Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems

By Vesselin M. Petkov and Luchezar N. Stoyanov

This book is a new edition of a title originally published in1992. No other book has been published that treats inverse spectral and inverse scattering results by using the so called Poisson summation formula and the related study of singularities. This book presents these in a closed and comprehensive form, and the exposition is based on a combination of different tools and results from dynamical systems, microlocal analysis, spectral and scattering theory.

The content of the first edition is still relevant, however the new edition will include several new results established after 1992; new text will comprise about a third of the content of the new edition. The main chapters in the first edition in combination with the new chapters will provide a better and more comprehensive presentation of importance for the applications inverse results. These results are obtained by modern mathematical techniques which will be presented together in order to give the readers the opportunity to completely understand them. Moreover, some basic generic properties established by the authors after the publication of the first edition establishing the wide range of applicability of the Poison relation will be presented for first time in the new edition of the book.

• Language: English
• Publisher: Wiley
• Release date: Nov 16, 2016
• ISBN: 9781119107699

Book preview

Chapter 1

Preliminaries from differential topology and microlocal analysis

Here we collect some facts concerning manifolds of jets, spaces of smooth maps and transversality, as well as some material from microlocal analysis. A special emphasis is given to the definition and main properties of the generalized bicharacteristics of the wave operator and the corresponding generalized geodesics.

1.1 Spaces of jets and transversality theorems

We begin with the notion of transversality, manifolds of jets and spaces of smooth maps. The reader is referred to Golubitsky and Guillemin [GG] or Hirsch [Hir] for a detailed presentation of this material.

In this book smooth means C∞.

Let X and Y be smooth manifolds and let f : X c01-math11 Y be a smooth map. Given x ∈ X, we will denote by c01-math-006 the tangent map of f at x. Sometimes we will use the notation c01-math-009 . If c01-math-010 (resp. c01-math-011 ), then f is called an immersion (resp. a submersion) at x. Let W be a smooth submanifold of Y. We will say that f is transversal to W at x ∈ X, and will denote this by c01-math-019 , if either c01-math-020 or c01-math-021 and c01-math-022 . Here for every y ∈ W we identify c01-math-024 with its image under the map c01-math-025 , where i : W c01-math11 Y is the inclusion. Clearly, if f is a submersion at x, then c01-math-029 for every submanifold W of Y. If Z ⊂ X and c01-math-033 for every x ∈ Z, we will say that f is transversal to W on Z. Finally, if f is transversal to W on the whole X, we will say that f is transversal to W and write c01-math-043 .

The next proposition contains a basic property of transversality that will be used several times throughout.

Proposition 1.1.1

Let f : X c01-math11 Y be a smooth map, and let W be a smooth submanifold of Y such that c01-math-047 . Then c01-math-048 is a smooth submanifold of X with

1.1 equation

In particular:

a. if c01-math-051 , then c01-math-052 , that is c01-math-053 .

b. if c01-math-054 , then c01-math-055 consists of isolated points in X.

Consequently, if f is a submersion, then for every submanifold W of Y, c01-math-060 is a submanifold of X with (1.1). Thus, in this case, c01-math-062 is a submanifold of X of codimension equal to dim(Y) for every y ∈ Y.

Let again X and Y be smooth manifolds and let x ∈ X. Given two smooth maps c01-math-069 , we will write c01-math-070 if c01-math-071 . For an integer k ≥ 2, we will write c01-math-073 if for the smooth maps c01-math-074 , we have c01-math-075 for every c01-math-076 . In this way by induction one defines the relation c01-math-077 for all integers k ≥ 1. Fix for a moment x ∈ X and y ∈ Y. Denote by c01-math-081 the family of all equivalence classes of smooth maps f : X c01-math11 Y with c01-math-083 with respect to the equivalence relation c01-math-084 . Define the space of k-jets by

equation

So, for each k-jet c01-math-087 , there exist x ∈ X and y ∈ Y with c01-math-090 . We set c01-math-091 and c01-math-092 , thus obtaining maps

1.2

equation

called the source and the target map, respectively. Given an arbitrary smooth f : X c01-math11 Y, let

1.3 equation

be the map assigning to every x ∈ X the equivalence class c01-math-097 of f in c01-math-099 .

There is a natural structure of a smooth manifold on c01-math-100 for every k. We refer the reader to [GG] or [Hir] for its description and main properties. Let us only mention that with respect to this structure for every smooth map f the maps (1.2) and (1.3) are also smooth.

