DEGRUYTER EXPOSITIONS
IN MATHEMATICS
Vladimir E. Nazaikinskii Victor E. Shatalov Boris Yu. Stern in
6
I DE
Contact Geometry and Linear Differential Equations
Contact Geometry and Linear Differential Equations by
Vladimir E. Nazaikinskii Victor E. Shatalov Boris Yu. Sternin
w DE
C Walter de Gruyter Berlin New York 1992
Authors
Vladimir E. Nazaikinskii, Victor E. Shatalov, Boris Yu. Sternin
Department of Computational Mathematics and Cybernetics Moscow State University Lenin Hills 119899 Moscow, Russia 1991 Mathematics Subject Primary: 58-02; 35-02. Secondary: 58015, 58G16, 58017; 35A05, 35A20, 35A30, 35B25, 35840, 35C20, 35L67, 35S05; 42810; 47030; 53C1 5 Keywords: Contact geometry, partial differential equations. Fourier integral operators,
Hamiltonian operator, Maslov canonical operator, pseudodifferential operators, symplectic structure ® Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Caialoging-ln-Publication Data Nazalkinskil, V. E.
Contact geometry and linear differential equations / by Vladimir E. Nazalkinskil, Victor E. Shatalov, Boris Yu. Sternm. cm. — (De Gruyter expositions in mathematics. p. ISSN 0938-6572 ; 6). Includes bibliographical references and index. ISBN 3-11-013381-4 (cloth ; acid-free)
I. Differential equations, Linear. 2. WKB approximation, I. Shatalov, V. E. (Viktor Eugen'evich) II. Sternin. B. IU. III. 'fltle. IV. Series. QA372.N39 1992 515'.354—dc20
92-24930
CIP Die Deutsche Bibliothek — Cataloging-ln-Publication Data
Nu*JkIisklj, VlsdIIr E.: Contact geometry and linear differential equations / by Vladimir E. Nazaikinskii ; Victor E. Shatalov; Boris Yu. Sternin. — Berlin; New York : de Gruyter, 1992 (De Gruyter expositions in mathematics 6) ISBN 3-11-013381-4 NE: Viktor E.:; Sternin. Boris J.:; GI
Copyright 1992 by Walter de Gruyter & Co., D-1000 Berlin 30. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Disk Conversion: D. L. Lewis, Berlin. Printing: Gerikc GmbH, Berlin. Binding: Lüderitz & Bauer GmbH. Berlin. Cover design: Thomas Bonnie, Hamburg.
Contents
Introduction
v
Chapter 1
Homogeneous functions, Fourier transformation, and contact structures 1. Integration on manifolds 2. Analysis on PP and smooth homogeneous functions on 3. Homogeneous and formally homogeneous distributions 4. Fourier transformation of homogeneous functions 5. Homogeneous symplectic and contact structures 6. Functorial properties of the phase space and local representation of Lagrangian manifolds. The classification lemma
I
11
24 30 44 63
Chapter II
Fourier-Maslov operators 1. Maslov's canonical operator theory) 2. Fourier—Maslov integral operators 3. Singularities of hyperbolic equations; examples and applications
78 78 112 137
Chapter 111
Applications to differential equations
149
1. Equations of principal type 2. Microlocal classification of pseudodifferential operators 3. Equations of subprincipal type
149 175 192
References Index
215
211
Introduction
The method of characteristics (known also as the WKB-method), which goes back to Peter Debye, is a classical method to solve differential equations. It has been
closely related to the geometry of the phase space since the very beginning of its development (Hubert's invariant integral, Bohr—Sommerfeld quantization conditions, and so forth). its true geometric interpretation, however, was not given before Maslov's canonical operator, an advanced global version of the method,
was developed ([M Il, [MF 11, [MiSSh in. Namely, Lagrangian manifolds as special submanifolds of the phase (cotangent) space are those very objects whose quantization leads to global asymptotic solutions for equations containing a small (or large) parameter (a so-called quasiclassical approximation). It has turned out that Maslov's works on quasiclassical asymptotics (which originally related to a certain field of physical applications) are applicable to the "pure" theory of differential equations as well, producing in particular asymptotics of solutions with respect to smoothness, existence theorems, and so forth. The corresponding techniques known as Fourier integral operators (see, e.g., ED 2], [H 31, [DH 11, [NOsSSh 11, [MiSSh I], and others) have undergone intensive development in the last two decades. It has implied essential progress in the theory of differential equations with real and complex characteristics (see, e.g., [E 1], [H 2], [MeSj 1,21, [MeIU 1], [Shu 1], [1 1), [Tr 1], [SSh 1) and other publications). The latter case (the "smooth" theory of differential equations) is quite different from the former one from the geometric point of view. Here the main geometric concept, the phase space, is homogeneous with respect to the multiplicative action
of the group R÷ of positive numbers in the fibers as well as all the other objects (Hamiltonians, Lagrangian manifolds, the "homogeneous" Maslov canonical operator and Fourier integral operators, and so on). The contact geometry (see, e.g., [Ar 2J,[Ly I]) of the quotient space with respect to the group action, however, is a more adequate geometrical framework in the "smooth" case. Since differential equations are being considered, the space to be factorized is the cotangent bundle with the zero section deleted, and the action of of nonzero real numbers is considered. The corresponding quotient the group space is endowed with a natural contact structure. We should emphasize that one *
This is corroborated, for example, by the fact that an operator of principal type (which
is one of the main objects of the study) is defined as an operator whose principal symbol does not have conutci fixed points.
Introduction
rather than This the quotient space with respect to the action of yields a more flexible theory capable of studying phenomena beyond the powers such as metamorphosis of discontinuities, lacunas of the conventional in hyperbolic equations. and so forth. From the geometrical viewpoint, the reason is that the phase space is a projective space rather than a sphere. Projective spaces are finer and more sensitive objects whose geometry is undoubtedly more adequate in the situation of smooth theory. That is why asymptotics, and they play an essential role in a special chapter is devoted to analysis on projective spaces. This chapter contains the presentation of one of the most elegant constructions of the theory, namely, of the "projective Fourier transformation." We show that the projective Fourier transformation may be defined by simple axioms and give important explicit formulas expressing this transformation via integrals of (residues) of certain closed forms over projective spaces. The same framework is used to construct the theory of Fourier integral operators (Chapter 2). An experienced reader might at first be puzzled to discover that rapidly oscillating exponentials (which are an integral part of the Fourier integral operator theory) are not used at all. However, after a short consideration, the reader is likely to come to the conclusion that integration over compact cycles is preferable.** Indeed, the problem of regularizing divergent integrals does not even arise in this context. It becomes possible to define the algebra of pseudodifferential operators precisely (rather than modulo infinitely smoothing operators), and so on. Let us say a few words about the applications of the theory, which we considered expedient to include in the book. They are concerned with two classes of equations, namely, with equations of principal and subprincipal types. Both notions may be defined naturally in terms of contact geometry. The equations of principal type are those with real principal symbols (Hamiltonians) whose contact vector fields vanish nowhere in the R,-homogeneous phase space. whereas for equations of subprincipal type, the contact vector field may possess isolated fixed points. These two classes are rather different. Thus, for example, any two Hainiltonians without fixed points are contact equivalent, while there is a collection of orbits of the group of contact diffeomorphisms if the contact fixed points are present. It turns out that the classification of Hamiltonians in a neighbourhood of a contact fixed point may be carried out, and the list of corresponding normal forms may be given. Further, the quantization procedure for contact transformations yields the classification and normal forms for the operators themselves, thus providing the possibility to prove solvability theorems on microlocal, local, and semiglobal takes
levels.'1 ** This
point of view was already stated by J. Leray in his works on complex analysis
IL lI—IL 41.
See Chapter 3 for precise formulations.
Introduction
In short, this is the outline of the book (see also the Contents for more detailed information).
Acknowledgments. We are grateful to Professor Victor P. Maslov for his support and express our gratitude to Mrs. Helena R. Shashurina for her dedicated work while preparing the manuscript. Moscow, March 1992
V.E. Nazaikinskii
V.E. Shaialov B. Yu. Sternin
Chapter /
Homogeneous functions, Fourier
transformation, and contact structures
1. Integration on manifolds This section contains some background results related to the integral calculus on manifolds, which are gathered here mainly for the reader's convenience. We mention primarily the topics which are either relatively less conventional (such as the theory of odd forms introduced by de Rham [Rh 1)) or specific for the theory developed in the subsequent sections (e.g., residues of forms with pole singularities on real manifolds). Most of the results discussed here are not new, and when possible we omit the proofs, which may be found elsewhere. 1.1 Orientations
and oriented manifolds. Differential forms
: S1 (M) M be the bundle of pseudoscalars on the manifold M, that is, a real one-dimensional vector bundle over M with the transition functions of the form
Let
(1, A) = (i(x), sgn Di/Dx A),
I
where x and I are coordinate systems on M. I = (x) is the corresponding change of variables, and A, A E R' are the corresponding coordinates in the fibre. Smooth sections of this bundle will be called pseudoscalars. Recall that M is called an orientable manifold if all coordinate systems on M split into two groups in such a way that the Jacobian rn/Ox is positive if both coordinate systems x and i belong to the same group, and negative otherwise. It is well known that this is equivalent to any of the following conditions. (1) The bundle S1 (M) of pseudoscalars on M is trivial. (2) The bundle A" (M) of n-forms on M is trivial for n = dim M.
Thus the orientation on M becomes fixed once we choose a pseudoscalar e on M with €2 = I or a nonvanishing n-form on M. We use pseudoscalars below.
1. Homogeneous functions. Fourier transformation, and contact structures
2
Let us introduce the following notation. Let
Vect(M) = r(M, TM) be the space of smooth vector fields on M. We set A°'°(M)
r(M, S1(M)) and (M) are the spaces of smooth functions and smooth pseudoscalars on M, respectively) and define ALO(M), A"(M) as the spaces of all
(i.e..
maps
Vect(M) —+ Vect(M) —f respectively. The elements of the space A'°(M) (A''(M)) are called even (odd) I-
forms on M. The even forms are nothing but conventional differential forms on M, so we drop the word 'even', provided that this will not lead to misunderstanding. We set
A°(M) = A°°(M) A1(M)
= A'°(M)
is evident that A°(M) is a ring and that the space A'(M) possesses a natural structure of a A°(M)-module. Let us consider the corresponding exterior algebra and denote by Ak) (M) the spaces of its homogeneous elements, with k being the degree and j being the parity of the elements. These spaces may be thought of as the spaces of alternating forms It
Vect(M) x •. x Vect(M) —÷ & factors
or
Vect(M) x
x Vect(M) —+
k factors
Akl (M) will be called the spaces of even (respectively, odd) k-forms on the manifold M. Note that elements of these spaces may be considered as sections of the corresponding smooth bundles. Now let us give a coordinate description of the introduced objects. Let (U, x), x") be a coordinate neighbourhood on M. Then any k-form c4 on x = (x' depending on the parity. The spaces
1. Integration on manifolds
M may be represented in the chart U in the form
,(x)dx" A••• AdxA.
c4(x) = iI<...
are smooth functions in U, and A denotes the interior product. The transformation law for the coefficients (x) under the change of variables x = x(i) is given by the formulas
where
(x(i))
(i) = a11
..
.
Dx ax'
=a1
Here and in the sequel, we use Einstein's dummy suffix summation convention. (x) are determined by (I) for ordered tuples The values of the coefficients 1k) and then by antisymmetricity for all other tuples. (i1 (M) (a = 0, 1), The exterior differential d acts in the spaces
d:
—÷
This operator is uniquely determined by the properties
dod = 0,
d(co1 A
Adw2,
w2) = dw1 A w2 +
df(x) = where the latter formula applies both to the elements of A°'°(M) and to those of A°'(M) (this means, in particular, that (6) is compatible with the transformation laws (2) and (3)). In the local coordinates, the operator d has the form
d4(x) =
A••• A dxk,
>
where 4(x) is given in the local coordinates by (1). Note that for an oriented manifold M, the spaces naturally isomorphic, with the isomorphism being given by AkO(M)
= e4(x)
cvk4.(x)
and A"'(M) are
E
is the orientation of the manifold M. Now let us study how smooth mappings of the manifolds act on even and odd forms on these manifolds. Let M1, M2 be smooth manifolds, and let where
M1
be
—+ M2
a smooth mapping. We define the induced mapping —+ AkO(M1)
4
1.
Homogeneous functions. Fourier transformation, and contact structures
.
of even forms by the equality *
w)(X1
Xk)del= w(ço.X1 .
where is the tangential map of at the point x E M1. —÷ Definition (11) makes no sense for odd forms. Indeed, the sign of its right-hand side depends on the choice of the coordinate system on M2 while the left-hand side
should depend on the choice of the coordinate system on M1. These two choices are independent of each other, and that is why the definition fails. In order to overcome this difficulty, we introduce the notion of oriented mapping. The mapping (9) is called oriented if a one-to-one correspondence is established between possible orientations of U and V for any pair of contractible charts U C M1 and V C such that co(U) C V. with these correspondences being compatible for any two pairs (U, V) and (U', V') with nonempty intersection U fl U'. If q is an oriented mapping, we define the induced mapping Ac '(M2)
Ak (M1) I
locally by the formula
= where r1 and
are orientations related by the said correspondence.
1.2 Integration of forms. Currents Let M be a smooth n-dimensional manifold, and let w E (M) be a finite odd form of maximal degree on M. For any domain D C M with a piecewise smooth boundary, we define the integral of the form w over the domain D as follows. First, let D be contained in some coordinate neighbourhood (U. x), and let the expression for w in U be
v=a(x1 Then dcl
Jo
n
n
Jo
It is evident that the integral (12) does not depend on the choice of the local coordinate system, since the rule (3) coincides with the substitution rule in the common integral calculus. In the general case, we set
1. Integration on manifolds where D
=
5
D1 is a Fartition of V into the union of nonintersecting domains
such that each of these domains is contained in some coordinate neighbourhood. Now let w be an even form of maximal degree. In order to define the integral, we assume that the domain D is oriented, and its orientation is given by a pseudoscalar e, e = ±1. We set
w=few.
f
I)
(1).r)
The requirement that w be finite is in fact irrelevant. It suffices to assume that the intersection of the domain D with the support of the form cv has compact closure. Now we intend to define the notion of integral for forms of dimension less than n. We need some preliminary considerations. a map Let us define an odd singular k-simplex
—+ M, where is the standard simplex determined by the relations
of dimension k,
that
is, the subset of
xl>O,xl+...+xk4I=1) endowed with the standard orientation ((x' coordinate system). A finite formal linear combination and
=
is regarded
as a positive
N
will be called an odd k-dimensional chain on M. An oriented map of the form (14) will be called an even singular k-simplex (1 of odd singular k-simplexes with real coefficients
on the manifold M.
A finite formal linear combination
= I
of even singular k-simplexes with real coefficients
will be called an even k-
dimensional chain on M. We define the integral of an even form cv E k-dimensional chain of the form (15) as the sum
c
i=I
of degree k over an odd
6
1.
Homogeneous functions. Fourier transformation, and contact structures
of integrals of the induced forms ça(w) over the oriented standard simplex of dimension k. The integral of an odd form w Atc '(M) of degree k over an even k-dimensional chain of the form (16) is defined similarly as the sum
Nj
=
j
Two chains of same dimension and parity are regarded as equal if for any form of appropriate degree and parity, its integrals over these chains are equal. We define the boundary of the standard simplex as the formal sum k+ I
do1'
= where
of
is
the ith face of the standard simplex as',
of
Note that of
a standard simplex of dimension k — in the space R" with the coordinates (x1 x' (x' is omitted). Now let be a singular simplex. Denote by Oj9, the restriction of the map (14) to the ith face of the standard simplex as'. The chain is
1
k+I do9,
=
1=1
will be called the boundary of the simplex It is odd when 09, is odd, and it is even when is even. The boundary operator d is extended to the space of all chains by linearity. The correctness of the above definitions is a consequence of the following theorem.
Theorem 1 (Stokes). Let be an odd (respectively, even) singular k-simplex, and let w be an even (respectively, odd) k-form. Then
I
Ja,
dw=
f
Jan,,
The relation (19) is also valid for integrals over any chain of appropriate dimension and parity.
Denote by (M) the space of k-forms of parity a with compact support on the manifold M. We introduce the notion of convergence in (M) as follows. The sequence E is said to be convergent to zero if the supports of all its members are contained in some compact set K C M, which does not depend
I. Integration on manifolds
on 1, and the coefficients of the forms w1 converge to zero in the C°6-topology in any local chart. I_0(M) will be called currents Continuous linear functionals on the space the space of all such of degree k and parity a on M. We denote by functionals and by (T, a) the value of the current T on the form a. The formula
(UAa,
aE
determines an embedding (note that the support of the C is compact, with the integral therefore being defined correctly). This form w A a embedding is dense in the weak' topology of the space (M). Now it is clear that in local coordinates any current may be written in the form (1) where the (x) are distributions. The transformation laws (2) and (3) for the coefficients a1 coefficients under changes of variables are valid for currents with no modifications. The notions of an exterior differential and of an exterior product may be extended to currents in the following manner:
da),
(dT, a) (T
above formulas correctly define the differential of a current T and its exterior product with a smooth form w. We finish the topic with the Schwartz theorem. the
Theorem 2. Every Continuous linear operator L
—÷ D'(M2)
:
may be represented in the form
L(f) =
I
J M1
L(x, y)f(y),
(22)
where L(x, y) E D'A"' (M1) 0 D'(M2). The kernel L(x, y) is uniquely determined by the operator L.
The proofs of the propositions presented above may be found in [Rh 1] (see also [Sw 1]).
1.3 Integration In fibre bundles Let us consider a smooth fibre bundle
p:E—+B
(23)
8
1.
Homogeneous functions, Fourier transformation, and contact structures
with a fibre F, which is a smooth manifold of dimension m. We assume that F is orientable and that an orientation is chosen in each fibre F,, = ((b)) whose dependence on the point b of the base space is continuous (i.e., for any trivialization p'(U) = U x F, the orientation of F doesn't depend on bE B). We intend to define integration over the fibre as a map
1:
(24)
Westartwitha =0.Letb E Bbeapointofthebasespace,andlet
Y1
Yk_m E
be an arbitrary system of vectors tangent to B in the point b (we assume that k in). If w is an even form of degree k on E, we define an even form w of degree in on
setting
Xm)
,y1
def
=
w(Y1
*
Y&_m,Xi
Xm)
for any tuple X1, ..., Xm of vectors tangent to at some point x. Now we are able to define the operator (24) by the equality (25)
It is easy to verify that (25) is a correctly defined even (k — m)-form on B. Theorem 3. The relation
1(do4=d(Iw)
(26)
holds.
Proof Using a local trivialization, we can split the exterior differential
on E the sum of those on B and F, dE = dF + d8 (of course, this is not invariant). Therefore, we have into
1(dEw) = I(d8w) +
= d81(w) + 1(4w) = dBI(w),
since
I(dFWXYI
Yk_m)
= Lb =
J
=0
w by Stokes' theorem
sinceaFb=ø.
Now consider the case of odd forms (a = 1). Note that the orientation of the fibre F,, determines an orientation of the projection p. Indeed, let an orientation of a domain W C B be determined by a coordinate system (x' x's). Then the corresponding orientation in p W may be determined by coordinate systems ym,xI ym) is a positive coordinate of the form (y1 x's), where (y1
I. Integration on manifolds
system on Fb. Thus, for odd forms, the operation I may be defined by dcf
(27)
1w = 611(62w),
where the orientations Cl and 62 are related via the mapping p. Thus, we have determined the mapping (24) both for even and odd forms. The-
orem 3 remains valid for the latter as well, since its proof is purely local with respect to the base space B. Calculations in local coordinates show that the formula (28)
AI(0(B), c Since the restriction of the mapping holds for (24) on the space of forms with compact support is continuous in the topology introduced in Subsection 2, (28) allows us to define the extension ph
(29)
:
of the induced mapping p*
in the following way. We set (P*T
1*).
is dense in The extension is unique since The current p*T is called the inverse image of the current T under the projection p.
1.4 Residues of differential forms of maximal degree with poles on a submanifold of codimenslon 1 Let X be an n-dimensional manifold, and let 1: Y X be an oriented embedding of codimension 1. We identify the manifold Y with its image 1(Y). Let s(x) = 0 be a local equation of Y. The form w E A't0(X \ Y) is said to have a pole of order m on V if the form [s(x)]mw has a removable singularity on V.
Let co be a form with a pole of order
1
represented as a ratio (0
(0= —, S
on V. This means that co may be
_____— I. Homogeneous functions, Fourier transformation, and contact structures
where th is a smooth form. Since dx 0 on Y, there exist a neighbourhood U of Thus, ds A Y and an (n — 1)-form in U such that & cv
ds =—A 5
We
define the residue of the form cv on the manifold Y by the formula
It is evident that resco does not depend on the choice of the function s(x), determining the manifold Y.
\ Y) have a pole of order m on the submanifold Y. Then there exist forms ä E (X \ Y) with pole of order I and aE \ Y) with pole of order m — 1 such that cv — = da. Theorem 4. Let the form cv E M'°(X
Proof Since the embedding i is an oriented mapping. Y is a two-sided (n — 1)-
dimensional surface in X. This implies that Y may be determined by an equation of the form s(x) = 0 globally, with s(x) being a real-valued function whose differential does not vanish on V. We have cv
for some th E
A"(X).
0 on Y, the relation
Since ds
&dsAfi 5m
use the notation rewritten as We
1
5m
holds for some (n — 1)-form
(32)
/
m—1
'
m — 1 Sm—I
which is determined by the equality & = dx
(33) A
= ã/ds, d/3 = dth/ds. In these notations, (33) may be d&
I
I
I
rn—I
di—j. \Sm-I/
(34)
Applying a similar procedure to the first term of the right-hand side in (34), and by repeating this m — I times, we eventually come to the formula I
(m—
where a has a pole of order m —
1.
ldmL.
1)!sdsm-l cv+da,
(35)
thus completing the proof.
Let cv be a form with pole of order rn on V. We define the residue of cv on Y as the residue of the corresponding form constructed in Theorem 4, Resw
dcl
=
resw.
2. Analysis on RP't and smooth homogeneous functions on
II
The proof of Theorem 4 also provides the formula for the residue; it reads
with the form & being determined by the relation (32).
2.
and smooth homogeneous functions on
Analysis on
In this section, we gathered some facts concerning the analysis and geometry of the projective space It turns out that homogeneous functions on play an important role in the "projective analysis." We introduce Leray forms and study via homogeneous forms on representations of forms on The action of the group GL(n + I, R) in the spaces of these forms is considered.
2.1 Notations the standard (n + 1)-dimensional Cartesian space with the coordinates (x° x's) and by its dual space with the coordinates (po. ..., The coupling of these spaces is given by the bilinear form We denote by
p x=
p0X0 + p1X1
+." +
We use the following notations below:
x=(x° =
(Po
. . . ,
=
(po,
+ ... + +. + = pox° +... + = PoXo = + ... +
. .
Pi+i' .
.
+
. . ,
... +
(hat) indicates that an object under it should be omitted. Similar agreements will be used with other (n + 1)-dimensional objects. However, we write Thus the sign
def o Adx A...AdX; dx=dx t
l)tdxO A
n
... A dx A ... A df =
ax'
I.
12
Homogeneous functions, Fourier transformation, and contact structures
where j denotes the interior product (see [St 1 1); dx
dx
=
(_l)i+k_tdxOA... AdxJ A••• AdXk A••• Adf,
j
0.
j=k, A••• A dxk A••• A dxi A••• A df,
(2)
j > k.
Thus, in the expressions such as dx1, dx jjc, the indices i. j,... are in fact superscripts. We use the notation = \ (0}. On this space, the multiplicative group
of nonzero real numbers acts according to the formula
The infinitesimal generator of these dilatations is the radial vector field
d
,d
on the space
Now let us consider some notions concerning the projective space RP". The space RP' may be thought of as the space of orbits of the action of the group
that is, RP = The structure of a smooth manifold on RP' which makes is compatible with the projection it —' into a with the fibre R.. Let us describe this structure explicitly. fibre bundle over with a system of open sets, R4' = For this purpose, let us cover R4 in
where V1 = (x 6 V,0 = {x E
I
jx' >
x 0)
0}. The set V1 consists of two connected components and V,1 = (x E = ,r(V,). The 1x1 < 0). Set
functions
u1=(u0....,uI form a coordinate system in U1. The chart (U1, u') will be called an affine chart
of the projective space RP1. As was already shown, an orientation of the projection in a fibre bundle is defined by an orientation of the fibre. Let us choose an orientation in each fibre in such a way that (d/dX} be a positive basis in the tangent space of the fibre. This defines the orientation of the projection it, which takes the standard orientation of = to the orientation in the corresponding (inherited from If n is odd, = e,1 = (—1)', and these orientations coincide in affine chart the intersections of the affine charts, thus defining a global orientation of RP". For even n, RP' is not orientable.
2. Analysis on RP' and smooth homogeneous functions on
The Euler identity
2.2 Spaces of smooth homogeneous functions on
We introduce the following spaces of smooth functions on R4
=
(f Here,
f(Ax) = Ak(sgnx)Gf(x), A
{f I
f(Ax) = AkJ(x), A >
k E Z, (YE (0,1).
Theorem I (Euler). A smooth function fix) on if and only if
belongs to the space 0k
I)
= kf(x). Proof Trivial. The elements of the spaces will be called homogeneous, odd-homogeneous, and positively homogeneous functions, respectively.
2.3 Leray forms and related identities We introduce here certain forms on of the analysis on RPM. Consider the differential forms
A••• A dxi
=
w=
which are important for the development
A df E
(4)
= with summation by k from 0 to n being assumed in the latter formula. The standard orientation of the space being fixed, these forms may be considered either even or odd, as desired. They will be referred to as Leroy forms.
Theorem 2.
(1) The identities
dw = (n + l)dx;
w=
=
dw,
=
=0 are valid. (2) The forms
... ,
are linearly independent, provided that x'
0.
14
1.
Homogeneous functions, Fourier transformation, and contact structures
(3) 1ff E
then •m.
(4) 1fF
then
d(Fw) =0. (5) The forms w,
are
Xw =
=
Proof We have
=
w=
=
so the second formula in (6) is valid. Next, the computation dw = d
=C*(dx)=Ilm =Iim
+
=d
An—
A.I A—i
A*dx_dx A—I
dx=(n+1)dx
proves the first formula in (6) (here
is the Lie derivative along the vector field X). We omit the proof of (7). since it is quite similar. It is also easy to obtain (8):
= xjxkdxjk = 0. since dx"
is symmetric. is antisymmetric in (j. k) and Now suppose that x0 0 and consider the linear combination
= = 0 for all j (this follows from the fact If this combination vanishes, we have that the dxi' are linearly independent), and consequently, cr1 = 0 for j = I, , n. Thus the forms w1 w,, are linearly independent. Then Let f E d(fw1) = df A w1 + fdwj
=
Adx'+
A x'dx" + nfdx'
AdXU+nfdxl,
2. Analysis on RP" and smooth homogeneous functions on
since dxk A dx" = 0 fork
dx'
A
1,1. Since
dx" = dx',
dx' A dx" = —dx',
we have
d(fui) = which proves (9) for j = proved in a similar way.
+ ni) dx' 1
—
= —11.w,
and, by symmetry, for any j. The relation (10) can be
0
2.4 The spaces
and
We intend to study the inverse images of n-forms and (n — 1)-forms on RP' under the projection
Theorem 3. We have
,r'(A(RP'7)) =
(folf E
=
(12) (13)
for odd n, and
ir(A(Rr)) = (fwlf =
(14)
If' E
(15)
for even n. We see that there is an essential difference between orientable (n is odd) and nonorientable (n is even) cases in the structure of the inverse images of the spaces
A:(Rr) and The proof of Theorem 3 is based on the following result.
Lemma 1. A form $7
is the inverse image of some form $7,
$2 = yr$2,
if and only if the following conditions are valid: = 0,
inv $7 = Here mv is the inversion operator in
$2.
invx =
1. Homogeneous functions, Fourier transformation, and contact structures
16
Proof It is easy to see that the conditions (18)—(19) taken together are equivalent to the single condition that for any A
E
=a (1) Necessity. Let (16) be valid. Then
=
=
=
since = 0 (recall that the field (20) is also valid, since
=
0,
is tangent to the fibres). Condition
=
(jr
cA = ,r). (we used the fact that (2) Sufficiency. Let (17) and (20) be valid. In a neighbourhood U of an arbitrary point z E RP', we may build a solution for equation (16) in the following way. Choose some smooth section s : U —+ of the bundle over the neighbourhood U and set
= in
this neighbourhood. Denote by w the difference
,r*ci
U)
=
— £2
— £2.
We have A*w — w =
A*lr*s*c
—
= (,T o A)*s*f
—
oA
= r; furthermore, d
d
— £2)
=0
—
by (20), together with the relation
d 7jw =
(A*c
= 0,
—
(22)
since
d
*
s
£7
= it *
£2)
—0,
and 4jc=o; =
—
= (ir o s)*Q
—
=
0.
