ 1

N ! it

²

2

^N+1

Z

1/λ 0

e

^2ist2

1

λ − s

^N

ds.

Hence, the following equality is valid:

+∞

Z

−∞

e

^it

2 2λ

b g

^k

(t, p, ξ) dt =

N

X

j=0

(2iλ)

^−j

j!

+∞

Z

−∞

(it)

^2j

b g

^k

(t, p, ξ) dt + r

^k_N

(λ, p, ξ), (4.31)

where p ∈ P( e p e

^k

) , ξ ∈ Σ(e e ξ

^k

) , λ > 0 and

r

^k_N

(λ, p, ξ) =

+∞

Z

−∞

1/λ

Z

0

(it

²

)

^N+1

N!2

^N+1

e

^i2st2

1 λ − s

^N

b g

^k

(t, p, ξ) ds dt. (4.32) Taking into account formulas (4.16), (4.21), we obtain the formula

Z

+∞

−∞

(it)

^2j

b g

^k

(t, p, ξ) dt = 2π ∂

^2j

g

∂z

₁^2j

(0, p, ξ), j = 0, . . . , N, (4.33) where p ∈ P( e p e

^k

) , ξ ∈ Σ(e e ξ

^k

) ,

g(z

1

, p, ξ) = f (x

1

(z

1

, p, ξ), x

2

(z

1

, p, ξ), ξ) ∂x

1

∂z

₁

(z

1

, p, ξ)×

×p

⁻¹₂

∂

₂

q(p

₁

x

₁

(z

₁

, p, ξ), p

₂

x

₂

(z

₁

, p, ξ))

−1

.

(4.34)

Note that functions x

1

(z

1

, p, ξ) (resp. e x

1

( e z

1

, p, ξ) ), g(z

1

, p, ξ) at xed p ∈ P , ξ ∈ Σ are dened, at least, for z

1

(resp. z e

1

) in a small neighborhood of 0 ∈ R and in this neighborhood formulas (4.22), (4.23) hold. This is a consequence of (4.11), (4.34).

Formulas (4.22), (4.23) imply that

∂

^s

x

₁

∂z

^s₁

(0, p, λξ) = λ

^−s2

∂

^s

x

₁

∂z

₁^s

(0, p, ξ), λ > 0, (4.35) where p ∈ P , ξ ∈ Σ , s ∈ N ∪ 0 . Using (4.4), (4.34), (4.35) we obtain the identity

∂

^2j

g

∂z

₁^2j

(0, p, λξ) = λ

^m−j−12

∂

^2j

g

∂z

₁^2j

(0, p, ξ), λ > 0, j ∈ N ∪ 0, (4.36)

where p ∈ P , ξ ∈ Σ .