For a non-empty set A and an integer s ≥ 1, define

equation

Note that if A is a topological space, then c01-math-106 is an open (dense) subset of the product space As. If f : A c01-math11 B is an arbitrary map, define c01-math-109 by

equation

Let X and Y be smooth manifolds, let s and k be natural numbers and let c01-math-114 . The open submanifold

equation

of c01-math-115 is called an s-fold k-jet bundle. For a smooth f : X c01-math11 Y, define the smooth map

equation

by

equation

We will now define the Whitney Ck topology on the space c01-math-120 of all smooth maps from X into Y. Let k ≥ 0 be an integer and let U be an open subset of c01-math-125 . Set

equation

The family c01-math-126 , where U runs over the open subsets of c01-math-128 , is the basis for a topology on c01-math-129 , called the Whitney Ck topology. The supremum of all Whitney Ck topologies for k ≥ 0 is called the Whitney C∞ topology. It follows from this definition that c01-math-134 as n c01-math11 ∞ in the C∞ topology if c01-math-137 in the Ck topology for all k ≥ 0. Note that if X is not compact (and c01-math-141 ), then any of the Ck topologies (including the case k = ∞) does not satisfy the first axiom of countability, and therefore is not metrizable. On the other hand, if X is compact, then all Ck topologies on c01-math-146 are metrizable with complete metrics.

In this book we always consider c01-math-147 with the Whitney C∞ topology. An important fact about these spaces, which will be often used in what follows, is that whenever X and Y are smooth manifolds, the space c01-math-151 is a Baire topological space. Recall that a subset R of a topological space Z is called residual in Z if R contains a countable intersection of open dense subsets of Z. If every residual subset of Z is dense in it, then Z is called a Baire space.

In some of the next chapters we will consider spaces of the form c01-math-159 , X being a smooth submanifold of c01-math31 for some n ≥ 2. Let us note that these spaces have a natural structure of Frechet spaces. Moreover, if X is compact, then c01-math-164 has a natural structure of a Banach space. Denote by

equation

the subset of c01-math-165 consisting of all smooth embeddings c01-math-166 . Then C(X) is open in c01-math-168 (cf. Chapter II in [Hir]), and therefore it is a Baire topological space with respect to the topology induced by c01-math-169 . Finally, notice that for compact X the space C(X) has a natural structure of a Banach manifold. We refer the reader to [Lang] for the definition of Banach manifolds and their main properties.

The following theorem is known as the multijet transversality theorem and will be used many times later in this book.

Theorem 1.1.2

Let X and Y be smooth manifolds, let k and s be natural numbers and let W be a smooth submanifold of c01-math-177 . Then

equation

is a residual subset of c01-math-178 . Moreover, if W is compact, then TW is open in c01-math-181 .

For s = 1, this theorem coincides with Thom's transversality theorem.

We conclude this section with a special case of the Abraham transversality theorem which will be used in Chapter 6. Now by a smooth manifold we mean a smooth Banach manifold of finite or infinite dimension (cf. [Lang]).

Let c01-Ascr-nor , X and Y be smooth manifolds, and let

1.4 equation

be a map, c01-math-187 . Define

1.5 equation

by c01-math-189 .

The next theorem is a special case of Abraham's transversality theorem (see [AbR]).

Theorem 1.1.3

Let rho have the form (1.4) and let W be a smooth submanifold of Y.

a. If the map (1.5) is C¹ and K is a compact subset of X, then

equation

is an open subset of c01-Ascr-nor .

b. Let c01-math-197 , c01-math-198 and let r be a natural number with c01-math-200 . Suppose that the manifolds c01-Ascr-nor , X and Y satisfy the second axiom of countability, the map (1.5) is Cr and c01-math-205 . Then

equation

is a residual subset of c01-Ascr-nor .

1.2 Generalized bicharacteristics

Our aim in this section is to define the generalized bicharacteristics of the wave operator

equation

and to present their main properties which will be used throughout the book. Here we use the notation from Section 24 in [H3]. In what follows c01-Omega1 is a closed domain in c01-math-208 with a smooth boundary c01-math3 .

Given a point on c01-math3 , we choose local normal coordinates

equation

in c01-math-211 about it such that the boundary c01-math3 is given by c01-math-213 and c01-Omega1 is locally defined by c01-math-215 . We assume that the coordinates ξi are those dual to c01-math2 . The coordinates x, ξ can be chosen so that the principal symbol of c01-squ has the form

equation

where

equation

and c01-math-220 is homogeneous of order 2 in ξ′. Introduce the sets

equation

The sets Sigma , H and G are called the characteristic set, the hyperbolic set and the glancing set, respectively. Let

equation

The diffractive and the gliding sets are defined by

equation

respectively.