(23)
since it os = id. We intend to show that w = 0. Let X1,... Xk E and therefore X = x E it 1(U). Then x = As (ir (x)) for some A E
j =1
A,Y3, where
w(X1
Xk)
= w(Yi
Yk) =
k. The relation (21) implies that
+
,...,sSJrsYk + 'ak),
2. Analysis on RP" and smooth homogeneous functions on where
crieR, Thus, we have Ic
+
Xk) =
w(X1
x
=
=
w(s.,r,Y1
s*w(Jr*YI
=0
(here we used (22) and (23)). Thus, we have proved that = in the neighbourhood U. Since it is an epimorphism, the solution of equation (16) is locally unique. Consequently. the local solutions coincide in the intersections of the neighbourhoods where they are defined and determine a global solution of (16) on RP". The lemma is thereby proved.
Proof of Theorem 3. Let c E
Then
= g'(x)dx' = Suppose that
Then the conditions (17)—(19) are valid. Condition
E
(17) gives
which implies that the vectors
= 0,
ax
dA
are proportional. This means that
and
g'(x) = x' f(x) for some function 1(x). and we have
= g'(x)dx' = f(x)x'dx' = f(x)w. Using condition (18), we obtain
0=
=
=
AW+ (n +
l)f
jdx)
=
+ (n + 1)1)
.
L
18
Homogeneous functions, Fourier transformation, and contact structures
since w =
have f +
therefore *jw 0. Since the form w does not vanish, we = 0 and, by the Euler theorem, f E Finally,
and
(n
+
l)f
(19) yields
f(x)w = inv*(f(x)w) = f(—x)
(—l)'w.
Thus, if n is odd, we have f(x) = f(—x), that is,f E For even n, we obtain f(—x) = (—1)°f(x), so that f We proved that the left-hand sides of the relations (12) and (14) are contained in their right-hand sides. The inverse inclusion may be proved by direct computation of the left-hand sides in (17)—(l9) for fw. Let us prove the relations (13) and (15). Any form E may be represented as a linear combination of the forms
= which may be rewritten as
=
A
ax' If that
dx)
'(RP")), the conditions (I 7)—( 19) are valid. Condition (17) implies
E
Id
d
d\
a
—jc = I — Ag's— A — ljdx =0. Since
dxJJ
dx'
dA
the form dx is nondegenerate, it follows that
d
a
a
dA
dx'
dx)
— A g" — A
= 0.
By the Cartan theorem on divisibility of the forms, a
d
dxi
dA
.a dx'
g"— A— = — A V = X'— dx'
for some v =
=
We conclude that
=
.a dx) so that
=
Using conditions (18), we obtain (24)
2. Analysis on RP' and smooth homogeneous functions on R.H
fr,), j = 0,
1. n
be a partition of unity subordinate to the covering (V1 (x) are homogeneous functions of order 0. Then such that e,
Let
= >ekcl = Expressing the form Wk as a linear combination of (coo
neighbourhood Vk, we may rewrite
as
in the
follows:
= The relation (24) is valid for the forms
A
+
0,
hence E
The remaining part of the proof is quite similar to the case of n-forms, and we 0 leave it to the reader. 2.5 DuaLity of spaces of homogeneous functions. Integration by parts
We adopt the following convention in order to simplify our notations. If is a instead of (jr form of the type described in Theorem 3, we write where S = a if n is odd and S = 1—a Let! E g so the integral if n is even. By Theorem 2.3, fgw E
defined. The bilinear form (25) is evidently nondegenerate and thus defines duality between spaces of homogeneous functions. The pairs of spaces related to each other with this duality are listed below:
is
I
g O_(fl+k+I)(Rfl+1)
nodd
Q_(fl+k+I)(Rfl+l)
O_(n+k+I)(Rn+I)
neven
(n+k+I)1
n+I', Ok' * /
I'
Proposition
1
0
(Integration by parts). ag (1.
=
af
st+I *
20
1. Homogeneous functions. Fourier transformation, and contact structures
provided that f and g are homogeneous functions such that either of the pairings is defined.
Proof From (9), we have
d(fgw1) =
=
(ff4 +
cv.
This implies
f
dx'
RP'
dx'
RP
by Stokes' theorem.
2.6 Action of the group GL(n + 1, R) The group G L (n + I, R) of invertible matrices of order n + I acts on to the formula
(A,x)i—+Ax,
according
AEGL(n+I,R),
(26)
may be realized as the subgroup of nonzero scalar matrices in GL(n + I, R); its action (A. x) —+ Ax may then be obtained as the restriction of the action (26).
Since scalar matrices lie in the center of GL(n + 1, R), we see that the action of GL(n + 1, R) commutes with the action of Thus, each linear transformation GL(n + 1, R) induces a projective transformation A : A RP' such that the diagram
R''
A
RP A
commutative. On the other hand, G L (n + I, R) also acts in spaces of homogeneous functions, namely, for any A e GL(n + I, R), we have the corresponding mapping is
A*
—÷ A*f(x)
f(x) for
any k,
a. Since
the actions of GL(n
+ I,
A*
dA
—
R) and
f(Ax)
commute, we have
2. Analysis on RP' and smooth homogeneous functions on
Let us study how the transformations A
G L (n
+ 1, R) act on the forms w,
Since the transformations do not necessarily preserve the orientation of have to distinguish between the cases a = 0 and a = 1.
(1) Let a =
0: then w E
A*wj
=
8xJ
In this case, we have
E
=
Aw =
=
= (—A'
we
.
dx) = detA w, (27)
detA . w = detA
8xi
(28)
.
with (A_t)si being the (s, j)-th element of the matrix A'.
(2) Let a =
1; in this case, we consider w, as elements of the spaces and Ar'. respectively. An auxiliary factor sgndet A appears in the
transformation formulas for w and w1,
A*w=IdetAI.w, = IdetAl
(29)
We finish this subsection with the study of the action of the transformation A on the pairing (25). We have
(A*f, A*g)
=
J
At(fg) .
w
=1 = IdetAI'J fgw = the integration is invariant with respect to variable changes. In (31), 0) is considered as an element of since
2.7 RepresentatIon of functions in the divergence form. Orthogonality conditions For any k, a the differentiation operators
act in the spaces
—p
(32)
Let us study the problem: What are the necessary and sufficient conditions for the to have a representation of the form function f
f=!,
i=0,l
n.
(33)
22
1.
Homogeneous functions, Fourier transformation, and contact structures
We consider the following cases.
=
xf
n+k+l
i
,
=0, l,...,n.
(34)
Indeed, by the Euler identity
Thus, no additional conditions arise in this case. B. k = —n — 1. By (9), the equality (33) is equivalent to a Pfaff equation
da = —fw
(35)
d(fw) = 0
(36)
with a = g'w1. Note that
(cf. (10)), hence (35) is always locally solvable, and the hindrance to global solvability is the cohomology class of its right-hand side. Two essentially different situations are possible, depending on the values of n and a.
Bi. The product ncr is even. By Theorem 3, we may consider fw as a form on RP. If n is even and a is odd or vice versa, we have 1w E The homology group If,, (RP") is a group with one generator, which is the class of RP" itself. By the de Rham theorem, the class [1w] E is equal to zero if and only if
I fw=0.
(37)
This may be considered as an orthogonality condition,
(f,l)=0.
(38)
If n is even and a = 0, we have fw E
and [fw] = 0 automatically,
due to parity considerations.
B2. n is odd and a = 1. Here we cannot consider fw as a form on but we may use the n-dimensional sphere condition
instead. Since 1(x) = —f(—x), the
f fw=0
Js,
is always satisfied, and equation (35) is solvable. Let us introduce the new spaces of the homogeneous functions as follows. Set
=
(39)
2. Analysis on RP" and smooth homogeneous functions on R+I
23
for k > —n — 1, and define recursively
=
{f E
If
for some g1 E
=
fork=—n—l—n--2,—n—3 =
We have proved that also that if n + a is odd, then
=
if n + a is even. We have proved
{f
I(f.
A description analogous to (42) exists for all k
1) = 0).
—n — I.
(42)
It is given by the
following lemma.
Lemma 2. Suppose that n + a is odd, and let k < —n — 1. Then
=0 for any multiindexa
={f E
such that laI = —(n + k + l)}.
(43)
Proof We proceed by downward induction on k. The basis of the induction is valid (see (42)). Next, let f E where g E Then f = Using (26) and the induction hypothesis, we obtain (xu,
f)
=
(xcL,
=
g')
= 0.
ax' Conversely, let f belong to the right-hand side of (43), and let (Xa, f) = 0 for Ial = —(n ÷ k + I). Define the functions g1 by formula (34). Then (33) is valid; on the other hand, we have =0
=
for any multiindex fi with lfiI = —(n + k + 2). Hence g' E lemma is thereby proved.
2.8 The hyperplane
Let p
and
and the
0
related orientations
be an arbitrary point. Define the hyperplane
C
by the
relation
={x E
Ip•x
= pox° + ... + p,,x" is the above-defined pairing between We call a basis B in positive, if (p. B) is a positive basis in This defines an orientation of The restriction of the projection on where p . x
and
is a fibre bundle whose fibre is R,; as above, the standard orientation of the fibre (the
24
1. Homogeneous functions, Fourier transformation, and contact structures
direction of the vector
is regarded as positive) defines the orientation of the
projection. We have a commutative diagram of oriented mappings:
RP' (The upper embedding is oriented, since the orientations of and are chosen and fixed. The corresponding orientations on and define the orientation of the lower embedding). in the following, we often denote simply by
3. Homogeneous and formally homogeneous distributions in this section. we introduce the spaces of homogeneous and formally homogeneous distributions. We study the structure of distributions of maximal and submaximal degrees and introduce certain regularizations of these distributions at the origin in K" . We also study the properties of the regularization operator with respect to
the action of the group GL(n + 1, R).
3.1 Definitions and notations determines a distribution f E Any function f E the same letter, according to the formula
denoted by
ço(x) E
For any A
E
R. we have
(f.ço(x/A))
=
J
f(x)ço(x/A)dx
=
f
= We
J
may rewrite this formula, using the notations (I):
(f,w(x/A))
AE
f(x)w(x)dx. * I
3. Homogeneous and formally homogeneous distributions
25
Formula (2) motivates the definition of the following spaces of homogeneous iistributions:
= {f E = {f€ = (1 Since
(2) is valid for any E is valid for any ço E
f(x) is smooth for x
(3) (4)
0).
there is a natural restriction map
C
-+ ço)
It is evident that the map
for p E
acts in the spaces
—+
/2100A(R
(7)
—÷
:
(8)
The kernel Ker of the map JL consists of the distributions, whose support is the origin. By the famous theorem of L. Schwartz, any such distribution is a finite linear combination of the Dirac 8-function and its derivatives. We have
=
(8,p(x/A)) and thus,
8(x) E
C
for odd n + a. Therefore,
=
C
E
for such values of a. As a consequence of (II), we have
=
(0), k > —(it + 1) or it +a is even,
=
{
E
otherwise.
IaI=—n—k—I
call the elements from D' and from D' formally homogeneous distributions, respectively. We
homogeneous and
26
I. Homogeneous functions, Fourier transformation, and contact structures
3.2 Regularization Let T E The element T1 E be a distribution on is called T (this terminology is commonly used in the case a regularization of T if when I is a smooth function with a nonsummable singularity at the origin). Of course, if the regularization exists, it is not unique—it is defined modulo An operator reg
—*
:
will be called a regularization operator if a reg = id In this section, we construct one of the possible regularization operators and fix it for subsequent usage. We set Let f(x)
I
f(x)ço(x)dx,
JR" (regf,qi) del =
r f(x) ç(x)
f
k> —n — 1, —n—*—l
dx
—
J
L
+ k
f
r f(x) I
-n—k-2 go(x) —
1 I
dx,
<—n — I.
Using the spherical coordinates, it is easy to verify that all integrals in (15) are convergent. Therefore, (15) defines a continuous linear functional on the space If then all its derivatives vanish at the origin, and (regf,ço) = (f,co), that is, (14) is valid.
3.3 Regularization and action of the group GL(n + 1, R) We recall that the group GL(n + 1, R) acts in the spaces according to the formula A *1(x)
Given a function f E
and
f(Ax).
let us try to evaluate the distribution
A*(reg f) — reg(A*f)
[At, regjf
(in other words, we wish to calculate the commutator [A*, reg] explicitly). From (15), we have
[A*,reg]f =0
3. Homogeneous and formally homogeneous distributions
27
fork> —n —1. Ifk <—n—i, then reg]f, p) =
I det Al
(reg
f(x). co(A'x)) —
(reg f(Ax),
co(x))
I
=
— I
a! I
—
Idetal
a!
IaI=—,,—k—I
xaf(x)dxj —J (we used the invariance of the form
a=const
a'
under the transformations x E-* Ax). In the spherical coordinates x = rO, 191 =
we have
([A*, regif, q) I
a!
— Idetal
f
f(8)
dr
—,
IA-'OI-'
being the volume element on the unit sphere
with
J'
r
Hence,
([A*. —I
12 ldetai
aj=—n—k—I
n+a is odd, I
0,
f
a!
= Idetal
n+aiseven.
a!
I
JRP'
xaln xI
f(x)w.
1,
______
28
1. Homogeneous functions, Fourier transformation, and contact structures
In terms of the pairing (25), the relation (20) may be rewritten in the form (IA*, reg)f, 2
"
Idetal
=
'
k > —n —
IA
xI
,
n + a is odd, or n + a is even.
k < —n — I, 0.
In
&
IaI=—n—k—1
I
3.4 Orthogonality conditions and relations between spaces of homogeneous and formally homogeneous distributions
Let A be a scalar matrix, A = A*f = AL(sgnx)(hf, and
C GL(n + 1, R). Then
A) E
diag(A
regf(Ax) — Ak(sgnxy regf(x) I (n+k+l)
21 A
=
lxi'
&
—
n+aisodd, k> —n —
0,
1
or n + a is even,
if and only if f
by (21). Thus, regf E
(22)
in other words,
the following assertion is valid:
The operator (23)
—+
is an isomorphism fork> —n — 1 if n + a is odd, and for every k n + a is even. Its inverse is given by the operator reg. if k —n — I and n + a is odd, the operator (23) possesses a kernel and cokernel of equal finite dimension. The kernel
consists of linear combinations of the derivatives of the Dirac ö-function of order —(n + k + I). The cokernel is defined by the orthogonality conditions (xa, f) = 0,
a = —n
—
k
— 1. The operator reg is an in verse operator to the operator (23) on
the space
nzodulo
We introduce the following functional spaces.
(1) The space
We define it as follows:
k>—n—lorn+aiseven, =
U I f = regg +
g IaI=—n—k—I
k'cz._n_l,
n+aisodd.
(24)
3. Homogeneous and formally homogeneous distributions
29
(2) The space of associate homogeneous functions (for which we don't introduce any special notation). Let fi and f2 be homogeneous functions of some degree k. We call the function f = fi + in 1x1 f2 an associate homogeneous function of the first order. Functions of the form reg f which satisfy (22) with nonzero right-hand side will also be called associate homogeneous functions. This is natural since associate homogeneous functions thus defined satisfy the identity
f(Ax) = A"(sgn A)c[f(x) + f(x) In Al], where
f(x) E
is a homogeneous function.
3.5 Structure of formally homogeneous distributions. Generalization of Theorem 3 We can easily extend the above results to the elements of the spaces with the help of the following lemma.
Lemma 1. Let f
Then for any function
E
we have
0
(f,w) = I
(25)
Jo
where f
ço,. is a function on
defined by
=
In particular, Theorem 3 can be reformulated for distributions in the following form:
Theorem 1. We have —
=
{fwlf E
(26)
If3 E
(27)
for odd n, and
= =
{fwlf E
(28) E
(29)
for even n. Recall that the operator was defined in Subsection 2.1, using the fact that —÷ RP' is a fibre bundle. We leave the generalization of some of the other results presented above to the reader as an easy exercise.
30
1.
Homogeneous functions. Fourier transformation, and contact structures
3.6 Homogeneous functions on R1. It follows from Theorem 3.1 that
= These spaces are one-dimensional spaces generated by the functions
k?0,
o=O,
k
a=O, a = 1.
a=l;
xksgnx, v.p.x", 8(l_k)(x)
3.7 Duality between formally homogeneous functions and distributions The bilinear form (25) may be extended to the duality
(f,g)=[
fgw
where 8 + a = n + I (mod 2). It of the spaces and follows from the Schwartz kernel theorem (see Subsection 1.2) that any continuous —* may be represented in the form linear operator K :
kf = I K(x, p)f(x)w(x), JRP'
(32)
where
K(x, p) K(Ax. p) =
x
(33) AE
(34)
4. Fourier transformation of homogeneous functions The Fourier transformation of homogeneous functions has been studied thoroughly, beginning with the fundamental paper of Gel' fand and Shapiro ([GSha 1J). However. we present here a result which does not seem to be covered by the existing investigations. We show that several very natural properties determine this transformation up to a scalar factor. Namely, for functions of fixed degree and parity, it suffices to require continuity of the transformation and its natural behaviour under linear variable changes. The still remaining freedom to choose different constants
4. Fourier transformation of homogeneous functions
for different degrees is removed if we assume "proper" commutation with partial derivatives The outline of the section is as follows. In Subsection 4.1, we give the precise statement of the problem, while the theorems solving it are formulated in Subsections 8 and 9 as the eventual result of our argument, which occupies the rest of the section. Along with the proof, we obtain explicit formulas expressing the Fourier transform of homogeneous functions in terms of the pairing (2.25).
4.1 Action of GL(n + 1, R) on the Fourier transform. Statement of the problem Let .r denote the usual Fourier transformation,
/
= (—)
I
JR'-
We claim that an arbitrary transformation A E GL(n + transform of a function f(x) in the following way:
(Ff)(p) = .F(Af)('A
.
I
1,
R) acts on the Fourier
det Al.
Indeed,
F(Atf)('A p). IdetAl =
/
.\(fl+I)/2
,.
. IdetAI
I
I
.
(n+I)/2
=
f
F:
—÷
(Ff)(p).
Problem. Let
be a continuous linear mapping, satisfying the condition (2). Show that F coincides
with the restriction of the Fourier transform onto
up to a scalar factor.
4.2 Structure of the homomorphisms from GL(n + Lemma 1. Let
GL(n+
=C\{O}
be a continuous homomorphism of groups. Then we have p.(A)
=
1,
R) to
32
1.
Homogeneous functions, Fourier transformation, and contact structures
for some a
C,
(0, l}.
PmoJ First, we note that any continuous homomorphism R. —*
L
has the form the representation (5) is valid for any diagonal matrix A with at most one diagonal element a, 1. The numbers a and fi are independent of i, since takes similar matrices to equal numbers. It follows that (5) is valid for any nonsingular diagonal matrix and therefore for any matrix with distinct real eigenvalues. It remains to note that triangular matrices with distinct real eigenvalues form a dense subset in the set of all real triangular matrices and that any real matrix A may be decomposed as A = LU, where L and U are lower-triangular and uppertriangular real matrices, respectively. The lemma is proved. 0 IAIu(sgn
A
Hence
4.3 Decomposition of distributions Lemma 2. Let X be a smooth manifold, and let f be a distribution (i.e. a current of degree 0) on X. Suppose that u X -+ R is a smooth function without critical points and that the following conditions are satisfied:
(I) For any c c R, the set fu = c} C X is connected. (2) For an point x E X. there exists a local coordinate system a
in
neighbourhood of x such that U=
df
Of
= ... = — = 0.
(3) The mapping ii is oriented. Then
there exist.c a distribution K E D'(R) such that f = K(u).
Pmof Conditions (2) and (3) imply that for any point xX,E we have = in the vicinity of x (it is a consequence of Schwartz's theorem on solutions of the equation = 0). We only have to show that if u(x) = u(y) = UO, in the vicinity of u0. It follows from condition (I) that there then = exists a path y : [0, II —÷ X such that y(O) = x, y(l) = y and u(y(t)) It is easy to see that = for t and t' close enough. Using the Heine—Borel theorem, we complete the proof.
0
Remark. We can omit condition (3) if f is an even current; the statement of the lemma fails even for smooth functions f if condition (I) is not valid.
4. Fourier transformation of homogeneous functions
33
4.4. Homogeneous solutions of a certain functional equation Lemma 3. Let n > 1, Q (p, x) E conditions are valid:
*
and suppose that the following
x
(I) ForanyAEGL(n+l.R). Q(p, Ax) = Q('Ap, x)Jdet where a and fi do not depend on A.
(2) For
a=
(R') such that
a function K (z) E
Q(p,x) = K(p .x). x is connected, and d(px) Proof The set fpx = const} C pdx + x Due to Lemma 2, it suffices to show that in a neighbourxdp 0 in
hood of an arbitrary point system (u = px,q1
there exists a coordinate
x
x(o)) C
...
such that
aQ
aQ
aQ
Note that the equality (6)is invariant with respect to variable changes of the form
x = Cx', p =
Using such changes if necessary, we may consider that 0 in our neighbourhood.
Set U
= pX,
q'=x',
i=l,2
n,
i=0,1
1?.
We have
x
=
Consider the matrix A of the form A = E + rB, where all elements of B are equal to zero except for the one in the sth column and the lth row, which is equal to 1. By substituting A into (6) and differentiating with respect to r at the point r = 0, we obtain
,aQ X—= ax.
13Q
Ps
+aö5Q,
34
1.
where form
Homogeneous functions. Fourier transformation, and contact structures
is the Kronccker symbol In the coordinates (9). equations (11) take the dQ dQ
1=1
aq1
By calculating d2Q/aq1dq' from (13) and (14) and equating the results to each other, we obtain
j,I=l,...,n. This is possible only if a
0. Thus, we can apply Lemma 2 and get Q(p. x) = K(p x). Substituting this in (6) gives = 0. The lemma is proved. 0 .
Remark. The statement of the lemma remains valid for n = 0, 1 under the additional assumption a = 0.
4.5 The transformation F for the case when n
=
In this case, (2), we obtain
(Ff)(p) =
+ a-
is even or k> —n
By substituting a scalar matrix A =
Af(Ff)(Ap) =
—1 A
into
(sgn
that is,
(Ff)(p) E
mod
and the statement of Subsection
Using the isomorphism
3.7. we come to the equality
(Ff)(p) = where Q
E
Q(p,x)f(x)w(x),
x
Q(p, Ax) = The
I
JRP.
Q(p. x).
relations (2) and (2.31) show that Q(p, Ax) = Q('Ap, x). By Lemma 3, 0, where K(z) E Thus,
Q(p. x) = K(p x) for p
(Ff)(p) = regj K(p .x)f(x)w(x) +
4. Fourier transformation of homogeneous functions
35
where T is a functional supported at the origin. Recall that we consider the disso the last term is inessential, and by omitting the symbol tribution Ff in reg. we have
(Ff)(p)= I K(p•x)f(x)w(x).
(20)
Jill'.,
T = 0 in the cases a =
Remark. If we consider Ff as a distribution in or k <0 for parity and homogeneity reasons.
4.6 The transformation F for the case when n + o is odd and k < —n — I For these values of n and a, any element f tation of the form
has
an unique represen-
f=regg+ aI=—u—k— I
Using the result of Subsection 3.7, we obtain
(Ff)(p) =
where
=
JIll',
Q(p, x)g(x)w(x) +
(22) IcrI=—n—k—I
and Q(p,x) E
geneity properties described above. Let us begin with the study of the functions
!L(Ax) = ax' as
ax" .. where
We have
ax
is the element of the matrix
where
s=
Hence,
8(Ax) = .
. . . .
—n — k —
.
I,
possesses the homo-
x
(A')'
I
.
ö(x),
(23)
. . .
I. Condition (2) shows that
1,)(Ap) =
. . .
(24)
here we wrote where a is the number of members in the sequence = (1, i5) which are equal to i. Since B = tA—I may be an arbitrary nondegenerate matrix, we have
I,)(Bp) =
. . .
(25)
______ 1. Homogeneous functions, Fourier transformation, and contact structures
36
for any nondegenerate matrix B. Since nondegenerate matrices are dense in the set of all matrices, (25) is valid for any matrix B. Set p = (1,0, . , 0), = = 0 for! > 0. Then (25) gives . .
= *o
(26)
that is, we have shown that = COP,
(27)
is some constant independent of a. Now let us turn to the first term on the right-hand side of (22). Having studied the second term above, we may assume that c0 = 0. From (21), we have
where
A'f(x) = A'regg = reg(A*g) +
2
lxi
idetAl
,
a!
k,I=—n—k—
hence,
F(A*f)(SAp) =
2CO
Q(Ap x)g(Ax)w(x) + <
IaI=-n-k-I
ldetAI
J)fl+k+I
( —
lxi
—--- In
I
=
JRP"
a,.a
L.d I
(28)
lxi
)g(x)w(x)
a!
I +
[—(n + k + I)]!
In
ixl
xi
}g(x)w(x).
Using property (2). we obtain from (28) the following equation for the function Q(p. x):
Q('Ap.
+
lxi
(—(n +k +
=Q(p,x).
This equation has the following particular solution:
Q(p,x)
= [—(n +k +
lxi
(29)
4. Fourier transformation of homogeneous functions Thus,
f
for I E
(Ff)(p)
37
reg g we have
=
f
n+L+I
2
1
(for homogeneity reasons, the regularization of the function Ff(p) at the origin is uniquely defined). Note that the expression for Ff(p) depends on two constants, Co and c1.
tithe orthogonality conditions (Xa, g) = 0, = —n—k—i, are fulfilled, the second term on the right-hand side of(31) vanishes, and Ff(p) E In the general case, Ff(p) is an associate homogeneous function.
4.7 Commutation with differentiation operators and choice of constants In Subsections 4.5 and 4.6, the expressions were derived for the set of transformations
k> Fk
:
0,
c=
0,
otherwise,
namely, Cka
j f
,
n+aiseven, V.p.
JRP"
k>—n, (Fk.af)(P) =
f(x)w(x) (xp)'2
n+aisodd,
j
(32)
k <—n—I,
n +a is even,
I
IpxI ln—f(x)w(x)
JRP
(xp) -n-k-I
IXI
+
f
f the
these are to be imposed), then
(Fk..7f)(p) E
case when
38
1. Homogeneous functions, Fourier transformation, and contact structures
The conventional Fourier transformation satisfies the commutation formulas
=
3Pj
(33)
It is clear from the above considerations that the F&a coincide with F up to the scalar factors, which may be chosen for different (k, a) independently. We make use of (33) to reduce dramatically the uncertainty in their choice. There are the evident equalities:
=
ap,
(34)
n+k+ 1) V.p. ()fl+k+2
(xp)fl+k+
(35)
(xp), a —(sgn(xp)(xp) 'I
)
ap1
=
k=—n—l, — (n
±k+ k < —n
— 1;
k=—n—1, In
ap1
=
—
(n
+ k + l)xi(xp)_n_k_2 in
IxI
(37) k < —n —
Formulas (34)—(37) yield the recurrence relations for the constants ck.a; by solving them, we obtain
jn+k A
k
—n;
k
—n —
=
—k—I]!'
4. Fourier transformation of homogeneous functions
39
for even n + a, and
+ k)! ,
—
=
jn+k+I
(2,r)"/2[—n
k<—n—1
—k—I]!'
—
for odd n +a, where A and are arbitrary constants. If we choose A = = 1, Fk,a becomes a usual Fourier transformation. This choice also provides the symmetry between the direct transformation and its inverse.
4.8 Reduction of the integral formulas to residues Assume that k as follows:
—n and n + a is even. The first formula in (32) may be rewritten
=
ck.0
f
f
=
{d [sn+k_1(Xp)f(x)w(x)]
d(px)
pp.
[f(x)w(x)]}
—
d(px) = — Ck,a
d
J
d(px)
[f(x)w(x)J.
Repeating this procedure, we get the formula
(Fkaf)(P) =
(
(n + k)! j
Thus we have proved the following theorem.
(38)
40
1. Homogeneous functions. Fourier transformation, and contact structures
Theorem I. Let the continuous mapping F
(39)
satisfy the condition (2) and the commutation conditions of the type (33). Then this mapping is given bvfor,nula (38) or one of the last three formulas in (32), depending
on k, n. and a.
4.9 The inversion formulas and the Parseval identity Just as for the usual Fourier transformation, the inverse transformation may be obtained by substituting —i instead of i in all the formulas. The following theorem is valid.
f
Theorem 2 (Parseval equality). Let E sponding dual space (see Subsection 2.5). Then
and let g belong to the corre-
(Ff, Fg) = (_f)k(f,g)•
Proof Due to the inversion formula, it suffices to prove that
(f. Fg) = (Ff, g). But this is evident. Indeed,
(f. Fg)
=
j f(x) [J
=
f
g(p)
w(x)
f(x)K(xp)cv(x)] w(p) = (FJ, g).
The theorem is proved.
In the following table, the formulas for the Fourier transformation of homogeneous functions are collected.
___________________
4. Fourier transformation of homogeneous functions
Table 1. Formulas for the Fourier transform.
n+aisodd
n+aiseven
f(p)=(Ff)(p)
f(p)=(Ff)(p)
(n+k)!
— —
—
X — — X
2
(22r)"2 C
)L,, R
X JRP'
(—ira ç
f(p)=(Ff)(p) —
i
— —
—
x JRP'''x
x
sgn(xp) (xp)_k
x
lxi
f(x) w(x)
JRP(xP)f(x)w(x)
f(x)w(x) —(n-f-k-fl)1
E
function f(p) in the right lower corner of this table belongs to the space the function I satisfies the orthogonality conditions (f, f) = o for al = —(n + k + 1). In this case, the second term of the right-hand side of The
if
this formula vanishes.
Remark 1. Investigating the Fourier transform as a mapping acting to the space D'(R.,÷1.) (without the origin!), we omit the functional T in formula (19) which is concentrated at the origin. This is why the transform F for k 0, a = 0 l) of homogeneous has a kernel which is nothing other than the space Pk polynomials of degree k. Thus, in this case the transform acts in the spaces Fka :
A similar consideration shows that the transform values of n + a can be considered as a mapping :
(40)
—÷
for k <
—n
—
1
and odd
—+
The mappings (40) and (41) are useful if we investigate only the regular part of
the Fourier transforms of homogeneous functions.
42
1.
Homogeneous functions, Fourier transformation, and contact structures
Remark 2. The formula of the Fourier transform for k of n + a can also be rewritten in the form
—n —
1
and even values
.n
=
+ k + 1)]!
f(x)w(x),
(42)
JC(p)
whose orientation where C(p) is the relative odd singular cycle in changes on the manifold We note that such cycles were used by J. Leray while generalizing the inverse Laplace transform in [L 4]. The formula (42) can be used for continuing Fourier transforms of homogeneous functions to analytic functions of complex variables (see [L 4]).
4.10 The Radon transform In this subsection, we shall show that the well-known Radon transform is a specification of the Fourier transform of homogeneous functions defined above for a special choice of n and a. Let f(x) = f(x°,x' x's) be a homogeneous function of order k = —n and of the type a, where a = 0 for even values of n, and a = I for odd values of n. We consider the mapping —+
= f(1,x') = f(l.x'
(43)
= (x' x") are the coordinates in R". Note that the mapping (43) is invertible on its image. The inverse mapping is given by the formula where x'
=
(44)
for even values of n, and
=
(45)
for odd values of n. Denote by F the image of the mapping (43). For each f(x') E F, we define its Radon transform R [f(x')J by the formula
R If(x')] = F
f]-
Using the explicit formula for the Fourier transform for k = we have
R [f(x')j =
—n
w(x)
I
d (p x)
and even n + a,
4. Fourier transformation of homogeneous functions
By calculating the latter integral in the local chart x0 =
1
43
of the projective space
RP', we obtain R
f f(x'). d (p.x')'
[f(x')J = 1(p)
p.x' = po+plx' +•••+
wheredx' =dx'
(47)
and the plane
in R' is determined by the formula
=
=O}.
The transform R coincides (up to a constant factor) with the usual Radon transform (see, for example, [R 1]). As a direct consequence of definition (46), we see that the transform R acts in the spaces (48)
The transform R can be extended to the functional spaces wider than F such that the integral (47) converges with respect to some topology for elements of these spaces; we shall not examine this topic here. The formulas for the inverse transform depend on the parity of the number with even n +c n. If n is odd, the function R[f] belongs to the space (ask = —l,a = 1), and hence, — .*
(
(
d (p
f
d (p
—
where
= {p*
RP'tlp x'
.
+
po + mx'
= O}. Similarly. of n is
even, we obtain —
f(x)=i1 — —
I)!
(2,r)1?/2
2i"(n
—
1)!
(2,rY"2
t
I
Jpp'
V.p.
f(p)w(p) (x . p)fl
f 3
The latter integral can be rewritten in the more usual form as an integral over the
unit sphere in the space The commutation formulas for the Radon transform can also be obtained from the corresponding formulas for the Fourier transform of homogeneous functions. To do so, we note that
=
44
1.
Homogeneous functions, Fourier transformation, and contact structures
and hence,
af(x')
(j*)_I {
j =
f(x') E F. Therefore, we have R
[af(x?)]j
=F
=F
= —pj— F[(ity'f(x')] = aPo
P1
aPo
R[JJ.
Note that we can construct a set of Radon-type transforms if we use values of k different from —n. Fork E (—n, —IJ. all these transforms have the same properties as the Radon transform constructed above.
5. Homogeneous symplectic and contact structures Homogeneous symplectic and contact structures are widely used in asymptotic theory of differential equations. The aim of this and subsequent sections is to present basic facts concerning these topics.
5.1 Main definitions We recall here some definitions and affirmations concerning homogeneous symplectic structures, contact structures, and the relationship between these two notions. We begin with the definition of a contact structure (see also V.!. Arnold [Ar lj. V.V. Lychagin ELy 1], B.Yu. Sternin and V.E. Shatalov [SSh 5], and others). Let C be a 2n — 1-dimensional smooth manifold. By we denote the cotangent bundle of C with deleted zero section. The multiplicative group of nonzero real numbers acts freely on the space in a natural way. Let us consider the bundle C.
It is evidently a locally-trivial bundle with the fibre
The natural mapping
determines a locally-trivial bundle as well, so each section a* of the bundle (I) may be covered locally by a section
a C -+
T0*C
5. Homogeneous symplectic and contact structures
45
a1 a2 are two of the bundle (2), that is, by a differential form of degree 1. Ifand differential forms covering one and the same section a in a neighbourhood of a point c0 E C, then there exist a nonzero smooth function f in this neighbourhood such that cr1
= I a2.
Definition 1. A section at of the bundle (1) is called a contact structure on the manifold C, if this section is nondegenerate in the following sense: — if a is a 1 -form which covers the section a locally, then the restriction da IKera of the form dci to the kernel Kerci of the form a is a nondegenerate 2-form on Kera.
Note that the dimension of the space Ker a for each point that nondegenerate forms do exist on this space.
E C is even, so
It can easily be seen that the notion of nondegeneracy of a section at of the bundle (1) does not depend on the choice of the form ci which covers the section To show this, suppose that a1 and a2 both cover the same section at. Then we 0. Hence, Kera1 = Kera2 and dciulKera = have cii = f a2 with 0, the form da1 is, and vice versa. Since f IKcrat is nondegenerate ff A contact structure cit on a manifold C naturally determines the distribution of 2n — 2-dimensional hyperplanes on C. Namely, one sets
a.
I
f
=
Lç =
where c E C is an arbitrary point and a is any form covering the section at. The distribution {LC} is nonintegrable, since the Frobenius condition
a Ada = 0 not satisfied. What is more, this distribution is in some sense "maximally noninis a nonzero form of maximal degree on C (here tegrable": the form a A denotes the (n — I )-th exterior power of the form da). One can verify this statement by choosing a special basis e1 e2,,_1 of the vector space form a basis of the space such that the vectors e1 and has the canonical form A + A + is
Lemma 1. Let L C C be an integral
of the distribution
Then
dimL
J, one has T(.L C
=d(aIL)=0.
The latter relation shows that the linear space 7.L is a Lagrangian manifold in L(.
with respect to the form dci
dimL
and hence,
=n —
46
I. Homogeneous functions, Fourier transformation, and contact structures
(see A.S. Mishchenko, B.Yu. Sternin, and V.E. Shatalov [MiShS I]). This completes the proof of the lemma. 0 Definition 2. An integral submanifold of the distribution of maximal dimension n — I will be called a Legendre submanifold in C (with respect to the contact structure a5).
Examples of Legendre submanifolds will be presented below. Recall that the symplectic structure on an even-dimensional manifold S is determined by a closed nondegenerate 2-form w on this manifold.
Definition 3. A symplectic structure (S, cv) is called a homogeneous symplectic structure if the group R5 acts freely on the manifold S in such a way that: (i) the space C = SIRS of R5-orbits admits a structure of a smooth manifold such that the natural projection S —+ C is a smooth mapping; (ii) the relation
F(co) = Xcv holds where the mapping FA : S -+ S is determined for any A E R5 by the formula is the induced mapping. FA(s) = As. S E S, and
Note that the projection S
C
S/RS
determines a smooth bundle with the base space C whose fibres are the orbits of the group R5. The action of the group R5 on the space S also determines the mapping
F5 : R -+
S
for any fixed s S by the formula F'(s) = As. Let us introduce the vector field X0 by the formula (X0)5 = F55(l)
(k).
Below, the vector field (5) will be called a radial vector field on S. It is evident
that X0 is tangent to the fibres of the bundle
5.2 The relationship between contact and homogeneous symplectic structures
We shall show that there exists a natural contact structure on the orbit space C, the homogeneous symplectic space S related to the homogeneous symplectic structure cv. What is more, the correspondence between contact structures and
5. Homogeneous symplectic and contact structures
homogeneous
47
symplectic structures constructed in such a way turns out to be a
one-to-one correspondence in some sense. The definition of the mentioned contact structure is as follows. Let c E C be an arbitrary point and S E S be a point such that sr(s) = c. We define a linear form by the formula & in the tangent space
=
wUX0).,, Y'),
Y' being a vector which covers Y, that is, = Y. The definition (6) is correct, = since if Y" is another vector which covers Y, then Y' — for some /2 E R, and hence w((Xo)3, Y') = w((Xo)3, Y"). Ifs1 and s2 are points of the space S such that = ,r(s2) = c, then there exists A L such that s2 = FA(sI) (since the fibres of are Re-orbits). Hence, we have F2.((Xo)51)
=
o
= FA318(l)
If Y E
= (FA a
= F5,4(l)
=
is an arbitrary vector, Y' is a vector at the point s1 E S which covers (it is evident that Y" is a vector at the point S2 E S which
Y, and Y" = also covers 1), then
ã5,(Y) =
Y") = VFA*(XO)c, = (Fo4((Xo)5, Y') =
1") =
We see that the form & does not depend on the choice of the point s E up to a nonzero factor.
S, ir(s) =
Now let co E C and let s = r(c) be a section of a bundle in a neighbourhood of c0. We determine a smooth 1-form a in this neighbourhood by the relation a(.(Y) = As
YE
was proved above, if TI (c) is another section of the bundle ir, then
=
f &Tt((•) for some smooth nonzero function 1. Hence formula (8) correctly defines
a section a* of the bundle (1) which is locally covered by each form a determined by formula (8) for an arbitrary smooth section r of the bundle jr. We shall prove determines a contact structure on the space C, that is, that a* is that the section nondegenerate in the sense of Definition 1. To do so, we calculate the form dafor the form a, which is determined by formula (8) on the linear space Kera. Let V Ti(S) be an arbitrary vector, c = sr(s), Y = ,r.(Y) E and Y' = with A = A(s) being a number such that s = F2(r(c)). It is evident that A(s) is a smooth nonzero function on S. and the vector Y' E TT(d-)(S) covers the vector Y.
48
I.
Homogeneous functions. Fourier transfonnation. and contact structures
We have
=
Y)
FA_l$(Y)) =
=
f(s) = (.s) being a nonzero smooth function. Here J denotes the interior product, Xjw(Y) = w(X. Y) (see S. I I). By differentiating the latter relation, we obtain
f(da) = d(jra) = = df A ((Xo)3jw) + fd((Xo)5Jw) ((Xo)Jw) + = d(ln f) A = d(ln A Jr*(a) + fr(Xl)),(w),
f
.
being the Lie derivative along the vector field X0. Due to homogeneity of the form w and the definition of the vector field X0, we see that (w) and hence, with
= d(ln f)
+ fw.
A
shall use expression (10) of the form ,r(da) with vectors Y0, Y1 E Kera. Let yr;. y; be vectors which cover Y0 and Y1 at the point r(c). By (10), we have
We
da(Y0.
Y1)
= lr'(da)(Y(, = d(ln f) A = Y1'), Ker a,
j=
Y,') +
Y;)
1. The right-hand side of relation (II) does not depend Y1' which cover the vectors Yo, Y1 Kercr, and on the choice of the vectors since
E
0,
hence,
+
Formula
Y+
= = =
Y) + jzow((Xo)6. Y) YJ') + jsoac(Y1) — Y).
(II) provides the required expression for the form da on the linear space
Kera. Suppose that the form dalKera is degenerate at the point c E C. This means that there exists a vector such that = Y0 Kera. Yo 0, and that E for any vector Y' such that W((XO)r(c). Y') = 0, we have Y1')
= 0.
5. Homogeneous symplectic and contact structures
49
As the form w is nondegenerate, the relation
= dim Ker((Xo)1(()Jw) = dimS
dim
is valid. Due to definition (6) of the form
—
I
and the inclusion Yo E Kercr, we
obtain the inclusion C
The dimensions of the two spaces and each other due to (12). Hence, these two spaces coincide,
are equal to
Ker((Xo)T(C)Jw) =
The latter relation shows that the vectors and Y( are linearly dependent. = 0, = 0, which is a contradiction. On the other hand, Hence, da IKtra is nondegenerate, and a determines a contact structure on C. The obtained contact structure will be called the con tactizazion of the homogeneous symplectic structure (S, w). Let us now consider the inverse process, that is, the construction of a homogeneous symplectic structure corresponding to a given contact structure C, (see V.1. Arnold [Ar 21). Suppose that an odd-dimensional manifold C, dim C = 2n — 1, with a contact structure : C -÷ is given. Consider the bundle ir : S —* whose fibre over the point c E C is the set of linear forms a E C (recall that ñ is the pmjection of the bundle (2)). The bundle such that ñ(a) = —* C, which consists of forms is a subbundle of the bundle belonging to the equivalence class of the element a with respect to the action of at each point c. Hence, it is evident that the group the group acts freely on the bundle S -+ C. Each point of the space S is a 1-form a,dx, on C which vanishes on the distribution We define the simplectic structure on the manifold S by the relation
S —* C
w = dw°', where the form
is given by the equality
= (recall
here that a' E
S
Y,,),
a' E S
is a 1-form on C). It is evident that the form w is a
closed form which satisfies condition (ii) of Definition 3. To prove that this form is nondegenerate, we consider an arbitrary form a which covers a* over some neighbourhood in C. Then for any vector Y E 7.(C), we have
a(Y) = a vector which covers Y. Y' = Ta'(S) for some point a' E S.
I. Homogeneous functions, Fourier transformation, and contact structures
so
Due to homogeneity, we have
=
wa') =
= (Xo)a.JdwW
a
a (0) = 0
Hence, the equality a(Y) = by definition (14) of the form valid for any vector Y' which covers Y. Similarly to (11), we can prove the relation
for
Y')
is
Yo,Y1 E Kera.
that the form w is degenerate at some point a4. This means that there exists a vector Z' 0 such that w(Z', Y') = 0 for any does not vanish. Indeed, if Z' = then vector Y'. The projection Z = cv(Z', Y') = Aw((X0)a, Y') = Aa(Y) = 0 for any Y E Ta and a(Y) = 0, which is impossible. Since, in particular, w((Xo)a', Z') = 0, we have Z 0, Z Kera. Suppose
Hence,
da(Z, Y) =
f
w(Z', Y') =0
for any vector Y E Ker a, which is impossible due to nondegeneracy of the form da I Kera We have proved that the form (13) determines a homogeneous symplectic structure on the manifold S. The constructed symplectic structure will be called the symplectization of the contact structure (C, a') (see V.1. Arnold [Ar 2]). Let us now investigate the uniqueneness of the introduced construction. We shall C and S2 C with one and the same prove that for two symplectizations S1 compact space C. there exists a unique symplectic mapping f Si —' S2 such that the diagram S2
S1
.1.
C
C
is commutative. To do so, we construct the diffeomorphism f for S2 = S, = being the symplectization C constructed above, and prove the uniquewith (S. ness of this diffeomorphism. The mentioned diffeomorphism
f: S1 acts
by the formula
f(s)
=
S
5. Homogeneous symplectic and contact structures
51
where the form is dctlned by formula (6). It is evident that f is an R1-invuriant diffeomorphism and that diagram (15) is commutative for S = S2, = ir2. Let w be symplectic forms on S and S,. respectively, and ww be the form on S and defined by equality (14). Then
co=dwW;
w1
Evidently, the relation
=
=
is valid. Hence
f(w) =
= = d((Xo)1jw1) =
= WI,
f is a symplectic diffeomorphism. To prove the uniqueness of the diffeomorphismf, it suffices to show that if S1 = S2 = 5, r1 = 3T2 = r, and f is a symplectic diffeomorphism such that diagram (15) is commutative, then f is an identity diffeomorphism. Since this statement is local, it is sufficient to prove it in a special coordinate system. Consider a local S —* C is locally in C. Since the bundle coordinate system (x' A E R. in S. trivial, we obtain the local coordinate system (A,x' Every diffeomorphism f : S -+ S which covers the identity mapping id: C -÷ C has the form and
f(A,x',
=
.. .
iz(x', ... ,
for some nonzero smooth function jt on the manifold C. Every Re-invariant 2-form w has the form w = dA A w' + Aw"
for some forms w' and a)" on C. If a) is the symplectic form on S and f is a symplectic mapping, we have a) = dA A a)' + Aw" =
= jz(x) dA A W' +
A cv' + cv"].
I, that is, that f is an identity mapping. The above considerations prove the following theorem.
The latter equality shows that je(x)
Theorem 1. (a) For any homogeneous symplectic structure (S, cv), the manifold C = SIR. athnits a contact structure naturally related to the symplectic structure w (formula (8)).
(b) For any contact structure (C, a), there exists a symplectic structure (S, cv) which induces the contact structure (C, cr) in the sense of(a). This structure is unique up to a homogeneous symplectic d(ffeomorphism.
1. Homogeneous functions, Fourier transformation, and contact structures
52
As will be shown in the following, there is a one-to-one correspondence not only between homogeneous symplectic structures and contact structures but also between all main notions concerning these structures. Here we mention only the correspondence between homogeneous Lagrangian manifolds in the space (S. w) and Legendre manifolds in the corresponding contact space (C, aS).
recall that a manifold L C S is called a Lagrangian manifold if WIL = dim L = n. It is evident that for any L-invariant Lagrangian manifold L, the factor L/R5 is a Legendre manifold in the corresponding contact space (C. a*). Conversely, the preimage of any Legendre man4fold I C C (with respect to the We
0,
canonical projection) is a homogeneous Lagrangian man4fold in S. The following statement is a consequence of Lemma 1.
Proposition 1. if L is a homogeneous manifold such that WIL = 0, then dim L
n.
Hence, we can describe a Lagrangian manifold as a manifold L C S of maximal dimension with 0.
5.3 The contact product We begin by showing that the product S1 x S2 of two homogeneous symplectic structures (S1, wI) and (52, w2) admits a natural homogeneous symplectic structure determined by the form — 2r25C02, with ir : S1 x S2 —* S1. i = 1,2 being the natural projections. On the other hand, the product C1 x C2 of two contact structures (C1, a) and (C2, does not admit any contact structure (note that the product C1 x C, is even-dimensional). However, Theorem I gives an opportunity of defining the contact product C1 x C2 of two contact structures (C1, a) and (C2, a) in the following way. Let (S1, and (S2, w2) be the symplectizations of the contact structures (C1, a) and (C2, ai), respectively. We set C1 x C2 =
S1
x S2/R5
and endow it with the corresponding contact structure. The mappings
ir7:C1xC2—*C1, C
C
i=l,2, xC
5. Homogeneous symplectic and contact structures can
53
be determined in the usual way as the mappings for which the diagrams
S1xS2
SxS
S
(20)
)CXC
C
are commutative. The mapping in the diagram (20) is the diagonal mapping is a Legendre submanifold in the iA(S) = (s, s). It is evident that the image contact space C x C. More generally, suppose that
1: Si -+
S2
is a homogeneous symplectic mapping. Consider the embedding ij determined by the formula i1(s) = (s, f(s)). The mapping i
can
S1 —÷ S1
x 52,
: C1 —+ C1 x C2
be defined as a mapping such that the diagram
"
SI
S1xS2
)
(22)
Ci
I;
C1xC2
)
C
is
commutative. The image i
is
also a Legendre manifold in C1 x
the image of the embedding i1 : S, —*
S1
C2
(note that
x S2 is a Lagrangian manifold in
x S2;
this image is the graph of the mapping f).
5.4 Hamfttonians, vector fields, and diffeomorphisms The aim of this subsection is to examine the notions of homogeneous symplectic diffeomorphism and of contact diffeomorphism and their relationships.
1 Homogeneous functions, Fourier transformation, and contact structures
54
Definition 4. A diffeomorphism G : —+ 52 of two homogeneous symplectic spaces (S1. cvi) and (S2, w2) is called a homogeneous symplectic diffeomorphism if it commutes with the action of the group and G(co2) = DefinitIon 5. A diffeomorphism g C1 -÷ C2 of two contact structures (C1, a contact diffeomorphism if = and (C2, The relationship between these two notions is established in the following theorem.
Theorem 2. (a) Any homogeneous symplectic
G: -+ 52 uniquely g : C1 —+ C2 between the contactizations Ci and C2 of the spaces S, and 52 such that G covers g. (b) Any contact g C1 -+ C2 uniquely determines a homogeneous symplectic d(ffeonzorphism G : 52 between the symplecrizazions Si and S2 of the spaces C1 and C2 which covers g.
determines a contact
Proof. Part (a) of Theorem 2 is evident. The proof of the uniqueness of the dif-
feomorphism G in part (b) of Theorem 2 was given in the proof of part (b) of Theorem 1. We only have to prove the existence of the diffeomorphism G in part (b) of Theorem 2. C2 be a contact diffeomorphism and Let g C1 C1, S2 C2 be the canonical symplectizations of the spaces C1 and C2 defined above. The mapping —÷ is a bundle diffeomorphism such that g*(4) = a. Hence, the restriction of the mapping g* on the subbundle 52 of the bundle is a bundle diffeomorphism S2 —' S1. Denote by G : S1 —÷ S2 the inverse diffeomorphism. It is evident that G commutes with the action of the group R. and that G covers g. We shall now prove that G is a symplectic diffeomorphism. In the following, elements corresponding to the bundle S, C, are labelled by the subscript i. Taking into account the commutative diagram G
Si
C2
we have a
(I)
(I)
G (w2
0 G)*(Ya1))
=
= a2((g 02T1)s((Ya,))
o jri*(Ya)) =
=
and a2 = (gs)_lai is the corresponding point a point in in S2. Since the symplectic structure on the space S1 is given by the relation
5. Homogeneous symplectic and contact structures w,
55
= 1,2), we have
= G
a symplectic mapping. This completes the proof of the
0
theorem.
The notions of symplectic and contact diffeomorphism give us an opportunity to define Hamiltonian and contact vector fields. DefinitIon 6. A vector field X' on a homogeneous symplectic space S is called a Hamiltonian vector field if its one-parameter group {G,} consists of homogeneous symplectic diffeomorphisms. DefinItion 7. A vector field X on a contact space C is called a contact vector field its one-parameter group (g,) consists of contact diffeomorphisms.
if
Theorem 2 establishes a one-to-one correspondence between one-parameter groups of homogeneous symplectic diffeomorphisms and contact diffeomorphisms and, hence, between Hamiltonian vector fields and contact vector fields. Recall that the Hamiltonian function corresponding to a Hamiltonian vector field X' on a symplectic space S is a function H such that dH = —X'jw. The Hamiltonian function exists locally for each Hamiltonian vector field X' and is unique up to an additive constant term. However, in the homogeneous case (i.e., when S is a homogeneous symplectic space) for each homogeneous vector field xl, there exists a unique global Hamiltonian function which is homogeneous of order I with respect to the action of the group This function can be determined by the formula
H = w(Xo, X') = wW(X'). The validity of the latter formula can be established by direct calculation in a local coordinate system such that w = A dx1, and the action of the group given by the formula A(p1 = x"). The existence of such coordinate systems follows from the Darboux theorem for
f)
contact structures (see, e.g., S. Sternberg [St 1]). Theorem 2 shows that any contact vector field X on a contact space C is determined by a Hamiltonian function H which is a homogeneous function of degree 1 on the symplectization S of the space C. The Hamiltonian vector field which
corresponds to the Hamiltonian function H will be denoted by V(H), and the corresponding contact field, by XII. The situation is much simpler if the contact structure a* on a contact space C is determined by a global 1-form a. Then, for the form defined by formula (6) belonging to the equivalence class a*, we have = X(s)a. with x(s) being a nonzero homogeneous function of degree 1 on the symplectic space S. We have Xojw =
x(s)
.
56
I.
Homogeneous functions. Fourier transformation, and contact structures
The existence of a nonzero function x(s) which is homogeneous of degree 1 gives us an opportunity to identify the contactization C of the symplectic space S with the submanifold in S which is determined by the equation = I. In this case, any Hamiltonian function H which is homogeneous of order I is uniquely determined by its restriction 1: on the space C:
The function It is said to be a contact Hamiltonian function. We denote the corresponding contact vector field by Xh. The vector field Xh determines the contact 1-lamiltonian function by the formula h = Xhjcw.
To
prove the latter relation, we note that at any point s such that x(s) =
1,
we
have
h(s)
H(s) =
V(H)) = ã.,(Xh)
X,,ja.
Note that it is useful to generalize the above theory to the case when a Hamiltonian
function has an arbitrary order of homogeneity. The reason is that in applications to the theory of differential equations, one has to consider Hamiltonian vector fields (or their generalizations) determined by the principal symbols of the differential equations under consideration. However, the conventional method of reduction of
an equation to that with a first-order pseudodifferential operator by multiplying the equation by an elliptic operator of the appropriate order does not work in theory. This is due to the fact that every function of odd-order homogeneity on a connected homogeneous space vanishes at some point. Below, we describe the generalization of the developed theory which enables us to consider Hamiltonian functions of arbitrary-order homogenuity.
Let H(s) be a homogeneous function of order m on a homogeneous symplec0 on the set charsH = {s SIH(s) = 0}. tic space S. We suppose that dH Therefore, the set chars H is a smooth submanifold in S, which is evidently invariant with respect to the action of the group In addition, we suppose that the vector field V(H) is not parallel to the radial vector field X0 (in this case, the Hamiltonian function H(s) is called a Hamiltonian function of principal type; see J. Duistermaat, L. Hormander [DH 1], Ju. Egorov fE 21, B.Yu. Sternin, V.E. Shatalov LSSII 5], and Chapter 3 of the present book). Locally, the function H(s) admits the factorization of the form
H(s) =
H471_1(s)
.
H1(s),
(23)
where Hi(s) is a homogeneous function of the first order and Jim-i(S) doesn't vanish. This factorization can be constructed in the following way. The bundle S C is a locally-trivial bundle with the fibre and hence, the space S is locally diffee-
.
5. Homogeneous symplectic and contact structures
morphic to the direct product C x Let (x on C. Then the corresponding coordinates on S are The function H(s) has the form
H =XmH(XI and
be the local coordinates
...
,
AE
x2u1_l),
we put H1 = AH. Hm_i =
The function H1 (s) determines a Hamiltonian vector field V(H1) and a contact vector field X,11 as was described above. Note that at any point of the set charsH, the vector field V(H1) does not depend on the choice of factorization (23) up to a nonzero factor as well as the vector field X11 in corresponding points of the set
charcH = {c E CIH(s) = 0 for ir(s) = c}. To prove this, we note that
V(H) =
H1 + Hm-i
and hence
= Hm_i(s) We have proved the following statement.
Lemma 2. If H is a homogeneous Hamiltonian function of principal type, then determines a distribution of one-dimensional the Hamiltonian vector field planes on the manifold char5(H) for any factorization (23). This distribution is homogeneous and hence determines a one-dimensional distribution on the manifold
chars(H). Definition 8. A one-dimensional distribution on the submanifold of the symplectic space S is called a Hamiltonian distribution if it is locally determined by a Hamiltonian vector field.
Definition 9. A one-dimensional distribution on the submanifold of the contact space C is called a contact distribution if it is locally determined by a contact vector field.
We denote by LH the Hamiltonian distribution determined by a Hamiltonian function H, and by I,, the corresponding contact distribution. Note that the Hamiltonian distribution given on a submanifold in S of codimension I is uniquely determined by this submanifold. It is due to the fact that this submanifold determines the Hamiltonian function H(s) of principal type uniquely up to a nonzero factor. A similar statement is true of contact distributions. The remainder of this subsection is devoted to a very important property of a Hamiltonian (contact) flow of a Lagrangian (Legendre) manifold. Let H be a
1. Homogeneous functions, Fourier transformation, and contact structures
58
homogeneous function of degree m and let L0 be a homogeneous submanifold in chars(H) such that
(I) (2) L0 is transversal to the distribution LH.
C charc(H) such that
Then there exists the corresponding submanifold
(I) (2)
a*I_O; is transversal to the distribution
denote by L the union of integral curves of the distribution LH originating from L0. Similarly, / is the union of the integral curves of the distribution 'H We
originating from lo.
Theorem 3. L is a Lagrangian Theorem 4. 1 is a Legendre manifold.
Proof of Theorem 3. Note that the statement of the theorem is local. Let a E L be an arbitrary point on L. Then a belongs to an integral curve y of the distribution LH with an initial point a0 E L0. Let (ao, a1, . . , a,v = a) be a partition of the - aj 1, the distribution curve y such that in the neighbourhood of each segment be transversal Let LH is determined by the Hamiltonian vector field submanifolds with respect to y of codimention I in L, with L01 intersecting y in By induction, we see that it is sufficient to prove the theorem under the point the assumption that LH is determined by the Hamiltonian vector field V(H). being a one-parameter group corresponding be the image g1 (L0), with Let is transversal to V(H). Since V(H). to the vector field V(H); then L = .
H = 0, we see that HIL = £V(lI)(W)
0.
The Lie derivative £v(H)(W) vanishes on L, since
V(H)Jd w + d(V(H)jcv) = d(d H) = 0.