Next, consider the Hamiltonian vector fields

equation

Notice that c01-math-225 on Sigma and c01-math-227 on G, so c01-math4 and c01-math-230 are not radial on Sigma and G, respectively. Next, introduce the sets

equation

The above definitions are independent of the choice of local coordinates. Let us mention that if c01-math3 is given locally by φ = 0 and c01-Omega1 by φ > 0, c01-phiv being a smooth function, then

equation

We define the generalized bicharacteristics of c01-squ using the special coordinates c01-math-239 chosen above.

Definition 1.2.1

Let I be an open interval in c01-math6 . A curve

1.6 equation

is called a generalized bicharacteristic of c01-squ if there exists a discrete subset B of I such that the following conditions hold:

i. If c01-math-246 for some c01-math-247 , then c01-gamma is differentiable at t0 and

equation

ii. If c01-math-250 for some c01-math-251 , then c01-math-252 is differentiable at t0 and

equation

iii. If c01-math-254 , then c01-math-255 for all c01-math-256 , t ∈ I, with c01-math-258 sufficiently small. Moreover, for c01-math-259 , we have

equation

This definition does not depend on the choice of the local coordinates. Note that when c01-math3 is given by φ = 0 and c01-Omega1 by φ > 0, then the condition (ii) means that if c01-math-264 , then

equation

where

equation

is the so-called glancing vector field on G.

It follows from the above definition that if (1.6) is a generalized bicharacteristic, the functions x(t), c01-math-267 , c01-math-268 are continuous on I, while c01-math-270 has jump discontinuities at any t ∈ B. The functions c01-math-272 and c01-math-273 are continuously differentiable on I and

1.7 equation

Moreover, for t ∈ B, x1(t) admits left and right derivatives

1.8

equation

The function c01-math-279 also has a left derivative and a right derivative. For c01-math-280 , we have

1.9

equation

while c01-math-282 for c01-math-283 . Thus, if γ(t) remains in a compact set, then the functions c01-math-285 and c01-math-286 satisfy a uniform Lipschitz condition. For the left and right derivatives of c01-math-287 , one gets

1.10 equation

Melrose and Sjöstrand [MS2] (see also Section 24 in [H3]) showed that for each c01-math-289 , there exists a generalized bicharacteristic (1.6) of c01-squ with c01-math-291 for some c01-math-292 . Since the vector fields c01-math4 and c01-math-294 are not radial on Sigma and G, respectively, such a bicharacteristic c01-gamma can be extended for all c01-math-298 . However, in general, c01-gamma is not unique. We refer the reader to [Tay] or [H3] for examples demonstrating this.

For ρ ∈ Σ, denote by c01-math-301 the set of those μ ∈ Σ such that there exists a generalized bicharacteristic (1.6) with c01-math-303 , c01-math-304 and c01-math-305 . In many cases c01-math-306 is related to a uniquely determined bicharacteristic c01-gamma . In the general case it is convenient to introduce the following.

Definition 1.2.2

A generalized bicharacteristic c01-math-308 of c01-squ is called uniquely extendible if for each c01-math-310 , the only generalized bicharacteristics (up to a change of parameter) passing through γ(t) is c01-gamma . That is, for c01-math-313 , we have c01-math-314 for all c01-math-315 .

It was proved by Melrose and Sjöstrand [MS1] that if c01-math-316 , then c01-gamma is uniquely extendible. If c01-math-318 for some c01-math-319 , then γ(t) meets c01-math3 transversally at x(t0) and (iii) holds. For c01-math-323 we have c01-math-324 for c01-math-325 small enough, while in the case c01-math-326 for small c01-math-327 , γ(t) coincides with the gliding ray

1.11 equation

where c01-math-330 is a null bicharacterstic of the Hamiltonian vector field c01-math-331 .

To discuss the local uniqueness of generalized bicharacteristics, let c01-math-332 be such a bicharacteristic and let c01-math-333 be the solution of the problem

1.12 equation

Then setting c01-math-335 , we have the following local description of c01-gamma .

Proposition 1.2.3

Let c01-math-337 . If e(t) increases for small t > 0, then for such t the bicharateristic γ(t) is a trajectory of c01-math4 . If e(t) decreases for c01-math-344 , then for such t, γ(t) is a gliding ray of the form (1.11).

A proof of this proposition and some other properties of generalized bicharacteristics can be found in Section 24.3 in [H3].

It should be mentioned that for k ≥ 3 and c01-math-348 , we have

equation

therefore the sign of c01-math-349 determines the local behaviour of e(t).

Corollary 1.2.4

In each of the following cases, every generalized bicharacteristic of c01-squ is uniquely extendible:

a. the boundary c01-math3 is a real analytic manifold;

b. there are no points c01-math-353 at which the normal curvature of c01-math3 vanished of infinite order in some direction c01-math-355 ;

c. c01-math3 is given locally by φ = 0 and

1.13 c01-math-358

for every z ∈ G. If c01-math3 is locally convex in the domain of c01-phiv , then (1.13) holds.

Proof

In the case (a) the symbols c01-math-362 and c01-math-363 are real analytic, so the solution c01-math-364 of (1.12) is analytic in t. Consequently, the function e(t) is analytic and we can use its Taylor expansion in order to apply Proposition 1.2.3.

In the case (c), using the special coordinates x, ξ, and combining (1.13) with (1.9), we get c01-math-368 . On the other hand, if c01-math-369 has a jump at c01-math-370 , then this jump is equal to c01-math-371 . Thus, the function c01-math-372 is increasing. If c01-math-373 for c01-math-374 , we get c01-math-375 for such t, so c01-math-377 is a gliding ray. Assume that there exists a sequence c01-math-378 such that c01-math-379 for all k ≥ 1. Then c01-math-381 for all sufficiently small t > 0. Now (1.8) shows that x1(t) is increasing for such t, therefore there is c01-math-385 such that c01-math-386 coincides with a trajectory of c01-math4 .

Let c01-math-388 and let c01-phiv depend on c01-math-390 only. Then

equation

and if the boundary c01-math3 is locally convex, we obtain (1.13).

Finally, in the case (b), for each c01-math-392 there exists a multi-index c01-alpha , depending on x, such that c01-math-395 . This implies c01-math-396 , which completes the proof.

According to Lemma 6.1.2, in the generic case discussed in Chapter 6 the assumption (b) is always satisfied.

Let c01-math-397 . We will again use the coordinates c01-math-398 , this time denoting the last coordinate by t, that is c01-math-400 . For c01-math-401 , let c01-math-402 be the space of covectors c01-math-403 vanishing on c01-math-404 . Define the equivalence relation c01-math41 on c01-math-406 by c01-math-407 if and only if either c01-math-408 and ξ = η, or c01-math-410 and c01-math-411 . Then c01-math-412 can be naturally identified over c01-math-413 with c01-math-414 . Consider the map

equation

defined as the identity on c01-math-415 . Then c01-math-416 is called the compressed characteristic set, while the image c01-math15 of a bicharacteristic c01-gamma under c01-math41 is called a compressed generalized bicharacteristic. Clearly c01-math15 is a continuous curve in Σb.

Given c01-math-422 , c01-math-423 , denote by c01-math-424 the standard Euclidean distance between rho and c01-mu . For ρ , μ ∈ Σ define

equation

Clearly, c01-math-428 if and only if ρ ∼ μ, and c01-math-430 provided c01-math-431 and c01-math-432 . It is easy to check that D is symmetric and satisfies the triangle inequality. Thus, D is a pseudo-metric on Sigma , which induces a metric on Σb.

For the next lemma we assume that I is a closed non-trivial interval in c01-math6 , c01-math-439 and c01-math5 is a neighbourhood of c01-math-441 in Q.

Lemma 1.2.5

There exists a constant c01-math-443 depending only on c01-math5 and I such that for every generalized bicharacteristic c01-math-446 we have

equation

for all t , s ∈ I.

Proof

It is enough to consider the case when c01-math-448 is small. Then we can use the local coordinates introduced earlier. From (1.7) (1.8) and (1.10), we get

equation

where c01-math-449 is a constant independent of t and s. Thus, if c01-math-452 or c01-math-453 we get c01-math-454 . The latter holds also in the case when c01-math-455 for all c01-math-456 . Consequently, c01-math-457 whenever either c01-math-458 or c01-math-459 only for finitely many c01-math-460 .

Assume that there are infinitely many c01-math-461 such that c01-math-462 is a reflection point of c01-gamma . Then there exists c01-math-464 with c01-math-465 . Hence,

equation

and using the triangle inequality for D, we complete the proof of the assertion.