= 0, we see that wIz. = 0 for each value of r. Consider now the tangent vectors X, Y on L. We have X = X1 + a V(H), Y = Vi + V(H), with X1, Y1 being tangent to Lr. Hence, Since wILt,
w(Xi, Y1) = w(X1, Y1) +aw(V(H), Y1) +
= since dHIL =
0.
V(H))
=0.
This completes the proof.
The proof of Theorem 4 can be reduced to the proof of Theorem 3 with the help of Lemma 2.
5. Homogeneous symplectic and contact structures
59
5.5 Examples In this subsection, we consider three examples of homogeneous symplectic structures and the corresponding contact structures. Namely, we consider the following symplectic manifolds. X of a smooth manifold X (without zero section). (I) Cotangent bundle (2) Cotangent bundle Tj (X x X) of the direct product of two copies of a smooth manifold X. (3) Cotangent bundle T*(X x R) with special action of the group
Example 1. Let X be a smooth n-dimensional manifold and x (x' x's) be a local coordinate system on X. We use the corresponding coordinate system (x, p) (x' on Tax. There exists a natural action of the group
on
A(x,p)=(x,Ap), A typical chart in the projectivization
on the symplectic space
0. Local coordinates in this chart are (x, p'), where
2,...,n.
=
is
pip,, i =
X is determined by the form w with the local
The symplectic structure in representation
w=dpAdx =>dp1 AdX'. The radial vector field
has
the local representation
= >2Pij.
X(x.p) = and
hence, the corresponding contact form in the chart p,
0 is given by the
equality
a
= X(X.p)J W1p11
= dx' + >2
If H(x, p) is a (local) Hamiltonian function, H(x, Ap) = A H(x, p), then the corresponding Hamiltonian vector field is
V(H,)=
-
=
-
and the corresponding contact vector field in the chart Pi Xi1
0 is given by
1. Homogeneous functions, Fourier transformation, and contact structures
60
Example 2. Let us consider the direct product Xx X. We denote by x = (x',..., f) the local coordinates in the first factor and by v=(v' the local coordinates in the second factor of this product.
Since T*(X x X) = T*(X) x Tt(X), one can choose local coordinates in xX) of the form (x,p;y,q), with (x.p) = (x1 being coordinates in the first factor of the product TX x T'X, and (y, p) = T*(X
,...,yhi; q, group
on
being coordinates in its second factor. The action of the x X) is given by the formula
A(x. p: y, q) = (x, A p: y. Aq). (X x X) and the local coordinates
The typical charts of the projectivization of
in these charts are listed as follows: — the local coordinates in the chart P1
0 are (x, p*;
q*), where
q=q1/p,,i=l,2 0 are (x, p*;
— the local coordinates in the chart q,
q'), where
x X) is determined by the form.
The .cymplectic structure on
w=dpAdx —dqAdy. The radial vector field has the local representation X(x.p:y.q)
= in the typical charts is determined in the
x
The contact structure on
following way: in
the chart p'
0,
dx' — in the chart q,
0,
—dy'
—
(X x X)/R., because the points of the latter space with p = 0 or q = 0 do not belong to the Note that the contact product
x
is not equal to
former one.
Suppose that H, (y, q) is a local Hamiltonian function on the second factor of x Then the Hamiltonian vector field corresponding to this
the product
5. Homogeneous symplectic and contact structures
function is given by the formula V(H1)
+
and the contact vector field XH1 is given by the formulas XIII
in the chart pi
=
+
0, and
= + in the chart qi
I, q*) —
I,
0.
Example 3. Let us consider the cotangent bundle T*(X x R). As above, we denote
by x = (x'
f) the local coordinates in X. We denote by: the coordinate in
R.
We have T*(X x R) = T*(X) x x R1. We denote the coordinates in the latter space by (x, p, r, E) and define the action of the group R1 on T(X x R) in the following way:
A(x,p.t, E) = (x,Ap,AI_m,,AmE). with m 2 being an integer. The standard .symplectic form on
(X x R),
w=dpAdx+dEAdt, is homogeneous of order I with respect to the action of
defIned above. The typical charts on the projectivization x R)/R. are: — the chart p, 0 with the coordinates (x, p5, E5), where
i=2 — the chart:
0 with the coordinates (x, p5, E5). where
= — the chart E
E'=E/pr;
n;
=1 0
=
n;
ES =
with the coordinates (x, p5.
=
i
,
where
:=
(24)
62
1.
Homogeneous functions. Fourier transformation, and contact structures
The radial vector field is given by +
X(rprE)
(1
=
—
m)t- + m
The contact structure is determined by the form
p7dx' + mE*dt* +
dx' + in
the chart p'
(m — 1)t*dE*
0;
p*dx + (m — 1)dE* in the chart t
0, and pSdx
the chart E 0. If H,(x, p, z, E) is a Local Hamiltonian function which is homogeneous of order 1 with respect to the action (24) of R, then the Harniltonian vector field is given by the formula in
-
V(H1) = The contact vector field on
-
+
x R)/R* is given by the formulas
the space
XH = 1,
—
l,p*,t*,E*)
+ —
in the chart p, XH1
E*)
p,
[H,z(x,
1,
—
1,
r',
— (m —
p*j* ES) — m
1, p5:5
0;
= 1)—
+
E(X,
1)—
(m—
1)]
p, 1, I)]
6. Functorial properties of the phase space
ID
thc Chin
63
0,
= I
rn—i
1=1
—
in
the chart:
[HI1(x,p*.
i,E*)_
1E*HIE(X,p*. l.E*)]
0.
6. Functorial properties of the phase space and local representation of Lagrangian manifolds. The classification lemma We are especially interested in the investigation of a homogeneous symplectic space in the case when this space is the cotangent bundle of a smooth manifold M. In this situation, we call the cotangent bundle (without its zero section) the phase space of a M. In this section, we investigate the properties of the phase space M with respect to smooth mappings of the manifold M. 6.1 Induced mappIngs
of the phase space
First, we note that there is no reasonable mapping between the cotangent bundles and of two smooth manifolds M and N induced by an arbitrary mapping f : M —÷ N. Such nonexistence of the induced mapping is caused by the fact that points of manifolds and covectors are transformed in the opposite direction: If f : M —÷ N is a smooth manifold, then the induced mapping acts from the fibre to the fibre TM of corresponding cotangent bundles. However, there are three cases when the induced mapping can be defined. (1) f: M -÷ N is a diffeomorphism. It this case, the mapping f' : 17(X)N -+
is an isomorphism of linear spaces, and we can define the mapping f. T'N by the formula f.(x, = (f(x), (here is a linear
T*M —+
form on TXM). It can be easily seen that the mapping is an isomorphism of the homogeneous symplectic spaces M and Note that the one-to-one nature of the correspondence f was used twice while determining 1.. First, the mapping f is in
the point x, and second, if there were two
64
1.
Homogeneous functions, Fourier transformation, and contact structures
points x' and x" in M such that f(x') = f(x") = y, two different mappings f. would be determined on the fibre TN. and The following affirmation is quite evident.
Proposition 1. If L C
is a homogeneous Lagrangian manifold and f: M N is also a homogeneous Lagrangian N is a diffeomorphism. then (L) C manifold.
(ii) I : M -+ N is an embedding. Since each point of N has at most one preimage, the ambiguity marked above does not occur. However, the mappings
f
are defined only at the points of the image f(M) c N.
T7(X)N
Thus, we can define the mapping f* on the restriction of the bundle T*N to the manifold f(M). This situation can be illustrated by the commutative diagram
I.
I
'I.
c-+
M
N
Evidently, the following affirmation is valid.
Proposition 2. The mapping f* defined above is a fibre-to-fibre projection.
The relationship between the mapping f* and the symplectic structures of the and T*N is shown in the following proposition.
manifolds
Proposition 3. Let
be
generate 2-forms) on
the symplectic structures (i.e., the canonical nonderespectively (see Section 5). Then the relation
(f*)*()
—
holds.
Proof Since the affirmation (1) is local, we can use the local coordinate systems (x' xm) and (y' ,.., yfl) in the neighbourhoods of the points x E M and f(x) E N, respectively. We suppose, that the coordinates are chosen in such a way that the mapping f is determined by the relations xm,O
0).
Let be canonical coordinates in the fibres of T*M, and q,, . .. , be Xm) and (y', ... , y"), canonical coordinates in the fibres of T*N induced by (x' respectively. We have WM
= dp A dx = dp, A dx' + ... + dpm A A = dq A dy = dq, A dy' + ... +
dXm,
of T*N is determined by the equations ym+i = ... = y11 = The submanifold 0; the set of variables (y1 ,...,ym, q, qn) forms a coordinate system on
6. Functorial properties of the phase space
is then described by the formula
The mapping
f*(ylymq1 It
65
qm).
is easy to see now that both sides of formula (I) have the same local represen-
tation: (f*)*(WM) = WNIrN = This
dqi A dy' +
+ dqm A dytm.
completes the proof.
This Now let L be a nondegenerate homogeneous Lagrangian manifold in means that L is a graph of the differential of some smooth function 4) on N. In is evidently transversal, this case, the intersection of L with the manifold N. The following affirmation and hence, L fl N is a smooth submanifold of
is almost evident.
Proposition 4. If L is a nondegenerate Lagrangian manifold in T*N, then the restriction of f* to the intersection L 0 N is a Lagrangian embedding. It easy to show that the obtained Lagrangian manifold is also a nondegenerate Lagrangian manifold determined by the restriction of the function 4) to the manifold M.
We note that a nondegenerate Lagrangian manifold L is never R.-equivariant. analogue of Proposition 4, we introduce the In order to present the notion of a nondegenerate homogeneous Lagrangian man jfold as such a Lagrangian manifold L which is a conormal bundle of some smooth submanifold X of codimension I. We shall call the manifold X a determining manifold of the corresponding Lagrangian manifold L. Now we can formulate the homogeneous analogue of Proposition 4. in T*N. with the deProposItion 4'. If L is a nondegenerate Lagrangian lennining manifold X C N being transversal to f(M), then the restriction of f* to the intersection L 0 N is a nondegenerate homogeneous Lagrangian embedding with the determining manifold L fl f(M).
(iii) f : M
N is a projection. More exactly, we suppose that I : M —*
N determines a smooth locally-trivial fibration over the manifold N. Under this
M is a monomorphism for each assumption. the mapping i; : T;(X) N x E M. We denote by TJM the union of images of mappings f for all x e M. actually, TJM is an inverse Evidently, is a subbundle of the bundle image of the bundle T*N with respect to the projection f. The introduced notions
1. Homogeneous functions, Fourier transformation, and contact structures
66
are shown on the following commutative diagram, T*M
TI;M
T*N
I
I
I
N = M is defined as (fY' on each fibre f'[T,'(X)M] of the bundle TI;M.
M
where
The following affirmation demonstrates the relationship between the symplectic
structures of TM and TN. Proposition 5. The relation
(f)a(w)
=WMIT,M
holds.
The proof of this proposition can be carried out by the direct calculation in local coordinates.
Now let L be a nondegenerate Lagrangian manifold in the symplectic space T*M with the determining function 4). We suppose that the intersection of L and TI;M is transversal. The following affirmation is valid. Proposition 6. The restric lion of the mapping to the intersection L fl TI; M is fl TI;M) is a Lagrangian man 4fold in T*N. an embedding. The image
Proof Since the affirmations of Proposition 6 are local, we can prove it with the help of the local coordinate system. Let (x, 9) = (x1, . . , X", . . . . 0m) be a coordinate system on M such that the projection f is described by the formula f(x, 9) = x; thus, x are coordinates in N, and 9 are coordinates in fibres of the .
I
fibration M —÷ N.
Denote by (x, 9, p. T) the corresponding coordinate system on TM; the expression of the structure form WM of the symplectic manifold Tt M is t0M
It
=dpAdx+dr AdO =dpi
is easy to see that the equations of TI;M are {r° = ... = = 0). Suppose = 0. If we represent in
to be a tangent vector of TI;M fl L such that coordinates as
=
= 0,
then we have = 0 Thus we have = (0, 0,0). Since the = intersection between L and TI;M is transversal, each tangent vector in TM can be represented as a sum of the vector tangent to L and the vector tangent to TI; M.
6. Functorial properties of the phase space
Hence, for any
there exists a representation
+
= with tangent
being tangent to L. Thus, for any there exists a vector Due to the fact that L is a Lagrangian
to L with the last component
manifold, we have
= (dpAdx +dr for any The latter equation yields = 0, and hence, = 0. We have proved the first affirmation of Proposition 6. The second affirmation of this proposition is a direct subsequence of Proposition 5. This completes the proof. 0 We note once more that the affirmation of Proposition 6 (as well as of Proposition
There are two 4) is not versions of Proposition 6. The first one is quite analogous to the above version of Proposition 4; we shall not present it here. The second version of this proposition arises when (a) the group acts effectively on the fibres of M N; (b) the function 4), which determines the Lagrangian manifold L, is a homogeneous function of order 1 with respect to this action. In this case, in addition to the affirmations of Proposition 6, the image fl L) is an Lagrangian manifold. Taking into account the importance of the last construction, we present its coordinate description. Let 4) (x, 9) be a coordinate representation of the function 4) above; 4)(x, A9) = A 4)(x, 9) for any A E R.. The equations of the manifold L are
d4)(x,9)
d4'(x,O)
dx
ao
The equations of the intersection L fl
M can be now written in the form
d4)(x.O)_0 I
we used the fact that the (x, 9) form a coordinate system on L. The set (6), when x Re', will be denoted by C,. considered in the space M is rewritten locally as The condition of transversality between L and
rank
d24)(x,8) d24)(x,9) dxdO 89d9
=m+1
68
1. Homogeneous functions. Fourier transformation, and contact structures
Finally, one can easily see that the restriction of the mapping to the intersection M fl L coincides with the restriction to the same set of the mapping —+
Mapping (8) will be widely used in the following. In the described local situation, the function 4)(x. 9) will be called a detennining fl TM); the latter will be denoted by L(4)). function of
6.2 Local representation of Lagrangian manifolds In this subsection, we show that the third construction of the previous subsection allows us to describe (at least locally) any Lagrangian manifold with the help of some nondegenerate Lagrangian manifold. As we treat the local situation, we can x R' and suppose f to be the canonical projection of the put M = Cartesian product Onto the first factor,
f Here x = (x'
:
x
x") and 0 =
—+ Om) are coordinates in the spaces
and
respectively.
Let 4)(x. 0) be a determining function of the Lagrangian manifold L C L is supposed to be Re-invariant. Then, as was shown above, L can be represented L fl as an image of the manifold C0 M under the action of the mapping a (see formulas (6) and (8)). The function 4)(x, 0) is supposed to satisfy condition
Proposition 7. Each
Lagrangian manifold L c can be locally described with the help of the determining function 4)(x. 0), which is homogeneous of' order 1 with respect to R 4) (x, A0) = A (I) (x, 0), A E To prove Proposition 7, we need the following affirmation.
Lemma 1 (Lemma on a local canonical coordinate system; see, for example, [MiShS 11). For any point (Xo, P0) of a Lagrangian manifold L. there exists a
collection of' indices I such that the coordinates (x", pj). / = (i,, ..., I I form a local system of coordinates on L in the neighbourhood of Proof We consider the forms dx' EL'"'' dx'11L at the point (x0, p°) (for brevity, we omit the sign IL below: all forms are considered on L). Let I = (ii. . .. , C (1 ,z} be a set of indices such that the dx1 = (dx" form a maximum independent subsystem of the system of forms dx' dx". In particular, we have
iE!; jE I
6. Functorial properties of the phase space
69
we shall write relation (9) in the form dx' = It is evident that the projection to projects isomorphically to the plane of the tangent plane generated by the coordinates x1. Hence, for any i E 1. there exists a vector tangent to L such that its projection to is (0 0. 1, 0 0) (1 is in the ith place). Due to (9), we have
0= dx A dp = dx' A dpi + dx' A dp1 = dx' A dpi +
A dp7 = dx' A (dpi +
in (10), we see that all forms dpi are linear combinations of the forms dx', dp7. Since L is an n-dimensional manifold. the (dx', dp7) form a basis in the cotangent plane of L in the point (x0, p°). This completes the proof of the lemma. 0
Subsituting the vector
Proof of Proposition 7. For the proof, we use the canonical system of coordinates
(dx',dp7).
Let
xt = x'(x', be
p, = pj(x', p7)
the equations of L. The form
p,(x',p1)dx' —x'(x',p7)dp' is
closed on L due to the equality (OIL = 0. Hence, there exists a unique function such that p7) homogeneous of order I with respect to the coordinates
dS(x'.
= pj(x', p7)dx' —x1(x1, p,)dp7.
Now we consider the function
= S(x',01)+x".07. The equation of the set
(given by mlation (6)) is 8cD(x,91)
The mapping
= —x'(x'97)+x' =0.
is determined by the formula
aS(x',07)
=p,(x
P1 = Relations (15) and (16) show that function (14) is a determining function of the
manifold L. The proposition is proved.
0
Remark 1. As one can easily see, given a manifold L and a set 1, there exists at most one determining function 4 of the form (14), since the function S has to
1. Homogeneous functions. Fourier transformation, and contact structures
70
satisfy equation (13). The function S(x', p7) is called the action in the canonical chart U, with the coordinates (x', p7).
Remark 2. Sometimes it is convenient to describe the Lagrangian manifold L (Or, which is the same, the corresponding Legendre manifold 1) not by the function c1(x, 0) itself but by its restriction to the plane 9 = I (a renumbering of coordinates
may be necessary). Evidently, D(x, 9) is uniquely determined by its restriction
0) The
(x.
=
equations of the set K, = C, fl (9o
1} in terms of the function
(x, 9')
are
since for
=
1,
we have (due to Euler's identity)
1,0') = 4'(x, 1.9')—
= c1)i(x.0')— The mapping a is given, as above, by the relation p =
9'). One can also
show that the condition (7) can be rewritten in the form
ran k
axae'
aO'ao'
— —m+
do'
Above, we have described a method of representation of a homogeneous Lagrangian manifold by means of determining functions. One can easily see that one and the same Lagrangian manifold can be described by different determining functions. In order to investigate the ambiguity in the choice of the determining function, we present here three transformations of the determining function which don't change the Lagrangian manifold itself. In the next subsection, we shall show that any determining function c1, of the manifold L can be transformed to any other
determining function 42 with the help of a chain of transforms of the described type.
-
I. Honwthetie transformation. Let 9) = x(x, 9) (D(x, 9), where 4)(x, 8) is a determining function of the manifold L, and x(x, 8) is a nonvanishing function which is homogeneous with respect to 8 of order 0. One can easily check condition
6. Functorial properties of the phase space
(7) for the function $ to be a direct consequence of this condition for the function
Now, if (x,9)E C,, then dci,
dcD
dxtm
34)
34)
9), and hence, and hence, C, C Ci,. Conversely, 4)(x, 9) = Ex(x, We see that C, = C,. C, C For any point (x, 9) C,, the corresponding point of L = L(4)) is (x, p), p = = 4) + and the corresponding point of L = L(4)) is (x, j,), ji = Due to the homogeneity of L. we have (x, L(4)), = xp. = and hence, L(4)) c L(4)). Conversely, L(4)) C L(4). Thus, we have shown that L(4))
L(4)), that is, the homothetic transformation does not change the
Lagrangian manifold. 2. Change of variables. Since the construction (iii) of the previous subsection is invariant, it is obvious that the Lagrangian manifold L (4)) doesn't depend on the choice of homogeneous coordinates in the fibres of the bundle f : M —* N. Hence, if 9 = 9(x, 9) is a change of variables such that O(x, A 9) = A O(x, 0) for every the function 4)(x, 9) = 4)(x, 9(x, 0)) determines the same Lagrangian AE manifold as the function 4)(x, 9). 3. Stabilization. Let cD(x, 0) be a determining function of a Lagrangian manifold
LL(4)),9(90,...,9m).SetO(0o,...,9m,Om+i)and4)(x,9)=4)(x,9)+ We shall prove that 4)(x, 9) determines the same Lagrangian manifold as the function 4)(x, 9). We have
0m+I
=
0j.
the manifold C, lies in the subspace x x of the space = 0 and coincides in this subspace with the set determined by the equality Hence,
c, =
=oj.
72
1.
Homogeneous functions. Fourier transformation, and contact structures
Since acD/ax(x, 0) = 34/ax(x, 9) on C,, the mapping Thus, we have 0 with the mapping 0m+I
coincides on the space
= a,(C,) =
=
Remark 3. We note that every homothetic transformation I can be obtained also
(for homogeneous functions t(x, 0) of order I) with the help of a change of variables. To show this, we consider the mapping f : given by the formula
x
x
—,
= x(x, 9) .9,, with x (x, that
(20)
being a nonvanishing homogeneous function of order 0. It is evident
0)}
=
x(x. 6)9) = x(x, 9) 4(x, 9)
0). Thus, the action of mapping (20) to due to homogeneity of the function a homogeneous function c1(x, 9) reduces to the multiplication of this function by x(x, 0). We have to prove that mapping (20) is invertible. To do so, we consider the Jacobian
det
ao.
=
-ax
-ax
a00
ao1
-ax —=ae0
ae1
X
+
•.. ôOm
-8x —=... a91
-ax
Oi
aom
.
(21)
.
-ax-
Om
ao0
-ax
9m
ao1
...
-ax
X + Om
aom
Each row of this determinant can be represented as a sum of two rows of the form Hence, the determinant decomposes into (0 0. 0) and the sum of determinants whose rows are either of the first form or of the second one. Note that if a determinant contains two rows of the second form, it vanishes.
6. Functorial properties of the phase space
73
Hence, we have
x
0
...
ax
0
-ax
6o
ae0
•..
aom
o
o
(
+
-ax
x
0
do0
dO1
0
0
... ... ...
0 dOm
...
x
x
0
...
0
0
x
•..
0
+...+ maô
=Xm+l +
FjOm
= Xm+I
due to the Euler identity. Hence, the determinant (21) does not vanish, and mapping
(20) is invertible (at least locally). 0), we can use Thus, when using the homogeneous determining functions only transformations 2 and 3; for example, only these transformations will be used in the proof of the classification lemma below. However, if we use restrictions of = 0, D(x, 0') 4(x, 1,9') (see Remark 2 determining functions to the plane above), we have to use all three transformations. Indeed, one can easily verify that if the functions (D(x, 0) and 4(x, 0) are connected with the help of a homogeneous a change of variables 0 = 8(x, 8), the corresponding functions (x, 8') and (x, 0') are connected with the help of a change of variables 9' = 0(x. 1,9') and a homothetic transformation with a reasonable function x(x, 9'). That is why we have to consider all three transformations described above.
6.3 The classification lemma
The goal of this subsection is to prove the affirmation (mentioned earlier) that the set of transformations 1—3 is a representative set for the description of the
1. Homogeneous functions, Fourier transformation, and contact structures
74
ambiguity of the choice of determining the function of a Lagrangian
manifold.
More exactly, the following affirmation holds.
Lemma 2. Let D'(x, 9), D"(x, r) be two determining functions of a Lagrangian L. Then there exists a chain of transformations 1—3 which transforms into 4)".Here, =(r1 ti). Proof (Compare Hormander [H 3].) We shall show that any determining function
c1 (x, 9) of a Lagrangian manifold L can be transformed to the normal form (14) with the help of transformations 1—3. This will complete the proof of the lemma, since such normal form is uniquely determined by L (see Remark 1 above). We shall carry out the proof for homogeneous determining functions; due to Remark 3, we can use only transformations 2 and 3. Let 4(x, 0) be a determining function of the manifold L and let (x0, 90) be a point of Ce corresponding to the point (x0, p0) E L. We carry out our considerations in a neighbourhood of (x0, 90)• Consider the matrix a nonzero element. Without loss of generality, we can suppose that this element lies on a diagonal of this matrix (if all diagonal elements (xo, 00) vanish and 0, we use the change of variables Oj = + 9,, = = k 1. j; the matrix a2clvae, d91 contains a nonvanishing diagonal element). By renumbering the variables, we get the case when a2D/aem aem 0. Then the equation (22) can be solved with respect to Om in a neighbourhood of (Xo, Denote the solution of (22) by Om = Om (x, 0o 9m _i). Due to the Morse lemma, there exists a change such that of variables On, = Om(X, 1). (b)CD(x,Oo
where
Om_i,Om(X,Oo Om-i)). The sign in depends on the sign of the element d2c1/aen, Hence, the function c1(x, 9) transforms into the function (V(x, Oo, . .. , 9m— i) with the help of transformation 3 (stabilization). By repeating this procedure, we reduce the function cX(x, 0) to a function ar1(xo, r°) = 0. As it was shown above, t) such that condition (7) is preserved during such a procedure. Hence, we have (b)
rank
d2c1(x, r)
arax
0 (xO, t ) = rank
a2'4'(x, r)
d24(x, r)
arax
arar
(xo, t
)
= k,
6. Funciorial properties of the
phase
space
withk being a number of variables r : r =(TO,...,tk). Let! c {l,2,...,n} be a collection of indices such that (xe, TO)
det (evidently,
Ill =
k). We
denote by r = t (x.
(23)
0
8x' 8r a
solution of the system of
equations
d4(x, r) ax
Such a solution exists in a neighbourhood of (xO, r°) due to condition (23). We write
=
r(x, a)).
(24)
We have
ax' a2ci,
3() ax' a24'
ac)
dr
at
ax
at ax'
at at
aö
j€1,a2r
arar Since
dr (xO, r°) =
0,
(25)
j,kEl.-
the latter relation yields
a23
k,j
E
Due to this inequality, the equations of
and therefore, the equations of C,
be solved with respect to in the form can
can
be written (26)
x&'
We use the notation x
=x (x
= We
shall prove that the functions 1 and determine one and the same manifold = C3 and that 4' and 4 coincide on this manifold up to the terms of second
1. Homogeneous functions, Fourier transformation, and contact structures
76
order. Indeed, we have on C3,
= due to (25)
a6ax' +
-
8s1
ax1
x =0
—
(26). Hence, C3 C C4. Since both C4, and C4 are n-dimensional manifolds, they coincide in a neighbourhood of (Xe, The functions 6 and 6 coincide with each other on C3, since and
= The
=
first derivatives of 6
and
6
also coincide with each other on C3:
0a6
—
—
C,
C,
a6
as C,
due
=
=
+
a6
ax1
=
—
-
to equality (25). Therefore, there exists a smooth matrix
such that (27)
We shall now prove that if the matrix B11 is sufficiently small, then the functions 6 and 6 can be transformed to each other with the help of a change of variables. To prove this, we can search for a change of variables in the form -
=
+ A11(x,
act,
i,).
For the unknown matrix A11, we obtain the equation
+A11
(28)
where the matrix F" is determined by the relation
6(x,
= 6(x,
+
—
+
—
—
F"(x,
6. Functorial properties of the phase space
The Jacobian of the system of equations (28) equals
I
77
for B,1 = 0, hence this
system is solvable for sufficiently small matrices Now we shall reduce the general case to the one just considered. Let k-,) = 4)(x, be
+1
c-,)
a homotopy connecting the functions 1 and 4). Since
vanishes at
the point (xO, k?). we have
rank
(xo.
= rank
(xo,
=
k.
Since
4,(x,
= 4,(x,
+ (1 —
+I
1], there exists a neighbourhood U such that for every I U, the functions and are connected with each other by a change of variables (transformation 2). Since the segment 10, 1] is a compact set, we see that the functions cIo = cb(x, and = c1(x, are connected with each other by a change of variables. Thus, we have proved that any determining function 8) subject to condition (7) can be transformed by the chain of transformations 2 and 3 to the function of the form (14), which is a nonnal form of the determining function of a Lagrangian manifold. we see that for any point ri
[0,
Remark 4. As was shown in Remark 3, if we use the function 8') = 1,9') instead of 9), we need to use transformation I as well as transformations 2 and 3. The corresponding normal form, evidently, is
S(x',p1)+x'p1—x',
IU1={2,3
n}.
Chapter 11
Fourier—Maslov operators
theory)
1. Maslov's canonical operator
The canonical operator developed by V. Maslov as early as the 1960s is a powerful tool in the asymptotic theory of differential equations. Here we present its version designed specifically for applications to the study of singularities of their solutions.
1.1 Local elements
Consider the arithmetic space (with deleted origin) with the coordinates the conventional action of the group and of nonzero real num9m) (Oo, bers. 0m) = (A00,A01
AOm).
the corresponding quotient space denote by Let the following objects be given in some homogeneous neighbourhood of the point E 0), smooth with respect to (x, 0) and belonging (a) a real-valued function for any fixed x; to the space (b) a function a(x, 0), smooth with respect to (x, 0) and belonging to the space
We
In the sequel, these functions will be referred to as the phase function (or simply the phase) and the amplitude function (or simply the amplitude), respectively. a] by the equalities We define the local elements pk+m
= for k + m
/ (2,r)m/
"I j
0. m + a even; I k4-m
=
—m —
I)!
j
m+1)
(3)
1. Maslov's canonical operator (R1-equivariant theory)
79
for k + m <0, m + a even; F,k [4), a] =
fork+m
(k-Fm)! f
V.p.
a(x,6)w(0) (4)(x, Ø))k+m+I
odd; and
—1)!
f
x a(x,9)co(9) for k + m <0, m + a odd. Just as in Section 1.1, one may verify that the integrands in (2)—(5) are preimages of odd currents on RP'. Therefore, it is possible to consider the integrals (2)—(5) as those over the projective space PP". On the other hand, since the coefficients of the forms in the integrands are distributions, the definitions (2)—(5) need validation. In fact, these definitions are generally no: correct for the arbitrary phase functions 4)(x, 9). Therefore, we impose the requirement that the phase functions satisfy the following condition:
4)(x, 0) does not vanish on the set (4)(x, 0) = 0) 1. The of the phase function 4).