The next lemma shows that any sequence of generalized bicharacteristics has a subsequence that is convergent on a given compact interval.

Lemma 1.2.6

Let c01-math-467 be a compact interval in c01-math6 , let K be a compact subset of Sigma and let

c01-math-471

be a generalized bicharacteristic of c01-squ for every natural number k. Then there exists an infinite sequence c01-math-474 of natural numbers and a generalized bicharacteristic c01-math-475 such that

1.14 equation

for all t ∈ I.

Proof

Using local coordinates, we see that the derivatives of c01-math-478 and c01-math-479 and the left and right derivatives of c01-math-480 and c01-math-481 are uniformly bounded for t ∈ I and k ≥ 1. Hence the maps c01-math-484 and c01-math-485 are uniformly Lipschitz, which implies that there exists an infinite sequence c01-math-486 of natural numbers such that the sequences c01-math-487 , c01-math-488 and c01-math-489 are uniformly convergent for t ∈ I. It now follows from Proposition 24.3.12 in [H3] that there exists a generalized bicharacteristic c01-math-491 of c01-squ such hat

1.15 equation

for all t ∈ I with c01-math-495 .

Let c01-math-496 be such that c01-math-497 is a reflection point of c01-gamma . Then there exists a sequence c01-math-499 with c01-math-500 for all j. Thus,

equation

By Lemma 1.2.5, the first two terms in the right-hand side can be estimated uniformly with respect to m, while for the third term we can use (1.15). Taking j and m sufficiently large, we obtain (1.14), which proves the lemma.

In what follows we will use local coordinates c01-math-505 and the corresponding local coordinates c01-math-506 . In these coordinates the principal symbol p of c01-squ has the form

equation

where c01-math-509 and c01-math-510 is homogeneous of order 2 in ξ′. Consequently, the vector fields c01-math4 and c01-math-513 do not involve derivatives with respect to c01-tau , so by Definition 1.2.1, the variable c01-tau remains constant along each generalized bicharacteristic. This makes it possible to parametrize every generalized bicharacteristic by the time t.

Given c01-math-517 , consider the points

equation

Assume that locally c01-math3 is given by c01-math-519 and c01-Omega1 by c01-math-521 . Let c01-math-522 be a hyperbolic point and let c01-math-523 be the different real roots of the equation

equation

with respect to z. Denote by c01-gamma the generalized bicharacteristic parameterized by a parameter s such that

equation

Then c01-math-527 along c01-gamma and the time t increases when s increases. Such a bicharacteristic will be called forward. For the right derivative of x1(t) we get

equation

since for small t > 0, γ(t) enters the interior of c01-Omega1 and c01-math-535 . Therefore, setting

equation

we find

equation

In the case c01-math-536 it may happen that there exist several forward bicharacteristic passing through c01-math-537 . Denote by c01-math-538 the set of those

equation

such that c01-math-539 and c01-math-540 and c01-math-541 lie on forward generalized bicharacteristics of c01-squ . In a similar way we define c01-math-543 using a backward bicharacteristic, determined as the forward ones replacing c01-math-544 by c01-math-545 . The set c01-math-546 is called the bicharacteristic relation of c01-squ . If c01-math-548 and τ < 0 (resp. τ > 0), we will say that c01-mu is a reflection point of a forward (resp. backward) bicharacteristic. Similarly, if c01-math-552 , then rho is a reflection point of a generalized bicharacteristic passing through c01-math-554 , and, working in local coordinates as before, the sign of c01-tau determines uniquely c01-math-556 . The sets c01-math-557 and C are homogeneous with respect to c01-math-559 , that is c01-math-560 implies c01-math-561 for all c01-math-562 .

Lemma 1.2.7

The sets c01-math-563 are closed in c01-math-564 .

Proof

Since c01-math-565 is homogeneous, it is sufficient to show that if

equation

for all k ≥ 1 and there exists

equation

then c01-math-567 . Let c01-math-568 be a generalized bicharacteristic of c01-squ such that c01-math-570 and c01-math-571 lie on Im c01-math-572 . If one of these points belongs to H, we consider it as a reflection point of c01-math-574 , according to theabove-mentioned convention by suitably choosing c01-math-575 . Assume c01-math-576 . Then there exists a compact set K ⊂ Σ such that c01-math-578 for all c01-math-579 , so we can apply the argument in the proof of Lemma 1.2.6. Consequently, there exists an infinite sequence c01-math-580 of natural numbers and a generalized bicharacteristic c01-gamma satisfying (1.14) and (1.15). Then for the Euclidean distance d we find

equation

If c01-math-583 , according to (1.15) and the continuity of x(t), c01-math-585 and c01-math-586 , we get

1.16 equation

as m c01-math11 ∞, which shows that c01-math-589 . If c01-math-590 , then by our convention, c01-math-591 and c01-math-592 have the same sign for large m, which implies c01-math-594 .