ConditIon of zeroes
ProposItion I. Assume that the phase function 4)(x, 0) satisfies Condition I. Then a] E the integrals (2)—(5) converge in the space D(R'), and besides, any positive e.
Proof First of all, note that by using a partition of unity, we can reduce the study
of the integrals (2)—(5) to the case when the image of suppa(x, 9) in PP" is arbitrarily small. By renumbering the variables if necessary, we see that it suffices to consider two cases: A.
on suppa(x,9).
B.
0)
Consider
first
0 on supp a(x,
the
8).
case of even m
+ a.
In case A, by omitting an inessential
constant factor, we have
&(k+m)(4)(X
=
f
RP x
0)) a(x, 0) w(9),
Adx,
II. Fouiier—Maslov operators
0. By (L2.9), we have
provided that k + m
f
RP'"xR'
=1RP'"xR'
a(x,9) W(x)wo(8)Adx} a4/aoo(x, 9)
t
a(x,
ao0
RP"xR'
A
dx.
Continuing this procedure, we obtain the formula
6)) a(x, 0)
J
A dx a
ö(4(x,
=
a
x a(x,
9)
With the integrand on the right-hand side of the latter formula belonging to the integral is defined correctly. a] It will be shown below that in this case, In case B, similar computations lead to the formula
I
a]. ço) =const j
9)]
Adx.
x Since the distribution e > 0. relation (6) implies
i acD/ax'(x.
a
6)) belongs to the Sobolev space a], ço)I
for any
const .
We eventually obtain the desired inclusion duality of HG and H G•
If k + m < 0. the integral (3) defining
E
using the
a] converges absolutely, with
To verify that a] therefore belonging at least to the space a] it suffices to prove that the derivatives of (3) of order —k — m A straightforward computation shows anyhow that the belong to
1. Maslov's canonical operator (R1-equivariant theory)
formula
=const.f 8(4(x,9))
a(x,9)w(8)+
ax
—(k + m), where the dots stand for a sum of integrals of the form (3) belonging to The first integral on the righthand side of (7) was shown to belong to as early as when we were considering case A. The case of odd m + a may be considered in a similar way. Proposition I is
is valid for any multiindex a, lal =
0
now proved.
We point out that the estimates of the integrals (2)—.(5) are in a sense the best possible ones, since for m = 0, we have
al =
. a(x);
the latter function belongs to and does not belong to provided that a(x) 0. Now let us study the location of the singularities of the integrals (2)—{5). For this purpose, we introduce the following notion, which is also of importance for the subsequent presentation. DefinitIon 1. The gradient ideal J (Ct') corresponding to the phase function ct (x, 9) is the ideal of the ring generated by the elements {a4(x, 9)/dO0 J(ct'). We also write d'Z'(X,O)/dOm}. Jk,o(Ct) is the submodule of the Thus, any element a(x, 9) E Jk,a(4) may be represented in the form
a(x,9) =
b(x,O)
(Note that the explicit indication of the dependence on x is omitted in the above
definition; we assume this dependence to be smooth.)
Lemma I (On the gradient ideal). If a(x, 9) E
is valid modulo the functions of the class
b(x,9) =
(Ct'). then the relation
and besides, 8b1(x,9)1d91,
where b, (x, 9) are the functions encountered in representation (8) of a (x. 9).
II. Fourier—Maslov operators
82
Proof First consider the case of m + c being even. If k + m
1, we have by
(1.2.9): —
a]
—
(2jr)m/2
f
(2,r)m/2
f f—d
— —
ö(k+m_I)(ci(X, 8))b1(x,
8b,(x,8) co(8))j
—
—
J
(2jr)m/
= F[c1,b],
which is exactly what we desired. Let now k + rn = 0. Then, similarly to above, we have
= (2,r)m/2
f
acl'( x,8)
w(9)
L
=
P
in —
2jr"' /2
sgn(c1(x,8))b(x,8)w(8) =
Finally, if k + m <0, then we have
(_.l)k+minj
=
(2,n)"2(—k — m
—
1)!
[
sgn(4(x,8))
J x
sgn (q)(x, 8)) (4)(x, 1=0
ak(x, 0)
w(8)}
1. Maslov's canonical operator (R.-equivariant theory) (,...
(
= 2. (2n.)m/2(_k — m)! JRP'
sgn
83
(4)(x, 0)) b(x, 0) cu(9) = Fr" [4), bJ.
The lemma is thereby proved for even m + a. With odd m + a, only the case k + m < 0 should be considered, with the computations for k + m 0 being similar to those held above. In this case, we have (_1)k+m
I4)(x,0)I
I
=
—rn—i)! ad'(x,o) ao,
f=I
lOl
w(9) 1
—d
= (27r)m/2(_k —rn — 1)!
I I
L
—k—rn
xln l4)(x,9)I b.(x.0)w(0)] 101
"
1
— —k
—
9))_(k+m)1fl
m k+m
— km
I
= (2,r)m/2(_k_m_l)! pj.. I4)(x,O)l 191
x —
w(0)
101
1
xln
l4)(x,9)l ab1(x,o)
fE
bt(x.0)co(9)J —
(4)(X,O))_(k+m)
(1)k+m_I b(x,0)w(9) + (2,r)m/2(_k —m)!
RP i=O (4)(x, 0))_(k+m)91
92
rdd'(x, 8)/dOj (l)(x,0) L
d4)(x,O) aoj
b(x,9)w(9).
The first form on the right-hand side of this equality coincides with
[4). bl,
whereas the second one is a function of the class CCC, being a proper integral with 0 smooth integrand. The lemma is proved. As a consequence, we obtain the following result already mentioned in the proof of Proposition 1.
Lemma 2. If the support suppa(x, 0) of the amplitude a(x, 0) does not contain the aJ E points such that d4)(x, O)/dOo = •.• = d4)(x, 0)/dOm, then we have C°°(R").
II. Fourier—Maslov operators
Proof Since at least one of the derivatives dd(x, 0)/aej is different from zero in each point of supp a(x. 0). J(4) is an improper ideal over some neighbourhood of the set suppa(x, 0). Hence, we have b(x, 0) Jk.0('Z') for any amplitude function b with supp 1' C suppa. Repeated application of Lemma I allows us to obtain the equality
= which is valid for any natural N with some homogeneous amplitude bN (x, 0) of aJ evidently degree k — N. On the other hand, the expressions (3) and (5) for Ck(Rn) for sufficiently large N = N(k). This proves imply that bN]
0
the lemma.
As Lemma 2 shows, the set
C, = ((x,0)149(x,0)=0} an important role in studying the singularities of the integrals of the form has appeared in Section 1.6 where we considered (2)—(5). Note that the set functorial properties of the phase space. This notion plays an important role also in the proof of the classification lemma. Using the notation (10), one may put the statement of Lemma 2 in the following plays
form:
Corollary 1. There is an inclusion
sing supp u denotes the singular support of a distribution u (i.e.. the com-
plement of the set of points x E x
—÷
in whose neighbourhood u E
,r
is the natural projection.
1.2 Comparison of the local elements
In this subsection, we consider the question, to what extent there is ambiguity in integral representations (2)—(5). In other words, we study here the transformaa] itself tions of the form of the integrals (2)—(5) preserving the function (either exactly or asymptotically). We need three kinds of such transformations, namely, homothetic transformation, change of variables, and stabilization (these three transformations have been discussed in detail while considering the determining functions of Lagrangian manifolds in Section 1.6). Homothetic transformation. Consider a function x(x, 0) E whose zeroes lie outside suppa(x, 0). Here and below, we assume that the set suppa(x, 8) is connected (this is not a restriction, since one may always use a partition of unity to split the integrals (2)—(5) into sums). Hence, the sign of x(x, 0) is necessarily constant on suppa(x, 9).
1. Maslov's canonical operator
theory)
Lenuna 3. The congruence
modulo smooth functions holds, where • sgn x a, b = + a is even,
b=X_k_m_I •a,
is odd.
Proof Letk+m ?O,form+o being even. Then we have (2Jr)m/2
f
S
We rewrite this fonnula by using the identity sgn x(x, 0) S(c1(x, 9))
6) x(x, 6)) = Ex(x,
(one may obtain the latter relation, e.g., from the representation I
=
2jri
(x
—
I
—
(x + jØ).c+I f
of the derivatives of S-function, which will also be used in the following). We have
=
x,al
(2ir)m/2 —
(21r)m/2
/
w(9)
[
JRP'
= just as we desired (recall that m + a is even). The case m + k <0, for m + a even, and s > 0, for m + a odd, are considered in a similar way. If s <0 and m + a is odd, we have
x, a] = (2,r)m/2(_k_m_I)!
xln
/
[(4(x, 9)) . x(x. 0)]_(k+m+I)
RP"
a (x.9)w(O) 191
[
=
(2,r)m/2(_k x lnIx(x,9)Ia(x,9)cv(9).
[c1(x, 9)]_(k+m+I)1 n
9)1
11. Fourier—Maslov operators
The first integral on the right-hand side of this equality coincides with b], whereas the second one belongs to C°° (indeed, it is a proper integral with smooth integrand). Thus, the lemma is proved. 0
Change of variables. By using a partition of unity and by renumbering the 0 if necessary, we may ensure that 0 on suppa(x, 8), so that the
variables
functions
i=l,2 will be coordinates on RI" in a neighbourhood of the set suppa(x, 8). Let us write the integrals (2>-(5) in these coordinates. With relations (14) defining the projection ir
—+
:
Ri", we have
ir{fi(xO')dO') = where
f(x, 6)
f1(x, 9')
E
=
f(x, 1,9').
We introduce the notations
ai(x.O') =a(x, 1,9,...,6,',,), and rewrite (2)—(5) in the form
f
( aI
for k + m
=
(2,r)m/2
(x. 9'))
(x, 9') dO'
0, m + a even; (
(2jr)m/2(_k
= x for
(4>1(x,
—1)! a1(x,
91))_(k+m+I)
O')d9'
k + m <0, m + a even;
(k+m)!
k
f
V.p.
ai(x,9')dO'
JR
fork +rn >0,m+a odd; 1
k+m —
1)!
x lnIcD,(x,0')Iai(x,O')d0'
1. Maslov's canonical operator
theory)
87
for k + m < 0, m + a odd, respectively. Note that integral (19) coincides with (5) modulo smooth functions rather than exactly, with their difference being equal to (
l)k+m
v'l +
(x,
(2Jr)m12(_k — m — 1)!
a1(x, 9') dO',
which is a proper integral with a smooth integrand. We denote the integrals (16)— (19) also by Now let 9' = 9'(x, r'). r = (t1,..., be a smooth change of variables.
Lemma 4. The equality (x, 9'), a1 (x, 8')J =
(x, 9'(x, r')), b1 (x, r')]
holds, where
b(x,r')
Dr
Pro of This is a straightforward computation.
1.3
Stabilization and stationary phase formula
We study here the relations between the integrals (16)—( 19) with the phase funcIt will be shown that tions 41(x,9') and 41(x,9', = +
the latter integrals may be reduced to the former ones, provided only that the number r of the additional variables i is even. With odd r, the integrals with 0') prove to exhibit different behaviour in phase functions 41(x, 9', and the neighbourhood of the singularity set (see formulas (56)—(58) below). We shall use representation (13) for the derivatives of the S-function along with the similar representation
=
_iO)k+l +
(20)
Both representations follow from the Sokhotskii—Pleumel of the function V.p. formulas. We introduce the following notations in order to use these representations effectively:
k+m+1 >0.
(21)
II. Fourier—Maslov operators
88
In order to make the subsequent argument more convenient, we admit half-integer values of k in the integrals (21), with the square root being evaluated according to the following choice of the argument: E (—,r,,r)
fore >
(22)
(this is a conventional choice when the functions (x±iO)a are considered). To begin with, let r = I (i.e., 17 is one-dimensional). Let us introduce a cutoff function which is a smooth finite function equal identically to I in a neighbourhood of the origin. Note that the integrals 0
[ i = JR" are
independent of the choice of
Lemma 5. Let k + m + 2>
±
modulo smooth functions (see Lemma 2).
The congruences
+ =
+?fl
-. —i
.FA
17-
) —
F' (k + m + 2)
ai(x. 0')].
(24)
[cii(x.O') + —
F(k+m+2)
= -
(23)
r(k+m+2)
—
(25)
Ii,
+m
I'(k+m+2) hold, modulo smooth functions.
(26)
theory)
1. Maslov's canonical operator
89
By the definition of the function x ± tO, we have
—
[
(x, 0') ±
— JR"-
± jOJk+m+2
I
= urn
I
=
I
urn
[Di(x,9') ± a1(x,0')
I
dO'.
J—oc
(27)
Thus, the proof of formulas (23)—(26) requires only that the inner integral on the right-hand side of (27) be evaluated modulo smooth functions with different combinations of signs in the integrand. We have —
0') ±
i—co [c11(x, 0') ±! ± j8JL+m+2 —
± j,cJk+m+2
— l)d7
±
-cc
(28)
The first of the integrals on the right-hand side of (28) converges absolutely due to the condition k + m + 2 > As for the second one, it becomes smooth as e —*
+0. Indeed, çoe
limi
I
1
k+m+2
dnL
I
JD1(x, 0') ±
±
dij, '1
J
this procedure may be repeated as many times as we like. Therefore, the inner integral on the right-hand side of (27) can be replaced by the first summand of the and
II. Fourier—Maslov operators
right-hand side of (28) to within a smooth function. Thus, we obtain the formula
± urn p—.+O
(29)
J a1(x,0')
[cP1(x,
9') ±
±
dO'.
It remains to evaluate the inner integral on the right-hand side of the latter formula with different combinations of the "+" and "—" signs. We begin with (23). Then
both signs are "+,"
and
we need to evaluate the integral
[41(x, 9') +
+
+
where the argument of the square = (x, 0') + is root is chosen according to (22), reduces this integral to the contour integral The variable change ij
(
+—
(1
(indeed, arg4 = 0') + iE] E Deformation of this contour into the real axis (shown by > 0). 0) for arrows in Fig. 1) puts integral (30) in the form where
is a contour shown in Fig.
1
l+
+ — [41(x. 0') + 2)
—
j4'i(x. 0') + i5JL+?n+312
r(k + m + 2)
9') + j5]k+m+3/2
Substitution of the latter expression into (29) gives
+ —
=
r(k+m+2)
f
ai(x,0')dO'
fR"
a1(x, 9')].
I. Maslov's canonical oper4tor
theory)
91
'I,
Figure 1
The case (24) corresponds to the integral
+
J
—
+ jp]k+m+2
This time, the variable change we choose has the form and reduces 1 to
+ — [4,,(x,01)+IE]k+m+3/2
with the contour
f (I
having the form shown in Fig. 2.
=
0') +
II. Fourier—Maslov operators
92
Y2
FIgure 2 Indeed, for
> 0, we have
=
—
Hence, we evaluate
[
F(k+m+2)
thus proving (24). A similar procedure applied to integral (25) leads to the equality
-
0') +
f—ac
— j6]k+m+2
f
— —
—
J
I'(k +m +2)
=
with the variable change
= for
ij
> 0. Thus, (25) is proved.
0') —
e
being used (see Fig. 3), since E
(o.
I. Maslov's canonical operator
theory)
93
—I
Figure 3
Finally, formula (26) follows from the relation
f-co [(D1(x, 9')
—
— [CD1 (x, 9') —
—
—
[ 'yI (I + 2) 9') — e, since for
> 0,
The lemma is proved.
Furthermore, let us apply the results of Lemma 5 to study the integrals of the form 9') ± 4, a1(x, Here we limit ourselves to the case k + m 0.
11. Fourier—Maslov operators
94
For even in + a we have, by (13), (16), and (21):
F& a
[cDi(x.9')
E
(_l)k+m+I(k+pfl+1)!
f
—
(2,r)(m+I)/2
—
XI
2jri
9') ±
j
—
x
x
{j'k [cti(x.9') ±
Using Lemma 5, we obtain the following congruences, modulo smooth functions,
+ —
+m +
(k +m + 1)!
=
f'(k + m + 2)
+m + —
F(k+m+2)
+ 1
- { [4'1(x, 9')
—
—
—
a,(x,9')dO'
[c11(x.9')
r(k+m+4)
—
—
I
—
(_1)*+mr(k+m+4)
—
(2,r)m/2
(k +rn + I'
3"
ai(x,9')
dO'
1. Maslov's canonical operator
where
is a distribution, x:a =
Ix
—a
for
-equivariant theory)
x <0, x:a = 0 for x > 0. Similarly,
—
—i
(
(32)
JR' [4)i(x,
—
for x <0. Note that
x is a distribution, there exist canonical regularizations of the functions where
for half-integer values of
a (see Gel'fand and Shilov [GShi II). For m + a being odd, we have, by (18), (20), and (21),
(x, 9') ±
a1 (x, 9')
± ±
+
The subsequent argument is quite analogous to that in the case of even m + a. We
eventually obtain
F
+ —
—
(2irY"/2
f
a1(x,9')dO' (33)
[4)i(x,
—
(_1)k+fhr(k+m+4) =1
(2n.)m/2
-
f
ai(x,9')dO'
JW [CD1(x,
91)Jk+m+3/2
(34)
The case k + m will be considered later.
Formulas (31) and (32) show that the functions F (x, 0') ± (x, do not admit the representation of the form (1 6)—( 19) with the phase function (x, 9). We have a representation of the form (31) and (32) instead, with the integrands containing distributions of different type. However, should two squares be deleted, we get the integrals of the form (16)— (19) again. More precisely, the following statement is valid.
H. Fourier—Maslov operators
Lemma 6 (The stabilization lemma). The congruences
+
+ (35)
(x, 0')),
9'),
—
(36) —
—
(37) hold. modulo SnlOOth functions.
Proof Consider the case k + in 0 first. As in the proof of (31) and (32) for even m + a, we obtain, using (31) and (32),
—
(_))
+
+
(_l)k+m+I(k+m+1)!
f
2jri
— 1
[41(x, 0') +
1
+
—
jØ]k+m+2
x
—
Ad112
14'i(x, 9') +
+
+ jQ]k+m+2
= 2(2,r)(rn+I)/2
x
(38)
—
We apply formulas (23) and (25) twice to the right-hand side of the latter relation (to the first and the second term, respectively) and obtain
9') +
+
a1(x,
—
=
9'). a1 (x, 9')];
I. Maslov's canonical operator
coincides with (ii). Thc proof of we obtain this
[cii(x*O')+
theory)
97
is cxactly the same. Similarly to above,
—
(k+m+l)! — 2(2,r)(m+I)/2
x
—
—
(39)
However, we use (25) and (26) to treat the first term, and (23) and (24) to treat the
second one. This leads to the occurence of the extra factor i (in comparison with the first case we considered) in the first term. The second term gets the factor —i. As a result, we have —
9'), a1(x, 9')j}.
9'), a1 (x, 9')] +
Here the inverse transformation produces the integral of the form (16) rather than
(18),
—. (k+m)! —' 2(21r)m12
+
(
J
I
9')+ jØJk+m+I
•(k+,n)! f =' 2(2,r)"'/2 =i
V.p.
1
— jO]k+nI+I
j a1(x, 9')dG' 1
/
which agrees with (36). In the above calculations, formula (20) for V.p.
has
been used. Dealing with formula (37) where both sings are minuses, we apply formulas (24) and (26) twice to the corresponding terms (similar to that on the right-hand sides of (38) and (39)). This gives an extra factor —1 (in comparison with (38)) in both cases. The remaining part of the proof of (37) coincides with that of (35).
II. Fourier—Maslov operators
0, for m + a being odd. In this case, we may cariy out the Let now k + rn calculation similar to (40) in the opposite direction,
±
±
(k+m+I)! [
1
1
— (27r)("+2V2
6') ± !i ±
2
— jØJk+m+2
I
+
[c11(x, 6')
±L±
+ jØJk+m+2
AdlJi
x
= 2(2yr)(m+I)/2
<
±
+
±
the only difference from the case of even m + a being the sign between the terms on the right-hand side. Except for this detail, the proof contains nothing new and is therefore omitted. In order to consider the case k + rn < 0, we proceed by downward induction on k + m. We describe here in detail the transition k + m = 1 —÷ k + m = 0 for having the sing "+." All other cases, as m + a even and for both squares k + m = —j, may be considered in a well as the transition k + rn = —j — I quite similar way. If k +m = —1 and rn +a is even, we have
(x, 9).
(x, 9)1
2.(22r)m/2
= =
2. F(x.
f f
sgn[41(x,6')]ai(x,O')d6' sgn[cD1(y,9')]a1(x,8') d9'I
theory)
1. Maslov's canonical operator
where the function F(x, y) defined by (41) is a smooth function of x whose values are distributions of y. Similarly, we have
+
= x a1(x,
=
A di72
A
(21r)(m+2)/2
x a,(x, 6')
f
+ 4)
+ 4 + 4)
S
dO' A
A
=
(42)
where the function G(x, y) defined by (40) belongs to the same class as F(x, y). Let us calculate the derivatives with respect to yJ of the difference F(x, y) — G(x, y). We have
[F(x, y) L dyi
=
— G(x, y)l
— (2jr)m/2
f
(it (y, 9'))
(y, 6') a1 (x, 0') d6'
!. +
— (2,r)(m+2)/2 x
A
=
ii2)
A di72
dyi
+
+
4 + 4,
However, it was already shown that the latter expression (with x standing for (x, y))
equals 0 for any j. Hence the difference F(x, y) — G(x, y) does not depend on v We obtain the required result by setting x and is therefore of the class y. The lemma is proved. 0
Let us also give the "homogeneous" versions of the equalities (35)—(37). For this
purpose, let us assume as above that suppa(x, 6) lies entirely in the set
0.
II. Fourier—Maslov operators
Then the following congruences are valid modulo smooth functions.
a(x,9)
2)
+
+
(c)] (43)
a(x.9)
0) +
—
i
(44)
(ni)
a(x.0)
+
—
(712)] (45)
Let us also apply the results of Lemmas 5 and 6 to the study of the singularity against the parity of m type of the integrals of the form (i.e.. of the number of the variables 0). We assume that the support suppa,(x, 8) possesses a nontrivial intersection with the set C, defined in (10) (otherwise, by Lemma 2. the integral defines a function of the class Cx). By Condition 1, in any point of C,, one of the derivatives is different from zero. Using a partition of unity and renumbering the variables if necessary, we may assume that the inequality (46)
holds on suppa1(x,O'). By the Euler identity for homogeneous functions and by the equality (t)(x, 1, the equations of C, may be rewritten in the form
9') = ... =
C, = ((x. 0')141(x, 0') =
9') = 0}.
8') = (47)
We conclude that the equation D1(x,6') = 0 possesses a smooth solution x' = 61') (here x' = (x2 ,...,x")) in a neighbourhood of C,, and D1(x, 0') may be represented in the form I D1(x,0)=X(x,9)(x I
where x(x.O') integral
f
I
(48)
0 on suppa1(x.9'). We can now use Lemma 3 to rewrite the
under study in the form
=
—q1(x',0'),b1(x,8')J,
(49)
where h1(x. 8') depends on ai(x, 0') and x(x. 0') and may be evaluated by means of the formulas given in Lemma 3. We shall study the simplest case when the inequality det
0'))
0
(50)
1. Maslov's canonical operator
theory)
holds on suppbi (x, 9'). By the Morse lemma, (50) implies that there exists a smooth change of variables 9' = 9'(x', n'). = (x', 6') 71m) reducing to the simple form
p1(x'.6') =s(x') — with the function s(x') being given by
=
(x'. 8'(x')),
(52)
where 9'(x') is the smooth solution of the simultaneous equations
j=
1,2,...,m,
existing due to (50) in a neighbourhood of C,. By means of Lemma 4. we represent the integral
(53)
a11 in the form "I
= where b1(x,
7)')].
—s(x') —
may be obtained from b1(x,
(54)
via the formulas given in Lemma
4. Let us expand b1 (x. 17') into the Taylor series in powers of 77' and apply the gradient ideal lemma (Lemma 1). We obtain (55) —
s(x') +
b,(x) fl
modulo arbitrarily smooth functions (this follows from Proposition I).
Note that bo(x) is nothing other than the restriction of b1 (x, 17') to the set C,, = whose equations are = ... = = 0 by virtue of (47) and (51). Now let us apply Lemma 6 to the right-hand side of rejation (55). One should consider the cases of m even and odd separately. For m being even, we get the following result:
i'
—s(x'),b1(x)],
(56)
1=0
modulo arbitrarily smooth functions. Here 1 is the number of minus signs in
expansion (51), which coincides by virtue of (48) and (51) with the negative index
of inertia of the matrix HessA4l (x. 9'). The expressions on the right-hand side of (56) do not contain integrals by 0' any more (m = 0 here), and therefore by
11. Founer—Maslov operators
102
(16)—(19) we have k—j+m/2
I
— s(x ), 8(k—j+m/2) (x' — s(x'))
?0,(a+J)mod2=O; —k
—
2
—
,
sgn (x' —
k—J + m
(k—J + —)!V.p. 2
(—+j k
(x'
s(x')) (x' —
—
(x' — —
2
—
k
in Ixt — s(x')I b(x),
,
1.
Suppose now that m is odd. Successive application of Lemma 6 allows us to cancel out all but one square. Further, the procedure described in the second part of the proof of Lemma 6 enables us to extend formulas (31 )—(34) to arbitrary rather than only positive values of k + m (note that in this case, + m + should be
considered as the value given by the analytic continuation of the function r'(cr) into the plane C with negative integers deleted). Using (31)—(34) to get rid of the last square, we obtain (56), the functions being defined for half-integer values of k by the equalities a1] —
(2 ,r )m/2
m
(x,
+
m + a odd. For lack of a natural interpretation of the integrals (58) in the R.-equivariant framework, we avoid using them (as well as formulas (31 )—(34)) except for the cases of extreme necessity. Let us also present the results of the evaluation of the principal term in expansion (56). Evidently, one may assume that > 0, then x(x, 0') > 0 on suppai(x. 6'). In this case, the expressions for the amplitudes given in Lemmas 3
I. Maslov's canonical operator
theory)
and 4 lead to the formula 9')a1(x, 0')
—
X) —
9'=9(x)
where O'(x) is a solution of the simultaneous equations (53), that is, of the equations m. By substituting the latter expression into the first 0') = 0, i = 1,2
summand in (56), we put it in the form
(x, O'(x)),
(59)
Hess:'(—4)i(x, 0'))
with the argument of the expression under the root sign being given by
arg Hesso'(—4)I(x,
argAk,
O'(X) =
where Ak are the eigenvalues of the matrix —
j
A
Ak
i.),
O'=8'(x)
The expressions (59) together with formula (56) give the stationary phase for-
mula for integrals of the form F,fl4), a].
1.4 Filtration connected with the Lagrangian manifold Let 4)(x, 9) be a phase function which satisfies Condition I of this section (see Subsection 1.1). We suppose that 4)(x, 0) also satisfies the following condition:
condition 2. The differentials
d4)9,, are linearly independent on the set
C, fl suppa(x. 0). Here the set C0 was defined by relation (10) and by relation (1.6.6) Chapter I, Section 6. Note that Condition 2 is essentially a transversality condition of Section 1.6. Obviously, this condition is equivalent (see relation (1.6.7)) to the equality 324)
rank
dx
324)
= m + I.
II. Fourier—Maslov operalors
104
Let us formulate condition (60) in terms of the function 0,,,).
To do so. let us fix
= (l>(x,
= I and multiply the rows of the matrix dx"800
dOodOo
a2ct
except for the first one, each by the corresponding 9, and add the results to the first row. This procedure gives the matrix
a2i
a2i
ax' ao,
dx" 80,
...
0
dx"
0
800 ao,
dOm 80,
dOo 86m
dOmdOm
d24 dx' aom
80,,,
having the same rank. By multiplying the columns of its submatrix except for the first one, each by Oj and by adding the results to the first column of this submatrix, we obtain the matrix dx
0
0
dx 80' being of the same rank. Omitting the zero column here, we obtain the relation
rank
86'
ax a2
dx dO'
= m + I,
80'dO'
valid at the points of the set
= ((x,
6') =
6')
6') = 0}.
(62)
Note that due to the Euler equality, the set (62) is nothing other than the intersection
of the set C,1 with the plane 00 =
1.
1. Maslov's canonical operator
theory)
105
As was shown in Section 1.6, Condition 2 implies that the set C4, (as well x as the set K, given by formula (62)) is a submanifold of the space (respectively,
x
We recall that the restriction of the map
a, :
x
x
—÷
/
(63)
a,(x,O) = Ix, —(x.8)
\
8x
(see relation (1.6.8)) to the submanifold C, is a Lagrangian embedding homogeneous with respect to the action of the group We denote a(C,) by x Let I (4) be the Legendre manifold in the space = x P, corresponding to L(c1). For each phase function D(x, 9), we introduce a filtration lq(4) in the space in the following way. We denote by the set of functions f(x) which may be represented in the form
f(x) =
a] + f(x)
(64)
for some amplitude function a (x, 9) and the smooth function 1(x), where q = It follows from the classification lemma, the gradient ideal lemma, and the stabilization lemma that one may assume that m n — I modulo Jq'(4) for q' large enough. Proposition 1 now gives the inclusion
Iq('l') C since k = —m/2 — q lemma,
C —
q
—
m.
for any e >
0,
(65)
Further, if q' > q, we have, by the gradient C lq((D),
so that the !q(CD) form a descending filtration which is in agreement with the filtration H3(R"). We shall show that the associated graduation Rq = which corresponds to the filtration Iq(4) depends only on the Lagrangian manifold L(4>). Namely, the following lemma is valid.
Lenuna 7. if L(4>1) = L(4>2), then = the number of 9 variables is the same for both
provided that the parity of and 4>2.
Proof Here and below, we work in the affine charts of the form tOo = 1 }. By the classification lemma, the relation L(cV1) = L(cD2) is equivalent to the relation 9' = 4>2. The latter means that there exist variable changes 0 = O(r,
II. Fourier—Maslov operators
•
such that (66) (67)
•
Let f E Iq(4'i). Then f(x) = omit the
Let
=
us apply Hadamard's lemma to bt(x, r,
Since
—m/2 (here and below, we
—q
remainders). By Lemma 4. we have
fl2r). We obtain
the latter sum belongs to J(cD), we see that by Lemma 1, we have
modulo
Using Lemma 6 and taking into account that q is not affected by the transformation described in this lemma, we see that
f(x) =
bi(x,
0)].
Proceeding again with the same argument, we obtain
f(x) =
c(x, 9')]
for some c(x, 9'), with q being the same. Thus, any equivalence class from contains some element of Jq('12). The opposite is also true, due to symmetry. The lemma is proved. 0
1.5 The local canonical operator Let L be a homogeneous Lagrangian manifold, U C L be a homogeneous open set on which L is defined by a phase function cV(x, 9), U fl L = L(4). Let denote the mapping (63), as above. Let be a homogeneous measure on L of degree r. Define the function F[4, p] by the equality —
dXAdO0A...d9m
we assume in its neighbourhood in an arbitrary to be continued from F[c1', way. Thus, F[c1. jr] is defined to within an arbitrary element of function of degree r — m — 1. is an (here
I. Maslov's canonical operator
theory)
107
Let us also give the formulas for evaluating F[c1. izl via the function 4'(x, 0') = 0 on suppa(x,0)). In 4)(x, 1,0') (i.e., in the affine chart; we assume that 90
order to do so, we define the set
K, = K, =
fl {Oo = I)
fl
in the space x = use the coordinates of the form
= ... =
= 0,
= {(x, 9')
= 0)
lOo = 1}. In the calculations, we shall m on Rm+19. i = 1.2
where 0,' =
Then
0') = (x, AG0, 9').
A(x,
be the radial vector field: in the coordinates introduced above, we have
Let
= 0o
Now we have
a form of maximal degree on C,. Here the fi are coordinates
on K,; since K, =
we may consider (Go. as the coordinates on the condition 00 = 1. Next, since a is homogeneous, we have C
i'(fl) dfl
A
= where
a1 =
We
=
dA
dA
dA
get the equality
F(cD. d19 A
— dO0 A
dx
A .. .
A
A do0 A do1 A
—
I—
A dOn,
I
. (—I)"
Taking into account the Euler identity, we have
_... on
the set tOo = I) fl C,. Finally, we obtain
F[D,
a* (Lj ,e) JA 1
A dcD1 A d4'10 A I
. .
(mod
dXAdO1A•••AdOm The right-hand side of the latter formula will be denoted by F[41, of homogeneit5' will always be clear from the context.
(69)
the degree
II. Fourier—Maslov operators
108
In what follows, we do not distinguish between C, and K,. Note that F(4), does not vanish on U. Thus, the square root 4[F[4'. may be defined correctly (we assume U/R. to be simply connected), though its definition is ambiguous. Since is not connected, U divides into two disjoint components. Therefore, there exist four possible ways of fixing the definition of the square root, two of them giving an element of the other two giving an element of be an integer). We denote any of the (we have required that former by (F[4). and any of the latter by (F[4), jtJ} Choose and fix a branch of the square root for each a = 0, 1. Now let be a homogeneous function of degree k on L, çt (L). Set = 2_m12 F
Definition 2. The operator type a in the chart (U, 4)).
[1,
(70)
(70) is called a local Maslov canonical operator of
It is evident that the inclusion E I—L—fr—I)/2(4))
holds.
Remark 1. There is a possibility also to consider the amplitude functions q which For such functions, we are homogeneous on L of the type a = I p modify definition (70) of the local canonical operator in the following way: = 2_m12
[4).
Since the theory for such amplitude functions is quite analogous to that for homogeneous functions of the type a = 0. we consider below only the latter case.
1.6 Globalization We introduce new notations which from now on appear to be convenient. We write
Thus, s indicates either the number of derivatives of the 8-function or the degree of 4) in the denominator of the integrand diminished by I. Next, we shall widely use the affine chart, assuming everywhere that a renumeration was made, so that 0 in the neighbourhood under study. be a locally finite covering of L with the neighbourhoods of the Let (Un, Thus, a type described above. We fix the choice of the square root in each chart is defined on the covering tua}. Here canonical cochain E we shall study the conditions for this cochain being a cocycle modulo the space I—A—(r—h2+I(4)).
I. Maslov's canonical operator
theory)
109
from onc may by affected by the chart to another (see the proof of Lemma 7). Evidently, the type of the resulting canonical operator depends on the total number of "—" signs in equalities (66) and Thus, we have we denote this number by (67). With the intersection t1a fl to require that the following relation be valid: First of all, note that the rypc of
(72)
One can see from the proof of the classification lemma that .1 •
•
ind_
—
(mod 2)
ind_
We divide the comparison procedure for and e'(r, Namely, let 9(r.
and
dcl
=
(73)
(p) into three steps. be variable changes such that (74)
'I'a(x,0(T,uli
a1]. for a1 at the moment
with
(1) Compare
> 0 (otherwise, we being unknown. Evidently, one may assume that if there are no variables at all, we simply make the variable change —+ introduce two additional variables By Lemma 4, we have
+
see
item (2) below).
= On the other hand, we have
t, {
j2
= F[4'1,
D(r,
since
D(r,
ii)
and
D(r, on the set
Therefore,
D(r,
=
/
II. Fourier—Maslov operators
110
=
and besides, arg
arg F[cD1,
ii]. Finally, we obtain
=
p{F[
(75)
= argF[41,j.t] at this stage. ,iJ},V2} and
(2) Now let us compare
a21 with some
a2(x. r. a'). Note that by Lemma I we may assume, as in the proof of Lemma 7, that aI (x, r, does not depend on By successive application of Lemma 6, we obtain j)J/2]
F
=
/a9a9—ind.
a
ço(F[c11,
JL]}I/2]
where we have taken into account equality (73). On the other hand, by virtue of (69), we have
F[41,
=
(—1)"
(*' d4
A
A
A
A
A
—
dxAdr (Indeed, the number of minuses coincides with the negative inertia index of the see the proof of the classification lemma.) Proceeding in the opposite order, we obtain Hessian
F[4)1, /tIIq=O =
,.t]I,1.=o,
.
{F[41. ,z]).i/2
= = =
F(4,
2(mQ—mfi)/2
(F[42,
I. Maslov's canonical operator
III
theory)
Thus we obtain co(FI6D1,
(76) da,g
=
(3) The comparison of similar to case (I). Finally, we get
a3] is quite
with
=
ct(F142.
(77)
Taking into account (75)—(77), we obtain the following statement.
Proposition 2. With condition (73) being valid, the congruence
=
(78)
holds, where [arg F[4a, s] — arg
d1p =
+
i
I ao ao
— ind_
ILl]
(79)
d21p 1 ae' ao']
is a one-dimensional cocycle of the covering j IJJ with coefficients in Z2.
Hence, the existence of the canonical cocycle is guaranteed once we require triviality of the cohomology classes c, d E H' (I. Z2) defined by the cocycles (72) and (79). We say that the manifold L is quantized if c = d = 0. Thus, we have proved the following theorem. Theorem 1 (on cocyclicity). Let L be a quantized manifold. There exists a choice of such that the operators (70) coincide the types (a0 } and the arguments arg F[40,, modulo '—k—(r—I)/2+1
DefinitIon 3. The operator !_k_fr_I)/2(L)//_k_fr_I)/24., (L),
defined in each chart on L.
(80)
by relation (70) is called the Maslov canonical operator
his easy to see that there exist exactly two ways to choose a0 satisfying (72). Therefore, there are exactly two types of Maslov canonical operators on 1... Remark 2. By using a partition of unity, one may represent function on R'1.
(ço)
by a global
11. Fourier—Maslov operators
Remark 3. With the choice of the types
and of the values of JFID, /LJ sat-
isfying the quantization conditions being fixed, one may apply the operator to functions that are homogeneous of degree 1 on L as well; the type of the result to a homogeneous will be opposite to that of the result of the application of function of degree 0. To finish the topic, let us also mention the following proposition.
Proposition 3. For any manifold L, the cocvcle c is a zem cocvc!e. Proof The formula
=
2dafl
is valid, where Therefore,
ji]
—
=
arg
— arg
is
ji])
—(fe
—
ffi) (mod 2)
an integer cochain of the canonical covering.
= so the cocycle Ca/i represents a zero cohomology class.
2. Fourier—Maslov integral operators 2.1 Main definitions; the composition theorem Definition 1. An integral operator with a canonically represented kernel,
(x, Y) f(y)dy,
=
x where L C is a homogeneous Lagrangian manifold, and are a homogeneous measure and function on L. respectively, is called a Fourier—Maslov integral operator on L. Fourier—Maslov integral operators (FlO) form too large a set (in particular, this set includes boundary and coboundary operators); we limit ourselves to the consideration of a certain subclass of this set. Namely. let
T'(R)
g: be a homogeneous canonical transformation, isomorphic to via the injection
(y,q)
= graph g. It is evident that Lg
(y,q:g(y,q)).
2. Fourier—Maslov integral operators
We choose a natural measure ji on L, setting
=
A
dqY");
= 2n. We denote by the FlO corresponding to these objects; here identified with L? by = p(y, q) is a homogeneous function of degree h on the foregoing injection. The set of FIOs with homogeneous amplitudes of degree k will be denoted by Opk. It is easy to see that the kernel K(x, y) of the operator E OPA belongs to the space
K(x, y)
y])
E l—k—(n—I)/2 C
for any e > 0. This implies the following draft estimates for the operator T (q):
k+n+e—
—*
: H I/2(*+n—
T' Indeed,
([y]) —+ H
—
J/2Uc+n— I/2+e
([xj),
> 0;
otherwise.
one has
(1 + p2Y(l +q)'
s>0:
C(t+p +q), 2
<
2
s<0.
Now let K(x, y) E
Q(x)
x
E
We have
f q)(l + p2 + q2)ffi/2
f
=
I*(p) (1 +
I
(1 +
q2
)_m/2
(1
+
p2
(1
(I + q2)h12] (2,r)2"
+
x
supremum is finite for a = if m > 0, and a = —m if m <0. Thus, the operator k with kernel K(x, y) is Continuous in the spaces
The
K
:
H_mi2 -÷
K:Hm—÷Hm, In the sequel,
we shall give more
m>
0;
m<0.
precise estimates.
II. Founer—Maslov operators
114
Let
be
canonical transformations. Let also E
homogeneous functions. Consider the composition To evaluate this composition, it is sufficient to consider the composition of the corresponding local operators. We have locally he
Ik
øp
1
2
=
f
P
j
2
1(z) dOd z.
y,
/
=
f J where
f(y)dO dv,
V.p.
and 42 are the determining functions for LR and Lg.. respectively.
Here we consider the most different combinations of the types and assume that k -' > 0 (one may always prove this by the stabilization lemma). Recall that + n + m, — I is assumed to be even (this is a restriction on m,). Let us write down the composition (ki T°2(
)oT'1'(
f x V.p.
I
).
.' —
+
n+mI_I)! 2
(k2
+
fl+fl12—I)!jfl 2
(2,r)(mu+m2)/24fl2
[cDi(z y, O)JkI+(n+m-I)/2+I
f(z) dO dO'dy dz.
Note that if d,. 'D, and Q2 are independent on the support of the integrand, then the integral over dv involved in (2) is in (indeed, the singular supports of the two v.p.-type functions intersect transversally). Therefore, one may assume that and Q2 are proportional to each other at some point of the support. Using
a partition of unity, we can make the supports small enough. It follows that we may only consider the case 0, 0. Then, by Lemma 1.3, we may
2. Fourier—Maslov integral operators
assume that
4'i(z,y,O)=y, —i/i,(z,y',O), 'D2(z, Y' 9') = Yi where
y' =
x, 9'),
—
(y2, . .. , y").
We make use of the following lemma.
Lemma 1. If S1(x'), x1 =
xlt),
(x2
i
=
1,
2 are functions with continuous
derivatives, then
i:v.p. [x' — Si(xl)Jk k—I
=
[x' — S2(xf)]m
2
8(k+m_2)(S2(X1)
(m
dx,
—l)!(k— I)!
—
Proof Let us evaluate first 1
1
_Si(xF)±iO]k
[x'
[x'
_S2(X,)±jO]mdX
_S,(xI)±ielk
[x' _S2(X1)±i€]mdX
= tim 4. The integrand of 4 may be continued analytically in x1 into the complex plane. The pole singularities occur in the points Si(x') ± ie, S2(x') ± ie. If both signs coincide, then 4 vanishes (One may close the integration contour). Therefore, it is sufficient to consider the case
6
=
[x' —
S,(x') +
— S2(x') — iE]m
[x'
dx'
Res
x'=S2(x')+ie [x 1 — S, (x')
+ ieJk
I
=
(m —
[xl — 1
I)!
[x' — S,(x')
+
—2)! (m — l)!(k
Passing to the limit as e too
—
—
[S2(x') — S,(x') +
+0, we obtain the equality
1
[x'
1)!
1
1
dx [x' — S2(x') — iOIm —2)!2iri (k — 1)!(m — I)! [S2(x') — S,(x') +
S,(x') +iO]"
—
IL Fourier—Maslov operators
By the Sokhotskii—Pleumel formulas, we have
V.p.
I
V.p.
[x' — Si(xt)]k
[x' — S2(x')]m
I
=
I
+
[ix'
dxi
Ext _Si(Xf)_iOlk
r
xl
Idx'.
[x' —
Thus, the integral is either 0 or given by
S2(x') — iOJmJ
(5) depending
on the ± signs, so we
obtain I
V.p.
I
V.p.
[x' — S1(x')]"
I
ixt — S2(x')]m
dx1
i
—4
(k — I)!(m — 1)! (_l)k_i(k+m_2)!2lri
+ (k —
l)!(m
—
1)!
[S2(x') — Si(x')
+ iOlk+m_i
1
[S1(x') —
S2(x')
—
4(k—l)!(m—l)!
— X
I
I
I
11S2(x') — S1(x')+
jØ}k+ifl_I
— 1S2(X') — S1(x')
— jØ]k+Sfl_i
Using (1.13), we finally get (4). The lemma is proved.
Let us rewrite the integral (2) by taking into account the equalities (3). We obtain (k
(k
2
(2,r)(mr4rn2)/2
I
[y' — *i(z, y', 9)]ki+(n+mi_i)/2+i 112)
x V.p. —
(y.
x, 8')
f(z) dO dO dydz.
Let us expand y', 8)) .,/F(4'2, 112) in powers of (y' — 111) and x, 8')), respectively. Higher-order terms may be neglected, since and (v' — they give more smooth functions. In the principal term, we use Lemma 1 to evaluate
2. Fourier—Maslov integral operators
the integral over Yl:
x
j)Li
(
o
—1)12
= 2. +m.)/2—
f
X, 0')
—
*i (z. y'. 0))
R"I
(z. y', 0), y', 0)
x
x, 0'), y', x, 6')
jL2)
x f(z)dOdO'dy'dz. x, 0') — Consider first the phase function (z. y', 9) of the integral = (6). Since y' is included in "type 9 variables," the equations of C, read 8*1
8*2
*2—*I=O,
8*2
Consider the diagram
C,
C,1
C,,
at! g
gj
x
x
I x
where all the arrows are embeddings, with the mapping C, by the formula (x, z. y', 6,9') '-+ (y, z,6)
and the mapping C,
C, being given
= (*i (z, y', 0), y', z. 9),
C,. being given by
(x, z, V. 0,8') i-÷ (x, y. 6') = (x, *2(y, x, 0'), y', 0'). Recall that the mapping C,, —+ (y, while the mapping C,, -÷
z,
6) x
x (z.
is given by
r) =
is given by
I The commutativity of the lower left triangle follows from the equality a(C,,) = graph The lower right triangle may be considered in a similar way. Here the
U. Fourier—Maslov operators
formulas for the mappings are
f (x,p)= Formulas (7)—(lO) also provide the commutativity of the upper square of the dia-
gram.
The composed mappings
are evidently given by the formulas
(x,z.y',O,O')i-÷ (z.r) =
=
(x, z. y', 6,8') i-+ (x, p) =
=
These formulas evidently define the canonical mapping a for the Lagrangian manHenceforth, the commutativity of the outer triangle proves that CD is ifold
° gi). a defining function for the Lagrangian manifold = graph Now let us consider the amplitude. The fraction F[CD, I is a density of the form
Adz))
—
A
A (—d*io)
with respect to the form dx A dz A dy' A dO A do'. Since A dz)) A (—d*18) is a form of degree 2n — 1 + m1 depending
only on the variables (z. y'. 0), the second terms in the differences d (*2 — — will not contribute to the exterior product. Thus, the function d FIcD, lic, is equal to (_I)mI
A dz)) A di/i2 A
dx AdzAdy'AdOAdO'
A *29' A dlfrIN] C.
2. Fourier—Maslov integral operators
Furthermore,
A dz)) A d(y' — *i(x. y' 0)) A (—d*16)
dvAdzAd9
L
— (_1)mu+n_1
and
dy'AdzAdO
[
—
C.
besides difr1 cancels out via the same argument, whereas dy' may be removed
by the relation y' = *,(z. y'. 9). Similarly, we have A dy)) A
F[4'2,,t2]Ic.=
d(y'
0')) A
dxAdyAdO'
L
A dy)) A
= (_1)flt2+fl
x,
—
A
C,
... A 2
I
dxAdy'Ad9'
[
C,
Taking into consideration the relation
A
A
A dy'.
we obtain from the latter equalities,
F[c!), Since m, + n —
1
=
FEC!),, 1Li1
F[4)2, 1L2].
is even, we have
F[4, IL,] =
F[(D,,
z1J . F[1)2,
Note that due to the commutativity of the diagram, the amplitude
of the HO
x obtained is equal to Thus, the following theorem in proved.
Theorem I (Composition). One has (97j
mod OPki-i-1c1_l,
where the sign depends on the choice of the arguments of measures in the definitions
of the operator involved.
Let us now consider the adjoint operator to the operator (cc). Since any HO T' can be represented as a sum of local Fourier—Maslov integral operators, it is sufficient to consider the adjoint operator to the local HO. We can suppose (as in
120
IL
Fourier—Maslov operators
> 0. We consider here only the proof of the composition theorem) that k + the 'V.p."-type of the local operator; the '8"-type is considered quite analogously.
Thus, we consider the local operator (k + n+m_I)tjI7—I
=
I fR'"'
V.p.
f(y)dO dy.
x,
To calculate the expression of the adjoint operator, we consider the scalar product
(k+ = x
f
f(y)g(x)dOdydx
V.p.
= (1. ii
g),
where (k
+ fl+?fl.I)ijn_I
X
f
= J
V.p.
g(x)d8 dx.
x,
As one can see from expression (II), the roles of the variables x and y are changed in the adjoint operator. Thus, it is evident that the canonical transformation correThe measure and the function are induced, sponding to operator (II) is
by convention, from the domain of the definition of the corresponding canonical transform. Hence, the measure is (due to symplecticity of the transform g) a and the amplitude function of operator (II) is natural measure on the graph Let us formulate the obtained result as the following theorem.
Theorem 2. The equality
= holds.
2.2 Pseudodifferentlal
operators
DefinItion 2. The pseudodifferential operator (PDO) is an FlO corresponding to the identical canonical transformation. In case of PDOs, there exists a global defining function 4)
= p(x
—
y),
2. Fourier—Maslov integral operators
where p plays the role of the "a" variables. for PDOs. We have Let us evaluate FID,
a(x,y,p)=(x,y,p,q)=(x,y,p,p);
C,=(x =y}:
thus,
F[4
dx A dy A dpi A••• A dp,,
—
dpAdyAd(x—y) dx A dy A dp
—1
p] would equal (—Lv'). The amplitude P(x, p) of the corresponding HO will be called the symbol of the PDO. The PDO itself will be denoted by Pa (x. —i The following corollaries are valid. (with a —1,
Corollary 1.
The
congruence o
—i
=
(P(x, p)
)
holds.
Corollary 2. The operator
•\
g
is
a PDO with the principal symbol g'(P) (modulo lower-order terms).
Remark 1. By evaluating the "radial part" of the integral defining a usual PDO. one can verify that the class of our PDOs is a subclass of conventional "classical" PDOs. To conclude this subsection, we present a more detailed result for the composition of two PDOs.
Proposition 1. Let Pa1 (x, —i
and be PDOs of order m and k, (x, —i respectively. Then the composition of these two operators is the sum of pseudodj/. ferential operators of the type a1 + (mod 2) with the symbols
V al—i
.a!
P(x, p) ö' Q(x, p) apa
j=O,l
N,
up to operators of the order m + k — N.
The proof of this proposition can be carried out analogously to that of Theorem 1. One has to take into account that for pseudodifferential operators. there exists a global determining function of the corresponding Lagrangian manifold. We leave this proof to the reader.
II. Founer—Maslov operators
122
2.3 ActIon of the FIOs on the canonical distributions
One may prove the following theorem, following the lines of the proof of Theorem
Theorem 3. The formula (SOIL
holds where s is the order of the FlO T" the measure
it'),
(mod J—*—r—I)/2--s)
and r is the order of
of
In particular, for PDOs we have
("IL
—i
Thus, the following statements are valid:
Corollary 3. A PDO
(x, —i
of orders has the orders in the scale Iq(L)
as well.
Corollary 4. If the principal symbol of the P00 PY (x, —i
of orders vanishes has the orders — I in the scale Iq(L).
on L. then P0 (x. —i
In the situation of Corollary 4, the statement of the last theorem can be improved.
Theorem 4. If the principal symbol of the operator P0 (x. —i
vanishes on L, and the measure is invariant with respect to the Hamiltonian vector field V(H), then the commutation formula holds: H01
(Pgo),
—i
(mod
where
Remark 2. Formula (IS) may be extended to a congruence modulo arbitrarily smooth functions.
The proof consists of two stages. First let us study the action of the PDO on the functions of the form V.p.
or
where dS(x) 0. The result depends on the type of PDO. We consider one of the cases, the other being quite similar.
2. Fourier—Maslov integral operators
Let the PDO H (x, —i
Iii =
H(
be
123
given by the formula
x,—i—)f(x) EJxj
— (— —
J
—
y) p) H(x. p) f(y)w(p) A dy
(we assume that H(x, p) is homogeneous in p of degree k, type 0 for n odd, type I for n even). Consider the action of operator (17) on the first of the functions (16). First = S(x) = — *(x'). We have consider the case
\
m!ço(x')
(VP
Ix' —
f
m!
*(xl)Jm+I) = x Vp
Iy'
8(fl+k_I)(p. (x — y)) H(x, p)
w(p)Ady. —
If p is not proportional to
the latter integral is in clear that one may assume 0 on supp H(x, p). Using the coordinates in which P1 = I, we have
\ [x'
in!
—
8
—
f (x'
+ p'(x'
—
)fl+LI
)<
V.p.
(—l)"'
—
x f H(x, 1, p')
— Y' + p'(x'
9,(y,) dp' A dy — *(yP)Im+I
x H(x, I, p') V.p. —
f
=
—
as above. It is therefore
ço(y')
ly' —
—
y')) H(x, I, p') dp' A dy
(ni+n+k— I)! V.p.
[x' + p'(x' — y') —
dp'Ady'.
—
y'))
_________
II. Founer.-Maslov operators
124
Let us expand H(x, 1, p') in powers of p' —
\
m!
k (V.p.
—
(—
—
)
av'
(x.l.— xV.p.
(m +n +k —
*(xl)Jm+I) ==
xJ{"(
We have
/
+
(
Lx
+
)
+...}
)
dp' A dy'
—
(_1)fl+k_J (m + n + k — 1)!
xV.p.
(
8y'
— ,
I
Ix + p'(x' — y')
=
,
1,
1)!
JH
1
—
a' )
dp' A dy'
+ p'(x' — y') —
(—I)"' ay' —
H,,,, ( x,l,—
xV.p.
, I
ay' (p(y')
Ix + p1(x' — y')
(
+
)}
ay'
dp'
A
—
(P' +
\ ay'
)
dy' = !i +
In the first integral on the right-hand side of (18), we may replace y' by x' in H and (simply expand in (y' — x') and integrate by parts, noticing that only the principal term survives, with all the functions except the phase function being independent of p'). Making the variable change
x'
— y'
+ p' + F(x', y') (x' — y') 2
v' = —x' + y' + p' + F(x', y') (x'
—
y'),
where F is defined by dp' A dy'I I
=
du'
A
dv',
we obtain
=
(m +n +k —1)! x
fH(
I
I .
+ (u)'2— (v)2 —
2. Fourier—Mas)ov integral operators
Lemma 1.6 gives
H
+k)!V.p.
li
1,
Lx'
—
for n odd, and
=
— *(x')) H
ax'
)
for n even. In the second integral on the right-hand side of (18), we integrate by parts:
(m+n+k—2)!
-
x
——
x
H,,,, (
12
—
a*(v')) x,l,— 8y'
+ p'(x'
and neglect the terms of lower Just as for we have
) a2*
'
ay' ay'
—
dp' A dy'
—
order with respect to singularity.
= (_1)k_l (m +k — I)! x V.p.
I H,,
(x '
i
'
a*\ i?)
= (_1)m_I
a even.
a
1
—
[x' —
for n odd, 12
a
I,
f V.p. {
(x'
—
(x '
I
'
ax'1 ax'ax
II. Fourier—Maslov operators
126
Finally, we obtain
(V.p.
m!
)=
[Xl —
(_1)k (m + k)! V. p.
H (x. i. [x'
—
—
+(_l)k(m +k)! i —
x V.p.
—
Ox
'H,,.,, (x ' i '— axe) Ox Ox j p(x')
[x' —
S(m+k)
for n odd, and an analogous formula containing
for n even. We consider
only odd ii in the following. Note that
/0 dxdx =
dx"
dx'
0
02* Ox'dx'
).
so that (19) may be written in the form
\
m!co(x') + (_J)k (m + k)!V.p. Ix' — *(x1)Jm+I ) + (_1)k (m +k — 1)! I
X V•
H,,(
i•)
r + 'H
Ox!
PP
(Note that (20) is now proved only for the case when
S=
x1
+•••
[S(X)]m+k
is independent of x1, and
—
Let us prove that (20) in valid for arbitrary S. p. First let p be arbitrary, S =
Then
x1 —
V.p.
m!ço(x) [xt —
= V.p.
m!ço(i/i(x'),x') [xt
—
+m . V.p.
(m —
[x
I)! ffi (*(x'),x') —
+
2. Fourier—Maslov integral operators
127
Applying (20) to both terms on the right-hand side, we obtain
(v.p.
\ +(_1)k(m+k)!V.p.
m!p(x') [x' —
H(x,
*(x')lm+I)
ç,(*(x'),x') [S(x)]m+k+1
(m +k —1)!
+ x V.p.
{ H ') I
a2sl
as 'i;)
a —
9,(*(x'),x')
[S(X)]m+k
+
(m
+k
—
1)!V.p.
(i/i(x'), x')
mH (x,
+
= (_1)k(m +k)!
(*(x'), x') + ffr
H (x, x V.p.
[S(x)
+ (_1)*_I (m +k — l)!V.p.
8S\
x
+
x')
H,,',,.
1
1
[S(x)]m+k
a
aS"
a2s dx' ax'
9,
(*(x'), x')
Furthermore, ço
(*(x'), x ) +
a9,
dx'
(*(x'), x')
—
(*(x'). x') +
x
dx
H,, (xx.
ço(x)
H,,.,,.
(mod ((*(x'), xP))2)
&s dx' dx'
42
x')
dx
a* +
*(x'))
dlr\ /
+
I
H,,,, (,,x,
a9,
)
dx dx
(mod ((*(x'), x'))),
x')
II. Fourier—Maslov operators
with (20) therefore being valid.
finally. to prove that (20) in valid in general. one should show that the right-hand side of this formula is not affected (modulo appropriate quantities) when S(x) is Indeed. replaced by Q(x) S(x) and by
Q
(_j)A (rn +k)!V.p. x V.p.
[Q
1)!
.