Therefore, c01-math-595 is closed. In the same way one proves that c01-math-596 is closed as well.

Using c01-math-597 we now define the so-called generalized Hamiltonian flow c01-math38 of c01-squ ; it is sometimes called the broken Hamiltonian flow. Given c01-math-600 , set

equation

In general, c01-math-601 is not a one-point set. Nevertheless, setting

equation

for c01-math-602 , we have the group property

equation

The flow generated by c01-math-603 is c01-math-604 .

Let c01-math3 be locally given by c01-math-606 and let

equation

be the principal symbol of c01-squ . A point

equation

is called hyperbolic (resp. glancing) for c01-squ if the equation

1.17 equation

with respect to c01-math25 has two different real roots (resp. a double real root). These definitions are invariant with respect to the choice of the local coordinates. If (1.17) has no real roots, then c01-sigmasmall is called an elliptic point. Clearly, the set of hyperbolic points is open in c01-math-612 , while that of the glancing points is closed.

Let c01-math-613 be the natural projection, c01-math-614 .

Definition 1.2.8

A continuous curve c01-math-615 is called a generalized geodesic in c01-Omega1 if there exists a generalized bicharacteristic c01-math-617 such that

1.18 equation

Notice that, in general, a generalized geodesic is not uniquely determined by a point on it and the corresponding direction. If the generalized bicharacteristic c01-gamma with (1.18) satisfies

equation

we will say that g (or Im c01-math1 ) is a reflecting ray in c01-Omega1 . Two special kinds of such rays will be studied in detail in Chapter 2. One of them is defined as follows.

Definition 1.2.9

A point c01-math-623 is called periodic with period T ≠ 0 if

equation

A generalized bicharacteristic c01-math-625 , c01-math-626 , will be called periodic with period T ≠ 0 if for each c01-math-628 the point c01-math-629 is periodic with period T. The projections on c01-Omega1 of the periodic generalized bicharacteristics of c01-squ are called periodic generalized geodesics.

Notice that if c01-math-633 , then c01-math-634 , since we can change the orientation on the bicharacteristic passing through c01-math-635 . A uniquely extendible bicharacteristic c01-gamma is periodic provided Im c01-math-637 contains a periodic point. If T is the period of a generalized geodesic g, then c01-math-640 coincides with the standard length of the curve Im c01-math1 .

Let c01-math11 Ω be the set of all periodic generalized geodesics in c01-Omega1 . For c01-math-644 we denote by Tg the length of Im c01-math1 . We call length spectrum the following set

equation

Lemma 1.2.10

The set LΩ is closed in c01-math6 and c01-math-649 .

Proof

Consider a convergent sequence c01-math-650 of elements of LΩ converging to some c01-math-652 as k c01-math11 ∞. Then for every k ≥ 1 there exists a generalized bicharacteristic c01-math-655 of c01-squ with period Tk passing through a point of the form c01-math-658 . If c01-math-659 , choosing a subsequence as in the proof of Lemma 1.2.7, we obtain c01-math-660 .

It remains to show that the case c01-math-661 is impossible. Assume c01-math-662 . Passing to an appropriate subsequence, we may assume that there exists c01-math-663 and for every t there exists

equation

provided c01-math-665 and c01-math-666 . If x0 is in the interior of c01-Omega1 , then xk is also in the interior of c01-Omega1 for large k. Then for such k, c01-math-673 is in the interior of c01-Omega1 for sufficiently small t > 0, which is a contradiction. If there exists t′ with c01-math-677 and c01-math-678 in the interior of c01-Omega1 , then we get a contradiction by the same argument.

It remains to consider the case when c01-math-680 for all c01-math-681 . Then for such t, c01-math-683 is an integral curve of the glancing vector field c01-math-684 . Since the latter is not radial, c01-math-685 has no stationary points for c01-math-686 . Given a small neighbourhood U of x0 in c01-math3 , there exist c01-math-690 such that c01-math-691 and c01-math-692 for c01-math-693 . Since c01-math-694 as k c01-math11 ∞ uniformly for c01-math-696 , for sufficiently large k there exists a natural number mk with

equation

Then

c01-math-699

, which is a contradiction. This proves that c01-math-700 and this completes the proof of the proposition.

1.3 Wave front sets of distributions

In this section we collect some basic facts concerning wave fronts of distributions. For more details, we refer the reader to the books of Hörmander [Hl] [H3].

Let X be an open subset of c01-math31 and let c01-math-703 be the space of all distributions on X. The singular support sing c01-math-705 of c01-math-706 is a closed subset of X such that if c01-math-708 there exists an open neighbourhood U of x0 in X and a smooth function c01-math-712 such that

equation

For a more precise analysis of c01-math-713 , it is useful to consider the directions c01-math-714 along which the Fourier transform c01-math-715 of the distribution c01-math-716 is not rapidly decreasing, provided c01-math-717 and c01-math-718 .

Definition 1.3.1

Let c01-math-719 and let c01-Oscr-nor be the set of all c01-math-721 for which there exists an open neighbourhood U of x0 in X and an open conic neighbourhood V of ξ0 in c01-math31 so that for c01-math-728 and ξ ∈ V we have

equation

The closed subset

equation

of c01-math-730 is called the wave front set of u.

It is easy to see that c01-math-732 is a conic subset of c01-math-733 with the property

equation

where c01-math-734 is the natural projection.

For our aims in Chapter 3 we will describe the wave front sets of distributions given by oscillatory integrals. Such integrals have the form

1.19 equation

Here the phase c01-math-736 is a C∞ real-valued function, defined for c01-math-738 , and c01-math5 is an open conic set, i.e. c01-math-740 implies c01-math-741 for all t > 0. We assume that c01-phiv has the properties:

equationequation

The amplitude c01-math-744 belongs to the class of symbols c01-math-745 , formed by C∞ functions on c01-math-747 such that for each compact K ⊂ X and all multi-indices α, β, we have

1.20

equation

We endow c01-math-751 with the topology defined by the semi-norms

equation

where c01-math-752 is an increasing sequence of compact sets with c01-math-753 .

Let c01-math-754 be a closed cone and let c01-math-755 . For c01-math-756 we will now define the integral

equation

to obtain a distribution in c01-math-757 . To do this, we need a regularization, since the integral in c01-theta is not convergent for c01-math-759 .

Choose a function c01-math-760 such that c01-math-761 for c01-math-762 and c01-math-763 for c01-math-764 . For c01-math-765 , the functions c01-math-766 form a bounded set in c01-math-767 . Then the functions c01-math-768 also form a bounded set in c01-math-769 and

equation

as ϵ c01-math11 0 for each c01-math-771 .

Consider the operator

equation

with

equation

and c01-math-772 . For each compact set K ⊂ X we have

equation

where c01-math-774 depends on K only. Clearly

equation

and the operator c01-math-776 formally adjoint to L has the form

equation

with

equation

The operator c01-math-778 is a continuous map of Sm onto c01-math-780 . Define the linear map c01-math-781 by

1.21

equation

For c01-math-783 the integral on the right-hand side of (1.21) is absolutely convergent, and it is easy to see that c01-math-784 becomes a distribution in c01-math-785 . Thus, we obtain the following.

Proposition 1.3.2

Let c01-math-786 and c01-math-787 be as above. Then the oscillatory integral (1.19) defines a distribution c01-math-788 given by (1.21).

We are now going to describe the set c01-math-789 .

Theorem 1.3.3

We have

1.22

equation

Proof

Let c01-math-791 . Then the Fourier transform

equation

is expressed by an oscillatory integral. Let V be a closed cone in c01-math31 such that

equation

By compactness, there exists δ > 0 such that

1.23

equation

for c01-math-796 , c01-math-797 and ξ ∈ V. To obtain (1.23) it suffices to observe that if the latter conditions are satisfied, then the left-hand side of (1.23) is positive and then use the homogeneity with respect to c01-math-799 . As above, consider the operator

equation

with

equation

Then

equation

and applying (1.23), we conclude that

equation

This implies (1.22).

For asymptotics of oscillatory integrals depending on a parameter c01-math-800 we have the following.