{ Hp(x,*EQS1) EQ. Qm+Iw
(_1)k (m +k)!V.p. EQ
+ (_1)k_l (m +k — l)!V.p.
/ +H,,(x,Q— ax
dQ
as)
+
ax
' -L7H,,,,
/ +H,,,, ( x. Q \
Qm+l
as
Q
dx)
ax as
—)Q aX,
ax ax —
IQ
= (_l)IC (ni +k)!V.p. I
'
as\iaQ
/
i+(m+l)HpIx•—) \
Qax
'P
a2s
dxx'P 1
j.
= (_l)L(n: +k)!V.p. H (x,
'P
+ (_j)k_1 (in +k — 1)! a2s
xV.p.
1
(P
Thus. (20) is valid for any S and ço, provided that dS following lemma.
0. We have proved the
2. Fourier—Maslov integral operators
129
Lemma 2. The formulas a
(
\
ax)
I
1V.p.
[S(x)]m+lj
= (_1)k(m+k)!V.p.
xV.p. H0
1)!
[S(x)]m+k+I
+•••;
[S(x)]m+k
0)
= 8(m+k)(S(X)) H
I
OS\
/
I
OS\ 02s
are valid for the operator k0 of order k, type 0, and the formulas
'V.p. axj 1
m!Q(x)
1
[S(x)Jm+I j
(
(_l)mS(m+k)(S(X)) H x,
OS\
—) Ox1 OQ
\
I)
/
025
= (_l)k+m(m +k)!
dx
x V.p.
1
[S(x)]m+k+I
+(_l)m+k_I(m+k_ 1)!
xV.p.
[S(X)]m+k
valid for the operator JIj of order k, type I. Here dS(x) dots denote the terms with singularities of lower order. are
+... 0 is assumed. The
Now let us go on to the proof of Theorem 4. To be definite, we consider the case OI = 0, s!(—l)3
' V.i. a(x,9)w(O)
= (2,r)m/2 J
[4(x,
a(x,9) = (it evidently suffices to consider local operators).
________________________
II. Fourier—Maslov operators
130
By Lemma 2. the congruence
f
=
dx
a(x,9)
" I
\
°"
X, — I
ax,,
w(9)
x{vp
H
a(x, 9)
1)!
(
+ Hpp (x,
a
xV.p.
a2.(x.e)]
i
ax
ax
axax
a(x, 9) j w(9) J
is valid modulo 'q 0,
forlargeq. Next, since H(x, p)j, =
and consequently, H (x,
0.
we have H (x.
(21)
=
E
I H(x,— =V'F,(x,O) OcD(x,9) dx '7'
(22)
ao1
By substituting the latter formula into (21) and by integrating by parts, we obtain
(_l)54k_I(s+k_ I)!
Ha,
xJ
84(x.9)) dx
II
8CD(x.9)\ d2(D(x,9) dx ) dxdx
— +211
—>m
dF1(x,9)l
]
i=O
Next we introduce the phase space A do. The function dp A dx +
V.p.
a(x,O)
w(9)
d4'
dx
The manifold C lies on the zero level of the Hamiltonian m
71(x,9, p.
(23)
x x x with the structure form 9) defines a Lagrangian manifold C in this
space, I
+... .
J
= H(x, p) — 1=0
2. Fourier—Maslov integral operators
(see (22); besides, L = jr (Cl)
is the natural projection. Note first that the set
= 01), where ir
x
:
x
x 9 —*
x
= 0) is invariant with respect to V(7-(). Indeed,
= {HP_. m
—
dF1(x,O)
+
3
i=O m
=
8F1(x,6k)
(24)
1=0
Since NIL = 0, V(fl) is tangent to C. Consequently, V(fl) is tangent to = 0) as well. The projection of this field onto x is evidently equal C fl
to V(H). It follows that F1(x, 6)
b(x, 0) =
hc,
—
(which we have identified with L with the help of the mapping a). Now let us take into account the equality on
The expression in curly braces on the right-hand side of (13) is equal to
(iip(x,
—
34(x.0))
—
—
Let
1/F[cljt](x,0).
(25)
= 0. We have
A ... AA ...
= m
8F1(x,0)
*
II. Fourier—Maslov operators
132
Therefore, in the coordinates (x, 6), where V(1-() = we have
A"
£V(ii)(a/1
(x,
—
F,
= £V(n){F['1, /2J dx AdO)
=
AdO. 1=0
Thus, a2cD
=
—
axax (26)
By substituting this into (25), we obtain the desired result.
2.4 Boundedness theorems for FIOs In this subsection we prove precise estimates for FIOs of the form T' (p), completing at the same time the proof of the composition theorem which was proved above only for sufficiently low orders of the amplitude functions. Namely, the following theorem is proved here:
Theorem 5. Let g:
be a homogeneous canonical transformation. Suppose also that the projection of supp onto is Let also E is continuous in the pair of spaces compact) Then the operator
T(ço): for any real
—*
s.
Proof First of all, let us gather all the necessary facts which have been proved consisting of the functions already. Denote by the subspace whose supports are taken into compacts by the projection on Denote by OPk (g), —+ the set of all operators A, satisfying the following where g condition: for any N, A can be represented in the form :
1(N)
(27)
A
j=1 I
This condition could essentially be weakened, but we do not carry out these considerations here.
2. Fourier—Maslov integral operators
where çoj
E 0cA/c(T*Rflx),
133
k,
=
-*
is a continuous operator. Clearly, there is a filtration C
C
C
C
,
(28)
and the intersection flL OPk (g) consists of smoothing operators. We know already that then A is a continuous operator in the spaces (1) If A (29)
A:
when k +
> 0, and A
II
—*
:
(30)
when k + ji <0. Here = n — + S. (1') If A E Opk(id), then A is a pseudodifferential operator, SO we know the exact estimates II
A
for any s, just as required in the statement of the theorem. (2) If A1 = Opk1(gI), A2 = E Opk,(g2), and —k1 — k2 + fL (so that the range of A1 be contained in the domain of A2), then
> (32)
A2A1
and besides, A2A1
=
.
mod 0PL1+k2—l
o
(33)
(2') If A1 or A2 are differential operators (this means, in particular, that = id then the above composition formulas are valid without any restrictions or = id), on then (3) If A = T;(p1) (34)
A5
A5 =
mod Opk_I(g').
Based on these facts, it is not difficult to prove the statement of the theorem.
(35)
0
Lemma 3. For any k. we have
=
(E Opk-l(g)
+ Opk-l(g)
(36)
II. Fourier—Maslov operators
134
+
Opk(g)
(36)
Proof By (2'), one has
= T'(ço. p1)mod Opk-,(g)
T(ço)•
(37)
for any ço E
Thus, it suffices to prove that for any of the form
there is a representation
Vi=>cojpj.
(38)
But this is evident: indeed, we may set (
LI=1 Thus, the first of the relations (36) is proved.
The second of the relations (36) is proved in a similar way; one should solve the equation
= setting —
— The lemma is proved.
Lemma 4. Let A
Op* (g).
Then
for any s, the operator A acts continuously in
the spaces A
:
-+ H$_k_n_h/'2(Rn).
(42)
Proof Successive applications of Lemma 3 yield the following result: A
,
= where
Opk-N,-,v,(g).
(f—)
(43)
2. Fourier—Maslov integral operators
135
= n — +e Choose S E (0, 1] such that 2s — k — be an integer (thus, for any s). Now choose N1, N2 to be nonnegative integers such that
n+
(44) (45)
This is always possible. Then (46)
and, by (1), we have the following sequence of Hubert spaces and Continuous operators A: Now we have
—÷ H_i_NI
H5 —
N2
—s
—
>s
—
k
11N1 -N2-s
(47)
= s — k — jt. so A is a continuous operator in the spaces Hi_k_Il.
A:
Since s — k
-i (&/Ox)0
B
—
n
—
we
(48)
obtain the statement of the lemma.
Lemma 5. The composition formulas (32)—(33) hold for any indices k1, k2 without restrictions.
Proof By Lemma 4, the composition A2A1 is always defined. Ne,t, represent A1 in the form A1 = IaI5N
where
Aia E
Opk1—N(gI),
(
with N large enough so as to calculate the products
A2A1a by
A2
=
E
=
E
A1a =
=
E
mod
H. Fourier—Maslov operators
We have
=
moI Opk1+k2-I(g2
= (by(2')} = og1).
= Tg.og,(ipi
Lemma 5 is proved.
Now let us continue with the proof of our theorem directly. Let A Opo(g). By (3') and Lemma 5, we obtain
AA E Opo(id).
(52)
that is, A'A is a pseudodifferential operator of order 0. We have, for u E L2, Vu
= (Au, Au) = (u. A*A a)
IIA*AII
Uu
(53)
Thus, II
IIAA 111/12 < 00,
A
(54)
that is,
A:L2—*L2
(55)
is a bounded operator. Now let A E Opk(g). To prove that A
:
—*
(56)
is bounded. it suffies to prove that the operator
A =(l
(57)
is bounded in L2. However, by Lemma 5, we obtain A E Opo(g), so that we can apply the previous result. The theorem is proved. 0
Remark 3. Our proof is essentially based on the fact that the mapping (g,çt) is, in some sense, a "unitary representation" of the group of homogeneous canonical diffeomorphisms (i.e., the transition to is equivalent to taking the adjoint operator. see Theorem 2). There is another type of proof which may also be generalized to FIOs, with complex phase functions not considered in this book. Briefly
3. Singularities of hyperbolic equations;examples and applications
137
speaking, one proves that any canonical diffeomorphism g may be included in a smooth homotopy g,, I E [0, 1], with = g, go = id or the inversion operator,
depending on the orientation of g. The corresponding operator family A, = T turns out to satisfy the Cauchy problem of the form —i = H,A,, A0 = where H, is a certain PDO with a "good symbol." Next, one estimates A0 directly and uses standard "energy estimates" for the Cauchy problem (see, for example, [NOsSSh I]).
3. Singularities of hyperbolic equations; examples and applications In this section, we apply Maslov's canonical operator theory to three classical problems. These are propagation of discontinuity, metamorphosis of the discontinuity, and investigation of lacunas of Green's function of the Cauchy problem.
3.1 Preliminary notes First of all, let us study a very simple example. Consider a hyperbolic Cauchy problem of the form 32u — = 3t2
ui,..o = uI,=o = 0.
We shall compare the solution given by the conventional theory with that given by the theory, especially keeping track of naturality of these solutions. First, we point out that the initial data of problem (I) may be represented in the canonical form
= (_!_V \ 2,r,,
= K'"(ço),
J =
where L = ((x, p)Ix = 0), = dp, the solution u(x, t) of problem (1) in the form
Therefore, it is natural to seek
u(x, I) = for some Lagrangian manifold C in the phase space x Formula (2.14) prompts that the manifold C should lie on the zero level of the Hamiltonian function
138
11.
Fourier—Maslov operators
therefore invariant with respect to the vector field — V(p2) = Hence, the manifold £— may be constructed in the following way. — 2p x One lifts the manifold L up to the space p.E in a way such that the lifting lies in the set {t = 0} fl {E p2}, providing the initial manifold E
—
p2 (and is
= {(x, p,:, E)lx = 0,1 = 0, E2 =
p2}
for the Hamiltonian system.
£ is obtained as the phase flow of the manifold along the trajectories of the Hamiltonian system corresponding to the Hamiltonian function E — p2:
E=0, p=0,i=2E,x=—2p. Since the manifold
is the union of two connected components
= ((x, p. t, E)Ix = 0, = 0, E = With the same is true of £, £ = natural to present solution (3) of problem (I) as the sum
theory, it is therefore
the
u(x. 1) = u+(x. t) + u(x, t),
u±(x. t) =
where each term corresponds to one of the connected components of the manifold
The equation in (1) being very simple, we can give explicit expressions of all the elements of the canonical operators involved in formula (7). Namely,
= j(x, t, p. are
= ±IpI, x =
lpl
the equations of the manifolds £±; the corresponding actions are given by the
formulas
= ±:IpI. The density of the invariant measure
equals
and the amplitude functions
depend only on p. Hence, we have t)
=
f
with some functions w±(p). These functions can easily be found from the initial data of problem (I):
=
3. Singularities of hyperbolic equations;examples and applications
139
Thus, the asymptotic solution of problem (1) given by the K4.-equivanarn theory
of Maslov's canonical operator is
u(x, 1)
f
=2
e' IpI+idx dp +
I
2
dp.
the equation in (1) is an equation with constant coefficients, solution (12) is rather exact than asymptotic. Formula (12) shows that the solution u(x, 1) is represented as a sum of two functions corresponding to the connected components of the manifold £. To examine the naturality of such a decomposition, consider the case n = I. We Since
have
= 1) =
{S_(x + t) +
—
where I?
I
=
(i;) f
L(x)
1
=
j
and
u(x,t) = We see that solution (15) is concentrated on the "light cone" x = ±p, while its components u±(x, t) are supported outside the cone as well. A similar observation is valid for odd values of n. Thus, the decomposition of solution (15) into the sum of the functions (13) appears to be rather artificial. This is not surprising, since is not natural. In fact, the principal even the initial decomposition £ = £÷ U symbol E2 — p2 of the operator (1) is a homogeneous function with respect to the and the connected components of the initial manifold are not group invariant with respect to the action of this group. As we have seen above, the theory of Maslov's canonical operator includes another decomposition of the function (q) (where L is an manifold).
This decomposition is induced by the decomposition of the amplitude function (which is allowed to be only into the sum + of its even and odd parts. The decomposition = in this case is + the decomposition into the sum of distributions of different kind; for example, the solutions of the Cauchy problem (1) with different initial data can be of the type S(S(x, t)) or V.p. This fact allows us to investigate such problems as the metamorphosis of the discontinuity and lacunas of Green's function of the Cauchy problems for hyperbolic equations.
II. Fourier—Maslov operators
3.2 Propagation of discontinuity of solutions of hyperbolic equations We shall consider the Cauchy problem
11 •a\m
L(' i)
I
a\1
a
= fi(x)...., I
= fo(x).
= fm_i(x)
U
for a hyperbolic pseudodifferential operator
av" a a /I—i—I / —A lx,t,—i—.—i—
\
81/
ax
8t
written in normal form.
We suppose that the operator A is differential with respect tot and includes only the derivatives (—i up to the order j = in — I: A
j
with the operators being pseudodifferential operators of order m — and of the type a = 0. We denote by p), k = 0, 1. ... , in — the homogeneous
j
components of the symbols of the operators these functions are supposed to be smooth on the space x = ;. = l}. We suppose that the Cauchy data of problem (16) can be canonically represented. that is,
= for some homogeneous Lagrangian manifold L C with the homogeneous measure of order r and for some homogeneous functions çoj of order k + j and
type a. We denote by Ak(X,
t, p. E) =
Ak_J(X, t, p) j=t)
the homogeneous components of the symbol A(x, 1, p, E) of the operator A (x. t. —ii, —ii); here Ao(x, t, p) 0. The hyperbolicity of the operator (17) means that all solutions E = E,(x.t. p) of the equation Em — Am(X,t,
p, E)
0
3. Singularities of hyperbolic equaiions;examples and applications
real and distinct. It is evident that the functions E,(x, 1, p), I = I, 2 m are of order I; we shall now show how to examine the symmetricity properties of these functions with respect to the action of the group First of all, we suppose that for each (x. t. p), the functions E,(x, t, p) are enumerated in ascending order, are
E1(x.t,p) < E2(x,:,p) <"' < Em(X,t,p). In this case, the number of the function E1(x, t, p) is the locally-constant (and x R, x therefore constant) function on the covering {E = E,(x. 1, p)} —* It follows that the covering decomposes into m disjoint sheets. The mapping (p. E) u—t (—p. —E), which preserves the equation (21), inverts the numbers of functions E,(x,t, p). Therefore, for even m, these functions form pairs (E1, Er,) (Em12, Em12+i), invariant with respect to the action of the group For odd values of m, there exist pairs (E1, Em) the "middle" root which is Re-invariant itself. (E(m_t)/2. E(m-i)/2+2) and x If we denote by Char(Em — the set determined z. p. E)) C by equation (21) and by char(Em — t, p. E)) the corresponding quotient set x IL). we see that the latter set has connected components for in even values of m and connected components for odd values of m. We seek a formal asymptotic solution of problem (16) in the form u
=
(22)
where C is a homogeneous Lagrangian manifold in the space x = is a homogeneous measure on the manifold C, and is a homogeneous function on C. By substituting (22) into equation (16) and using formula (2.14), we obtain
1/
aV"
—A
I
a\1
a
t, —ii--,
-
u
.
L
(mod I—k—(r—1)/2—m), where
1-i(x, 1, p.
E) =
—
Am(X,
1, p. E).
So, in order that function (22) be an asymptotic solution of equation (16), we need to require that
In particular, the manifold C must be invariant with respect to the vector field
V(fl) (see [MiShS I]).
142
11. Fourier—Maslov operators
We choose the measure Theorem 2.4, we have
to be invariant with respect to V(7-t). Hence, due to
[(,a)rn — A
t,
_ij)]
(mod 1—k—(r..1)/2—m.-.I),
u
with P being the transport operator. (23)
Next, we require that (24)
Thus, for function (22) to be an asymptotic solution (of order 1) of equation (16), it is necessary that (I) £ be invariant with respect to the Hamiltonian vector field and flIt =
(ii)
be invariant with respect to V(fl); satisfy the transport equation (24). The requirements (i)—(iii) give us the equations for all three objects necessary to construct function (22). In order to obtain the initial data for these objects, we consider the initial data in problem (16). Let (iii)
—+
be a natural projection, ,r(x, p. E) = (x, p). Due to hyperbolicity of the operator (17), the intersection = 4, is a smooth manifold. As was shown above, the projection ir 4) —÷ L is a covering over L. The sets of this covering are in one-to-one correspondence with the solutions E,(x, I, p), I = I, 2 m of equation (21). We suppose that the intersection C fl (z = 0) coincides with This requirement gives us the initial data for the manifold £, and £ is determined uniquely. Namely, We choose £ is the phase flow of the manifold 4) along the trajectories of the measure as the (unique) invariant measure on C which coincides with the measure Ad on C0. Now we denote by the values of the restriction of the function to . . . L. Due to the choice the sheets E1 (x. 1, p),..., E1(xj, p) of the covering 4) of £ and jz, we have
=
=
+
+
+
Similarly, for the derivatives of function (22) with respect to:, we obtain
=
=
(25)
3. Singularities of hyperbolic equations;examples and applications
for any j; here we used formula (2.14) for the operator (—i
143
whose symbol is
equal to E. Therefore,
initial data
to satisfy the initial data of problem (16), we have to determine the for the function as a solution of the system of equations . . .
of)(x,t,p) (26)
[E1(x, 9,
p)
p) 1, p)
p) on the manifold L.
We note that the solution point (x, p) E L, since E1
. .
.
i,,, of system (26) exists and is unique at any
Em are
distinct, the determinant
... ...
1
1
E1
E2
em-I
cm-I
1
Em
of the system does not vanish. By substituting —p for p and using the relatio E
we obtain —
1,
—i J)) —
f, ,O
1)
.. determined by a homogeneous function of order k and of type a (the positive homogeneity of is evident). Thus, we obtain the initial data for equation (24), and hence, the function is uniquely determined. We see that the asymptotic solution of problem (16) is given by Maslov's canonical operator of the same type as that of the initial data. We note also that if the Lagrangian manifold L is nondegenerate, then the manifold £ is also nondegenerate in a sufficiently small neighbourhood (I: e(x)) of the initial surface t = 0. The latter relations mean that the function
on
.
Remark 1. By iterating the described construction, one may obtain the asymptotic solution of problem (16) up to an arbitrary order.
3.3 Metamorphosis of the discontinuIty In this subsection, we continue the investigation of problem (16). We suppose that the initial data have the discontinuity of some fixed type on the smooth manifold
II. Fourier—Maslov operators
144
with the equation
S(x) = 0
0 on S(x) = 0).
(dS(x)
(27)
and investigate what type of singularity this solution will have for various values of (1. x). First, we consider the simplest kind of singularity,
j =0.1
f1(x)
in — I.
(28)
Let us denote by X the manifold (27) and put L = N*(X). Then the functions (28) can be written in the form
j = 0,
f,(x) =
I,..., in — I
(29)
for some choice of the measure ji. We also suppose that the manifold L constructed in the previous subsection has the general position. and we denote by X the projection of this manifold on the space We also denote by EL the set £ —* of points on L at which the projection is degenerate (the cycle of singularities of C) and by E X the projection of the set E £ onto E X C X. The set E X divides the set X into connected components. Due to the consideration of the previous subsection, the asymptotic solution (22) has the form
u(x.t) = at
=
the points of the connected component of X\E X containing the intersection
X fl (i = 0) = X. On the contrary. if the point (xe, to) lies in another connected
component of X\E X. the type a of the corresponding local canonical operator is not necessarily the same. The type a of the local canonical operator in the neighbourhood of the point (xo. to) can be described as follows. Let! be a curve on the manifold £ originating from {t = 0) such that its endpoint projects into the point (x0, ta). Suppose that! transversally intersects the cycle of the Uk) be a sequence of open sets on C which singularities EL. Let {Uo. U1
covers ! and such that in any U3. the manifold £ is described by a determining 6). We assume that the charts U0 and are nonsingular (here U0 contains the originating point of 1, and contains the endpoint of!: we suppose U1 fl 0). Denote by c1 the value of the cochain c in the intersection determined by equality (1.73) with and instead of U, and cDv. We define the index of the curve! by the formula
function
md! =
c1. j=1)
follows from relation (1.72), the type a of the local canonical operator in the equals zero if md! is even and equals unity if neighbourhood of the point (xo, md! is odd. Thus, for even values of md!. the asymptotics of the solution u(x. t) As
3. Singularities of hyperbolic equations:examples and applications
in the neighbourhood of the point (xo, pa) is given by formula (30) and for odd values of md!, the solution u(x,t) has the asymptotics of the form I
In
(32)
S(x,t)
To represent the obtained result in a convenient form, we point out that the func-
tions
V.p.
and cS(z) are related via the Riemann—Hilbert operator
V.p.
=
where
Rf(z) =
p00
j
,TI
J-00
V.p.
(33)
Since 1Z2 = id, we get the formula
u(x. 1) =
(34)
(S(x, 1))
1)
which is valid for any nonsingular point (Xo, lo).
Remark 2. More exactly, on the right-hand side of formula (34), one has the sum of such expressions over all charts of the manifold C projecting into a neighbourhood of the point (xo, ta). Let us now consider the general initial conditions for problem (16)
j=0.I
rn—I.
with k(z) being a distribution with singsuppk(z) = (0) (i.e., k(z) E point except for z = 0). We represent function (35) in the form
j=0,I
C°0
rn—I.
in every
(36)
The integrals on the right-hand side of relation (36) are well-defined distributions, since the singular supports (z = S(x)} and (z = 0) of the terms in its integrand intersect each other transversally. Taking into account relation (34), we see that the
asymptotic solution U(x,t,z) of problem (16) with the initial data
f1(x) =
— z)
can be written in the form
U(x. 1. z) =
:)
8) (S(x, t) — z)
in a neighbourhood of any point (x0, to) lying in the projection of the nonsingular part C\E £ of the Lagrangian manifold £. By multiplying formula (37) by k(z) and by integrating the resulting relation by z. we obtain that the asymptotic solution
146
11.
Founer—Maslov operators
of problem (16) with the initial data (35) can be written in the form u(x. t)
=
f
U(t. x, z)k(z)dz t)
=
J
I)
8) (S(x, t) —
k) (S(x. 1))
(38)
a neighbourhood of any nonsingular point (xo, t0). Formula (38) describes the metamorphosis of the discontinuity in the general case. in
3.4 Lacunas of Green's function of the Cauchy problem In this subsection, we investigate Green's function of the Cauchy problem (16), that is, the solution G(x, y, 1) of this problem with the initial data (39) f1(x)=0, j=O m—2; with y = (yI,•••,yfl) being an arbitrary point of R'7. First, we have to present
the function 8(x — y) in terms of Maslov's canonical operator. Since 8(x — y) is a kernel of the identity pseudodifferential operator, we have 8(x
—
y) =
.
f I
—
y))w(p)
for odd values of n, and —
=
1
f
V.p. . I 2(2,riY2 JpJM-I [p(x
w(p)
—
for even n. with w(p) being the Leray form (see Section 1.2). One can easily
verify that the integrals (40) and (41) are nothing other than local representations of Maslov's canonical operator corresponding to the Lagrangian manifold L c x which is the graph of the identity canonical transformation,
L = ((x,y, p.qflx = y. p =q} C
x
=
The corresponding measure is equal to (dpAdx)", and the corresponding amplitude
function is = I. As above, the asymptotic solution of problem (16) can be constructed in the form (22). Let t be sufficiently small. For t = 0, the set of variables (x, q) forms a canonical coordinate system on the manifold L = £ fl ft = 0}. The corresponding action S(x, q) is equal to (42)
Due to hyperbolicity, we have at z = 0,
t= an 0
3. Singularities of hyperbolic equations:examples and applications
along the trajectories of the Hamiltonian system. Hence, the canonical coordinate
system in the neighbourhood of : = 0 can be chosen in the form (x, q. t). If S(x, q, t) is the action on C in this canonical chart, then the equations of £ are —
dS(x,q,t)
—
as(x,q.:)
E—
dS(x,q,t)
—
Relation (23) gives the Hamilton—Jacobi equation
/
I8SV"
as
as / as\ SI,,,o=xq —=EjIx,t,—J, a:
(43)
x the jth sheet of the covering {E = E,(x,:, p)J —* Note, that due to the considerations of Subsection 3.1, the Lagrangian manifold C decomposes into Re-invariant components for even m and into such components for odd m. Due to the relations (43), for each component, we have on
S(x,q,z)=x •q+
0(t2).
(44)
For simplicity, we suppose that for each j, the equality a2E1(x, t, rank V
dq,
q)
=—1
(45)
As was shown in Section 1.6. under such an assumption, the Lagrangian manifold £ is nonsingular for small values of i and: 0. We denote by the negative inertia index of the matrix (45). The type of singularity of the function G(x, y,:) depends on the parities of the numbers n and yj (for each Re-invariant component of the manifold C). holds.
Case I. n is odd and
is even. Using Proposition 1.2 and formulas (1.72) and (1.73), we have for the term G1(x, y,:) of the function G(x, y, 1) corresponding to the jth component of £,
Gj(x, y, t) = for
—m
:) 8(n+I)/2—m (S(x. z))
? 0, and G1(x, y, 1) = j.?(x, t) sgn(S(x, y)) [S(x. ,)Im_(s7÷3)/2
(47)
for — m <0. To derive formulas (46) and (47), we used the fact that for the initial data (39), the solution of the system of equations (26) is homogeneous of order —(m + I).
1!. Fourier—Maslov operators
148
Case 2. Both n and yj are odd. In this case, due to formulas (1.72) and (1.73), the type a of the local canonical operator is changed and we have
G(x. y, 1) = for
—
1)
. V.p.
[S(x, y)](fl+l)/2_flI+I
m > 0, and
v, 1) = j(x, t) [S(x, for
—
in IS(x, 1)1
m <0.
Case 3. n is even. Using formula (1.58), we derive for y,
where
(49)
z) =
y, 1) the expression
t) [S(x,
the ± sign depends on the parity of the number
The difference between formula (46) and formulas (47)—(50) is that the righthand side of (46) is concentrated in the projection X of the Lagrangian manifold £, whereas this is not true of formulas (47)—(50). To formalize this notion, we introduce the following definition. Definition 1. The function f(x) is said to be concentrated in the set X moduio the space if there exists a function f(x) such that f(x)—f(x) E and suppf(x) C X.
Thus, we see that Green's function is concentrated in the "light cone" X near ç = — m ? 0 modulo the space — + m + if yj is even. Note that for arbitrary values of t, we must consider the parity of the number + mdi rather than of yj.
= 0 for odd values of n and
Chapter III
Applications to differential equations
1. Equations of principal type The study of equations of principal type has a long history (see Introduction). Here we present results on microlocal. local, and semiglobal solvability for these equations, assuming that the principal symbol is real-valued.
1.1 Statement of the problem
Let k
be
a differential operator
ii
—i
with smooth coefficients aa(x) E
= In this section, we shall investigate the
solvability of the equation
Hu=f in the Sobolev spaces H5 (Rn),' under the assumption that H is an operator of principal type. We recall here the definition of an operator of principal type. Three versions of this definition are presented here: a microlocal version (in a neighbourhood of a
point of the phase space). a local version (in a neighbourhood of a point in the space Rn), and a semiglobal version (in an arbitrary compact set in the space Rn). Consequently, there are three notions of solvability: microlocal solvability, local solvability, and semiglobal solvability of equation (2). The exact definitions are as follows.