Lemma 1.3.4

Let c01-math-801 , c01-math-802 and let c01-math-803 be a real-valued function. Assume

equation

Then for each c01-math-804 we have

equation

Proof

Choosing a finite partition of unity, we can restrict our attention to the case c01-math-805 . Set

equation

Then

equation

Here W is a closed conic set such that c01-math-807 ,

equation

and c01-math-808 is interpreted as an oscillatory integral. For c01-math-809 and ξ ∈ W we have

equation

with δ > 0. Using the same argument as in the proof of Theorem 1.3.3, we see that c01-math-812 for all c01-math-813 . For c01-math-814 we use the fact that if c01-math-815 and c01-math-816 , then c01-math-817 is rapidly decreasing. This proves the assertion.

Now let c01-math-818 be a closed conic set. Set

equation

Using an argument similar to that in the proof of Lemma 1.3.4, it is easy to see that c01-math-819 if and only if for each c01-math-820 and each closed cone c01-math-821 with

1.24 equation

we have

equation

This makes it possible to introduce the following.

Definition 1.3.5

Let c01-math-823 and let c01-math-824 . We will say that the sequence c01-math-825 converges to u in c01-math-827 if:

a. c01-math-828 weakly in c01-math-829 ,

b. c01-math-830 for every c01-math-831 , every c01-math-832 and every closed cone V satisfying (1.24).

For every c01-math-834 there exists a sequence c01-math-835 converging to u in c01-math-837 . To prove this, consider two sequences χj, c01-math-839 such that c01-math-840 on Kj, c01-math-842 , c01-math-843 and c01-math-844 . Then

equation

and c01-math-845 in c01-math-846 . Moreover, the condition (b) also holds, so c01-math-847 in c01-math-848 .

For our aims in Chapter 3 we need to justify some operations on distributions (see [Hl] for more details). For convenience of the reader we list these properties, including only one proof of these – namely that of the existence of the pull-back c01-math-849 . We use the notation from [Hl].

Let c01-math-850 and c01-math-851 be open sets and let f : X c01-math11 Y be a smooth map. Consider a closed cone c01-math-853 and set

equation

For c01-math-854 , consider the map

equation

Theorem 1.3.6

Let c01-math-855 . Then the map c01-math-856 can be extended uniquely on the space c01-math-857 such that

1.25 equation

Proof

Using a partition of unity, we may consider only the case when X and Y are small open neighbourhoods of c01-math-861 and c01-math-862 , respectively. Set

equation

Choose a small compact neighbourhood X0 of x0 and a closed conic neighbourhood V of c01-math-866 so that

equation

Next, choose a small compact neighbourhood Y0 of y0 with c01-math-869 for all c01-math-870 .

Now let c01-math-871 and let c01-math-872 be a sequence such that c01-math-873 in c01-math-874 . Choosing c01-math-875 with φ = 1 on f(X0), we have

equation

where

equation

For c01-math-878 and η ∈ V we obtain

equation

Using the operator

equation

we integrate by parts in c01-math-880 and get

equation

for all c01-math-881 . On the other hand, there exists M > 0 such that

equation

Thus, I1 is absolutely convergent, and we can consider the limit as j c01-math11 ∞. To deal with I2, notice that c01-math-886 . For c01-math-887 , (b) yields the estimates

1.26 equation

uniformly with respect to j. Thus, we can let j c01-math11 ∞ in I2.

To establish (1.25), replace χ(x) by c01-math-893 and write

equation

Choose a small open conic neighbourhood W of the set

equation

so that c01-math-895 and η ∈ V imply c01-math-897 . As above, for c01-math-898 , η ∈ V and ξ ∉ W we deduce the estimate

equation

For such c01-xi and c01-eta we integrate by parts in c01-math-903 and obtain

equation

For η ∉ V, ξ ∉ W we choose a function c01-math-906 with c01-math-907 for c01-math-908 , and consider the operator

equation

Then c01-math-909 , and, as in the previous case, for η ∉ Vand ξ ∉ W, we get the estimates

equation

Combining these estimates with (1.26), we obtain

equation

for ξ ∉ W, where the constant CN does not depend on j. Letting j c01-math11 ∞ proves (1.25).

By an easy modification of the above-mentioned argument, one proves the following modification of Theorem 1.3.6 for distributions depending on a parameter.

Corollary 1.3.7

Let Z be a compact subset of c01-math-917 and let

equation

be a continuous map. Under the assumptions of Theorem 1.3.6, the map

equation

is continuous.

Next, consider a linear continuous map

equation

By Schwartz's theorem (cf. Theorem 5.2.1 in [Hl]), there exists a distribution c01-math-918 , called the kernel