DefinItion 1. Suppose the principal symbol H(x, p) of the operator H to be a real-valued function. Equation (2) is said to be 1
For simplicity, we consider equation (2) in the space R"; the results can easily be generalized to equations on an arbitrary smooth manifold.
ill. Applications to differential equations (I) (microlocally) an equation of principal type at a point a if the contact distribution 'H (see Section 1.5) corresponding to the Hamiltonian function
H(x,p)Hm(x,p) a a char H); (locally) an equation of principal type at a point x0 R" if no complete trajectory of the contact distribution 'H is contained in the set (ii)
chars H
fl
= (a E
I
H(a)
= O}
fl
over the point xo. is the fibre of the bundle (iii) (semiglobally) an equation of principal type if for any compact set K C there does not exist a complete trajectory of the distribution 111 lying in char5 H fl with ir : —÷ W' being a natural projection. where
To formulate notions of solvability corresponding to microlocal, local, and semiglobal situation, we introduce a modification of the definition of the wave-front set of the distribution. wave-front We recall that the set of a distribution u can be defined as a maximal set WF(u) such that the intersection WF(u) fl U is empty for an
open set U C S*Rfl if for any pseudodifferential operator P supp P(a) C U, we have Pu E
UflWF(u)=ø
of
order 0 with
VP(a)fsuppPCU
can show (see, for example, M. Taylor IT I], L. Hbrmander [H 2]) that the set W F(u) can also be defined as an intersection of the characteristic sets char P of the pseudodifferential operators P of order 0 such that Pu COG: One
WF(u)
= fl (charP ord
k =0, Pu
We need a slightly more detailed notion of the wave-front set. Namely, the wavefront set of the distribution u E HS(Rfl) of order z is a maximal set WFt(U) such that
UflWF'(u)=ø
VP(a){suppPCU
P(x,j)u€H'(R")}
(7)
(the operators P are of order 0). One can easily formulate an equivalent definition of WF'(u) similar to (6). It is evident that: (a) WF'(u) = e for u Hs(Rn); t; (b) WF"(u) C WF'(u) for:' WF'(u); (c) WF(u) = so, the collection of the wave fronts W F' (u) of different orders t E R provides more detailed information for singularities of the distribution u than WF(u).
1. Equations of principal type
151
We can now give the definition of solvability of equation (2) in the microlocal, local, and semiglobal sense.
Definition 2. Equation (2) is called S*Rfl if for any s E R and any f E (i) microlocally solvable at a point a there exists a function u E H5+m_I(Rn) such that
WF'(Hu — f)flU = 0 for some neighbourhood U of the point a, and IIUIIs+m_I (ii) locally solvable at a point xO E if for any f function u Hs+m_I(Rn) such that
there exists a
Flu-f =0 in a neighbourhood U of the point x0 and flufl5÷,,1_j
there exists a finite(iii) semiglobally solvable if for any compact K C dimensional subspace N C (which does not depend on s) such that for with fJ..N, there exists a function u E such that any f E
Hu=f in a neighbourhood of K, and 11u113+,,1_i
Here the space
K) is a space of distributions u
(Re) whose supports
lie in K, suppu C K. We are now in the position to formulate the main statement of this section.
Theorem 1. If equation (2) is microlocally (respectively. locally, semiglobally) of principal type, then it is micmloca/ly (respectively, locally, globally) solvable. The proof of this theorem is based on the construction of the regularizer for the operator (1). In the next subsection, we carry out the proof under the assumption that the regularizer of the necessary type (microlocal, local, or semiglobal) is already constructed. We present the explicit constructions of such regularizers in Subsections 1.3 and 1.4 of this section.
1.2 Proof of the main theorem To prove the microlocal part of Theorem 1, it is sufficient to construct a right microlocal regularizer of operator(l), that is, an operator R : such that
where I is the identity operator and
Q:
is an operator of order 0 in the scale HS:
Hs(RIJ) -+
Ill. Applications to differential equations
such that there exists a neighbourhood U of the point E S*Rfl with the following property: A. For operator P of order 0 with supp P(x, p) C U, the co?nposition P c Q is an operator of order — I in the scale Pc
Q:
say that operator (12) is of order —l in the neighbourhood U of the space S'R" if it satisfies (13). To prove the microlocal part of Theorem I with the help of the regularizer R, we note that the function u = Rf is a microlocal solution of equation (2) up to order 1, that is, that this function satisfies relation (8) with t = s + I. Indeed, for any pseudodifferential operator P of order 0 with supp P(x, p) C U, we have We
P(H u — f} = P{H c and hence.
Rf — f) = '(1 + Qf — f} f) = 0.
= Po Qf E
—
Furthermore, once a solution u of equation (2) up to order k is given, we have
flu = i + v. where
fl U
that suppS(x. p) c U'
= 0. Let S be a pseudodifferential operator of order 0 such = 1. with U' being a neighbourhood of and S(x.
U
U. Then
Hu=f+Sv+(l—.u)=f+vj+v2. It is evident that v1 E and = 0 for any N (this follows from Proposition I in Section 2 of this chapter). Thus, if we set Ui = u — R v1, then
=
PL'i
f}
HR + PV, — P11 + Q}vi =
P{Hu1 —11 = P{Hu
—
—
P
Pv2
—
'Qv1 c
for any pseudodifferential operator with supp P(x, p) C U'. Hence, microlocal solution of equation (2) up to order k + 1.
is a
Remark 1. We can avoid using the recurrent procedure described above if there exists a regularizer RN of order N, that is, an operator RN : H5(R") such that
II 0R,.,j = 1 + QN, where
possesses the property A with (13) replaced by
Po QN:
1. Equations of principal type
for any N. We shalt not construct such a regularizer in this book. The construction of the regularizer R is carried out in Subsection 3.
We shall now carry out the proof of the semiglobal part of Theorem I. Suppose that for any compact set K C there exists a neighbourhood U of K and an operator
k:
H0fl)P(K) •-.+
(a right regularizer for operator (I) in K) such that
Q : Homp(K) —*
where the operator satisfies the following condition. B. For any function with supp C U. the operator E Q is of order —1 in the Sobolet' scale H5. In what follows we use some auxiliary functional spaces connected with the Sobolev spaces Let K C W' be a compact domain in with the smooth boundary We denote by K) the subspace of the space H5(W) containing the distributions u E on the interior part K of the compact K. One can easily see that K) is a closed subspace of H5(W). We denote the quotient space by
= Thus, the elements of for some open set U
K).
can be represented as the distributions u e K (in general, U can depend on U), and we identify the
distributions which coincide on K. As above, we denote by K) the subspacc of the space H5 (Rn) which consists of the distributions u with supp u c K:
= (u E
suppu C K}.
One can easily verify the relations [Homp(K)]* = We note that with operator (1) being a differential operator, the spaces H5 (R". K) are invariant with respect to H. More exactly, H:
H
K)
K) = (20)
Ill. Applications to differential equations
154
is well-defined. Furthermore, the space (16) is a Hubert space, since the subspace K) is closed in H3(R"). We denote by lull3 the norm in this space. Let 11 :
he an (unbounded) operator with the domain D(H) = {u E One can easily see that (21) is a closed operator. The following affirmation is the first step in the proof of the semiglobal part of Theorem 1.
Proposition 1. Suppose that there exists a right regularizer (14) for operator (1). Then there exists a finite-dimensional subspace N C (K) (which does not with JJ..N, there exists a function depend on s) such that for any f E '(K) such that Hu = f (H being operator (21)). E Proof of Proposition
1.
Let
H* :
(22)
—*
be the adjoint operator to operator (21). Due to (15), we have (23) —* is the adjoint operator to the operator '(K), which is naturally determined by operator (14), and similarly. Q* is an adjoint operator to the operator Q : —÷ and hence, Due to Condition B, it is evident that Q
where
k:
:
—*
—+
Denote by N = N(K) the kernel of operator (22). By applying relation (23) to an element u N(K), we obtain
u+Q*u=O. Relation (24) yields that any element u E
(24)
N(K) is infinitely smooth, since u
gives u = In particular, we see that N(K) does not depend on s. Further, relation (24) shows that the identity operator id : N(K) —+ N(K) can be represented as the composition
N(K) of a continious and a compact operator. Hence, the identity operator on N( K) is compact. and the space N(K) is finite-dimensional. the estimate We note that, due to relation (23), for any element v E llvll-_, < C (
+ llvll_,_,)
(25)
I. Equations of principal type
155
be a closed subspace such that the direct sum holds. Let L C The following affirmation is valid. coincides with the space
Lemma
1.
L
There exists a constant C such that for any v in L. the estimate < C
(26)
holds.
Proof of Lemma I. Suppose, on the contrary, that estimate (26) does not hold for L, I such that any C. Then there exists a sequence Since the embedding is compact. we —+ 0 as n —* oo. c the sequence (va) which converges in the space can choose a subsequence of we have Then }
IIVnL —
C(
II—s—m+I
+ converges in the space I —÷ oo, and therefore, the sequence The limit element belongs to L (since L is a closed subspace) = K) (K) is closed, = I. Since the operator H and II the relation
= urn
urn H* v,,1 = 0
yields E D(H) and 11* = 0. Hence, obtained contradiction proves the lemma.
N(K) fl L,
0. The
Continuation of the proof of Proposition I. As a consequence of Lemma I, we is closed. Due to the Banach theorem (see obtain that the space H is closed in K. Yosida tYo II, Chapter YH, Section 5), the subspace the space
and
= N(K)-'-.
(27)
Relation (27) proves the proposition.
Proof of the semiglobal part of Theorem I. Let K be an arbitrary compact set in such that K C Consider a compact set K1 with a smooth boundary Let N(K1) be, as above, the kernel of operator (22). The space
N(K) = {u E Homp(K)
I
= 0J
is, evidently, a subspace of N(K1) and hence finite-dimensional. Let u1 Ut (r be a basis in the space such that the first r functions UI k) of this basis form a basis in the space N(K). Let xl E K1\K be a point such 0. and let K2 be a compact set with a smooth boundary such that Ur+i(Xi)
Ill. Applications to differential equations
K2, K C K2 C K,. Then the space N(K2) is a proper subspace of N(K,) containing N(K), and hence, dim N(K2) < dimN(K,). Continuing this that x1
procedure, we shall construct a compact set K with a smooth boundary such that
N(K) = N(R) and K C R. 1ff
and f±N(K)
E
I. there exists an element u
then, due to Proposition
such that
flu=f
(28)
Hence, equation (28) is valid in the interior part k of the compact k, that is, in some neighbourhood of K. The semiglobal part of Theorem I is proved. To conclude the proof. we note that the local part of Theorem 1 is a particular case of its semiglobal part for K = lxo}. D in
13 Construction of the microlocal regularizer Let a S*Rfl be a point in the phase space such that H(a) = 0 and the contact distribution 'H does not vanish at this point, with H being the principal symbol (3) of operator (1). Without loss of generality, we can assume that the coordinates of the preimage of a in are (x0, p0) where 0. We denote by H1 the function H(x. p) which is determined in a neighbourhood of a. Since Pi 0 in this neighbourhood, we see that XH1 determines the contact distribution
and, hence, does not vanish. In terms of the Flamiltonian are linearly
vector field V(H,). this means that the vector fields V(H,) and p independent at any point of the neighbourhood of a.
Let g, be the local one-parametric group corresponding to the vector field V(H,). As X,, (a) <0, there exist the homogeneous neighbourhoods U, C U2 of the point a and the numbers T, < such that (i) g,(fl) is determined for any fi E U,, fri T2; (ii) g,(fi) U2 for any fi U,, Ill 1'2; T2. (iii) g,(fl) E U2\U, for any E U1. T, fri Let L denote the embedding
(U, flcharH) x (—T2, 7'2) -÷
L
L
x Rn),
can also be described in the following way. Denote by Lo the
manifold
Lo = where
C
x
a function on T*(Rn x
= T*Rfl x
fl
is the diagonal and H1 is considered as which is independent of the first group of variables.
I. Equations of principal type
Then L is a part of the phase flow of Lo along the trajectories of the vector field V(H,). Hence, due to Theorem 1.5.3, we see that L is a Lagrangian manifold with respect to the symplectic structure
w=dpAdx with (y.
q) being
—dq Ady,
the coordinates in the first factor of
x
= T*Rfl x T*Rn,
and (x. p) being the coordinates in the second factor. We shall distinguish between two cases.
Case I. H,,, Case
0. We call this case a nonsingular case.
II. He,, =
Lfl(i
=
We write
p. We call this case a singular ease.
0,
>01,
First, we suppose that H1,,
L_ =Lfl(i <0). 0. Then the set of variables (x', .. ,x",
.. ., q,,) Since, due to the Euler identity, pi Hi,,1 + forms a coordinate system on p2H, P2 + + ,,,, = 0 on L, there exists a number i0 such that H1 ,,, io> 1. For definiteness, we suppose that 0. 0, Then the variables (x' x". q3, ... form a coordinate system on L0 = H, since the variable
q,,... qn) = 0
can be expressed from the equation H,(x'
neighbourhood of the point a. Taking into account that V (H1) . = H,,,2 0, we obtain a coordinate system (x' x", y2, qi, q,,) on the manifold L near the point (a, a) e L0. We suppose that the neighbourhood U2 is small, so that (x' q,,) is a coordinate x", y2,qj,q3 in the
system on L fl (U2 x LI2). Taking into account the fact that L is a homogeneous Lagrangian manifold, we obtain (32) and
hence, the function (33)
the action on L in the canonical chart with the coordinates (x. y2, that is, is
q3
dS, = [q1dy' (34)
= pdx —p2dy2+y'dq, Thus, the function
4,
—y3q3—-"-—y"qn
(35)
III. Applications to differential equations
158
is a determining function of the manifold L in (U2 x U2) fl L (see Section 1.6). Let be a homogeneous measure on L. We search for a regularizer in a nonsingular case in the form n—rn—I
=
,
f
9(x2
— %.2) 5(n—rn—l)
(36)
x
for even values of n, and
2i""(n —
— 1)!
=
1
j
0(x 2
—
y
2
ço(x,v2,qI,q3
xV
[t,(x,y,q,,q3
q,1)]n_m
qn)
(37)
for odd values of n. Here 0(z) = 0 for r < 0 and 0(t) = I for I 0. Using the chart q, = 1 of the corresponding projective space, we can rewrite the formulas (36) and (37) in the form '\fl—flt—l
'I
(2ir )fl_2
f 9(x2
—
qn))p(x.y2,ql,q3,...,qn) for evenn;
=
—
m
—
1)!
0(x2
—
(38)
2)
xV.p. q3
is] f(y)dy Adq3 A••• A for oddn.
Remark 2. For simplicity, we consider here only the case m
such that
0,
0,
and
—
0
I. Equations of principal type
To prove this affirmation, we consider the set I C (1, 2. .. . , n } of indices for which = 0 for i E I, and 0 for i I. If there exists a number 0. then the pair of numbers (i0. Jo) satisfies the required i0 I such that conditions for any Jo E I, and the affirmation is proved. If, on_ the contrary, i 1) are linearly = 0 for every i E I. then the vectors (p,. i I) and independent. Hence, there exist two numbers Jo. j, I such that Pj11
Pu
HX)U
Hx'I
— —
x'
—
Pj1
iixiu
Denoting by lo that of the numbers (Jo,
for which 0 and by i1 the other. we construct the required pair (i0, i1). By renumbering the variables if nesessary. we can assume that (a) p1(a) 0, (b)
Hf(a)
(c) P1(a) H,'(a) — p2(a) #0. Then the coordinate system on L0 = flcharH is (xI. x3
f,
since the variable x2 can be expressed from the equation H1 (xl.
,
x's, qi
= 0. Similarly to the previous case, we obtain a coordinate system (x', p2, x3, .. . ,x'1,qi qn) on L and the determining function of L
4',, =xp2—y'qI
x",qj
qn),
(39)
where
S,,(x',p2,x3
qn)=
(40)
is the action on L in the chart under consideration. We shall look for the regularizer in this case in the form (
=
_q2)nm+I)
I xq(x,y2,qp,q3
qn)
x.JF[4'j,,
dq2... for evenn; A dp2 A
— m + I)!
A
fO(p2—q2)
,p2,x ,...,x
,
,q2.....qn)
Y. P2. 1.
(y) dy A dp2 A dq2 for odd n.
A
.. A
III. Applications to differential equations
160
We suppose that the function of the manifold (29).
in (38) -(41) has its support in the set U1 x(—T2, T2)
Proposition 2. The relations
H
Ii (x.
o
=
+ S1
=
+
(42)
(x.
(43)
are valid up tot/ic lower-order terms. Here P is the transport operator (see Theorem 2.4). and S,. S,i are pseudodifferential operators with the principal symbols
(44)
S,,(x,p) = By s,,,
and
we
H(x,p)
(45)
denoted the difference derivatives with respect to the
corresponding variables on char H:
H(x. p) =
H(x p) ,
P2 — P2LharfI
H(x. p) =
H(x p) X
2
2
X Icharll
Proof. Let us calculate the left-hand side of formula (42) for even values of n. We have
=
j
(
x
dx'
(x, y2, q")a(x, y2, q") f(y)dv' A dy" A dq" n—ni— I
+ x
(2jr)fl—2
j
dy2 f
(x, y2. q") f(y)dy'
Y,
A
dy"
A
dq".
q")) i
2,
(46)
I. Equations of principal type
(_j)n-rn-I
a
P
(4,(x,
= (2ir)"2 J
•
161
I
y •x
xa(x,y 2 •q" )f(y',x2,y")dv1
+
2 •
y
ii
q ))
Ady"Adq"
(27rY'2 T2 dy2J
as,
—(x,y ,q ")a(x,v •q )f(v)dy' Ady"Adq 'I 8x2
x
2
2
(_j)n_m_l
+ x
I dy2 f 8(n-m-I) aa — (x, y ax2
f(y)dy'
2 ,
y. q"))
A dv" A dq".
(47)
In the last two formulas, we used the following notation:
...
y=
yfl);
,
q" = (q3.
.
a
,
. .
=
/AJ.
Besides, the variable = 1 is omitted for brevity. Since x2 = y2 is the equation of the manifold L0 in L. we have
y . x 2 , y " q") = S,(x, x2, q") 1
.
— y"
—
•
IF
— y — y q = (x' I
=
—
y') + (x" —
to relations (33) and (35). Therefore, the first term in the right-hand side of relation (47) is a pseudodifferential operator with the symbol a(x, x2, q") = due
lulL. Using formula (47), we obtain for higher derivatives.
'a
in—I
=
jI
k=O •)
x ((x' — y1) + (x" — y") q"a (x, x, q )) k ( as, a 2
x —(x,x q ox2 (_j)fl_in_l
+ x
+ x
J-__ as
1
(—i)n—m—I
I ios,
2
\jk1
_X
f(y ,x ,y " )dy' Ady"Adq" I
) ( dy2
,q" ))
/
F,
2
I,
y, q ))
J a(x, y2. q") 1(y) dy'
A
dy" A dq"
dy2fo(n_m4i_2)(cD,Lx,y,q I,
\)1
Oa
x•.
2
,q")
))
III. Applications to differential equations
+
(aSi
j (j — 1)
q"))
(x, y2
(dx2)2
(x, y2, q") a(x. y2
q")]
f(y) dv1 A dv" A dq"
x
(48)
up to the lower-order terms. The first sum on the right-hand side of the latter formula is a pseudodifferential operator with the symbol
i
f—I
í
2
'P )j\ /
k
f—k—I
,,
P2
i
a(X,x,p )
pIP)])
=
—
(x, x2, p")
1)2 —
a(x.x.p)
By differentiating formula (48) with respect to the variables x1, x" according to formula (46) and by multiplying by the coefficients of the operator (1), we
obtain
k
—i
=
f±
(x. x H
f
— (x, y, q")) dy2
+
IaH f as,
f 2
LdP \ +
H' (x.
a(x,
8d1J2)
\ q"))]
q") f(y)dv' A dv" A dq" y. q"))
a
J dx (x. y2.
(c1,(x, y, q"))
I
82H ix,--—(x,v,q)j / as, \
2 dpdp \
2
-
a(x, y2. q") f(y) dy' A dv" A dq"
up to the lower-order terms where the principal symbol of
a2s,
/ axax (49)
is given by (44). H'(x, p) is a homogeneous component of the symbol of the operator (1) of order m — I. Taking into account that H (x. (x. V2. q")) = 0, since L C charH,
Here
I. Equations of principal type
163
we come to the relation
Ii
= 5!
a
—i
f
+ 18H L
ap
I
—i
y.q"))
a i 8'-H I aS, fix,—(x,y,q ,, )j —+— ax 2 ap ap \
\
aS, ax
/ ax
+ I H'
(x. y2,
q"))] a(x, y2. q") f(y) dy'
)j\/ 3x ax
2
dv"
A
A
dq".
Using formula (11.2.26) similarly to the proof of Theorem 11.2.4, we obtain formula (42). The proof for even values of n is quite similar. For a change, we shall carry out the proof of formula (43) for odd values of n. (for odd To begin with, we rewrite expression (42) of the operator values of n) in the form jn_m+I(fl
—
m
+
J
=
dy A dp2
A dq"
W(,P2,X";(!'),
j V.p.
fe') dq2.
where q' = (qi qn). By differentiating expression (50) with respect to x and by multiplying the result we obtain (similarly to the calculations in the previous case) the formula by
Ii
a
—i
I)!
= x
/
as,,
ax'
f dy A dp2 ,p2,x
I'
A
dq" 1: V.p.
,q),p
2
as,, ax
[4,,(x, Y.P2. ,p2.x
"
x a(x1,p2,x",q')f(y)dq2 P2
—
f dy A dp2 A
Vp.
y.
' .q).p
2
Ill. Applications to differential equations
[dli I
as,,
\
L8P
,,
/
aS,, tx,—j-(x dx
/
,,
,
dx
8X
I
+iH
aS,,
,
,,
\8 ,dx
dS,,
,
.P2,X ,q) ,p2.x ,q).p2,—-(x dx
y, dx dx
aS,,
aS,,
/
,,
,P2,X ,q),p2,—(x
dx
,q)
dx
x a(x'. p2,x,q)f(y)dq2, up to the lower-order terms. The absence of the term of the type S,f is due to the fact that the upper limit P2 does not depend on the variables x. Contrary to formula (49), the first term on the right-hand side of formula (51) does not vanish, since the argument x2 of the function H(x, p) in the integral is not equal to (x1. P2. x". q'). However, the amplitude function H a of the integrand of this ierm belongs to the gradient ideal J(
explicitly extract the factor x2 +
P2. x", q') from the integrand of the
considered term; this factor is one of the generators of the gradient ideal
/
aS,,
.p2,x
dx
/ =Hix
,,
aS,,
/
dx
aS,,
,,
I
H(x,
—
=
as,, dx
,p2.x
H
(i'. +
I,x- +as,, — \
,,
.p2.x .q
+ x
,
,—-—-—(x .P2.X ,q),x
as!,
"
,q),p2,—-(x ,P2.X ,q)
8P2
P2. q') (x2 +
P2. x",
q')),
since
/ Hix.——x .—j-,P dx dx / 8P2 \ 8SF1
,,
2
HI, =0.
(52)
I. Equations of principal type
165
By substituting expression (52) in the integrand of the first term on the right-hand
side of (51) and integrating by parts, we reduce formula (51) to the form
Ii
=
—i
in+I .n,
f dy A dp2
A
dq"
,,
V.p.
—
I
X
—
aS,,\
aS,1
(s,,, H) (x, p2. q') i
82H j'
2
apap
+—
+
P2.
(x. P2. q') 8P2
8S,,\ 82t,,
as,,
8x
ill'
I
a
—
81)2
f
—i
/ dxdx a(x1,
1)2.
p2,x.q)f(y)dq2.
(53)
We note that the variable x2 in the integrand of the second term on the righthand side of (53) can be replaced by up to the lower-order terms. After this substitution, the difference derivatives will be replaced by the usual ones, and we can calculate the integrand of the second term on the right-hand side of (53) with the help of formula (11.2.26). This procedure leads us to formula (43). This proves the proposition. Li
ProposItion 3. The operators c1,. and act in the spaces
determined byforinulas (38) and (41),
—' /499)
:
11c nt—I
(54)
(R?t).
Prnof We shall carry out the proof (for definiteness) for the operator (38) in the case when n is even. The proof in the remaining cases is quite similar. First, we consider the operator tj and rewrite its definition in the form
=
I xa(x,y,q )f(y)dy Ad)' Adq JS(n_m_I)
y. q")) .
(55)
III. Applications to differential equations
166
The inner integral on the right-hand part of (55) can be considered as a set of operators with x2, y2 as its parameters: (
n—rn—I
(cD,(x, y,
J
=
f(y)dy' A dy"
xa(x, y2,
A dq".
(56)
As one can see from expressions (34) and (35), the Fourier—Maslov integral operator (56) corresponds to the Lagrangian manifold (57)
is locally difWe note that, since 0, the space charH fl = with the coordinates (x'. x2. feomorphic to the symplectic space x", (the coordinate p2 can be calculated from the equation H = 0 pj, ..., of charH). As we can see from the description (29) of the manifold L. the Lagrangian manifold (57) is the graph of the canonical transformation : char H fl —÷ char H fl (x2 = which is determined by translations along the (y2 = trajectories of the vector field V (H1) (we take into account that V (Hi) is transversal to the manifolds (x2 = Then, due to Theorem 11.2.5, we and jy2 = obtain that the operator (56) is a bounded operator in the spaces
Since the operator smoothly depends on the parameters x2, y2. we see that the operator (56) acts as a bounded operator from (W) to 11 rn1 (R't). To
(f(y)) has a compact support, and the
conclude the proof, we note that operator
ff(xI.y2,x")dy2 This completes the proof of the proposition for
acts from to the operator of the type
we introduce a associated with the homogeneous symplectic
To prove the proposition for the operators of the type special Fourier integral operator diffeomorphism
It:
qi
= qi
pi=qI; (58) Pi
= qiy2.
One can easily verify that formulas (58) determine a symplectic diffeormorphism, that is, that dp A dx = dq A dy and that the (global) determining function on the
1. Equations of principal type
167
graph of this diffeomorphism is equal to
=qi(x'
cD(x',x2,
—
y' +x2y2).
(59)
We put
3:(f)=Jo(xI_yI+x2y2)f(yI,y2)dyldy2=f f(x'+x2y2.y2)dy2. (60) Using the results of Section 2, Chapter 11, we see that operator (60) has the order
in the Sobolev space scale. On the other hand, we see that, up to the notation, operator (60) coincides with a specification of the Fourier transformation of homogeneous functions in affine charts of the corresponding projective spaces. Namely, if the function F(x°, x1. x2) is an element of the space such that F(l,x',x2) = f(x',x2), then, due to formulas (1.4.32), we obtain —
f 6(po+xt +p2x2)f(—x',x2)dx'dx2
(2,rY'/21
(3:1)
=
(Po. P2).
Thus, we see that operator (60) is invertible. The result of Section 1.4 allows us to
calculate the inverse operator
f(y',y2)dy'dy2
2
[x' — y' Operator (61)is a Fourier integral operator associated with the homogeneous symplectic diffeomorphism J
—
xl
qi q2
qi
p2=—qly,2
which is an inverse isomorphism for (58). In particular, we obtain that operator (61) has the order 1/2 in the Sobolev space scale. Operators (60) and (61) play the same role in the theory of Maslov's canonical operator as the usual Fourier transformation does in the theory of Maslov's canonical operator with a parameter (see EM II, [M1ShS I ]). Now one can verify that the composition 3:_I is the operator of the type '1,; this affirmation can be proved similarly to the proof of the composition theorem in Section 11.2. For this proof, one has to take into account that, due to property (c) above, we have 0 for 11 = h(H). 0
Remark 3. It is possible to define the operator as a composition 3:o in the singular case. For such a definition, we slightly have to change the proof of the composition theorem (Proposition 2) above.
III. Applications to differential equations
By applying the result of Proposition 3 to the remainders in formulas (42) and (43), one can easily obtain that these formulas are valid up to the operators of order — I in the Sobolev scale H6(R'1). We are now able to construct microlocal regularizers in both singular and nonsingular cases. Let us consider the nonsingular case in detail. Let = where is a solution of the transport equation .
(63) with the initial data = C. is a function depending only on t (the parameter along the trajectories of the field V(H1)) being equal to 1 in [0, T1] with support in [0, 1'2], and e being a function with support in U2 equal to unity in U1). Let us choose a measure on the manifold L which is invariant with,respect
where
to the vector field V(H) and such that its density is equal to (EL) -
at the
points of the subinanifold L0. Under these assumptions, the symbol (44) of the is equal to eIL( at the points of charH. Let T (x. —i operator S, (x. —i be a pseudodifferential operator with the symbol T(x, p). which is equal to I
— Sj(x, p)
H(x, p) in
+ t (x. —i
a neighbourhood U1 of the point a. Put R =
Then
we have
ii (x.
ok=
—i
(x. —i
+
where
Q
+
=
\
(67)
—i
\
—
point out that is a Fourier—Maslov integral operator on the Lagrangian manifold (i) 4'j L. since the support of its amplitude function does not intersect the diagonal We
(
•
(ii) the projection of suppPp on the second term of the product x TR" does not intersect U1;
x
=
L Equations of principal type
169
(iii) the term in the square brackets on the right-hand part of (67) is a pseudodifferential operator; the symbol of this operator vanishes on U1. Hence, the operator Q acts boundedly in the spaces (12), and for any pseudowith supp P(x, p) C Ui, the composition Po Q differential operator P (x, —i is equal to
By calculating the first term on the right-hand side of (68) with the help of the
composition theorem (Theorem 11.2.2), we obtain the FlO of order 0 with zero symbol. Hence, this operator has order —l in the space scale H5. Similarly, the second term on the right-hand side of (63) is the operator of order — I in the scale W due to Proposition 11.2.1. Thus, we see that the operator (67) satisfies Condition A of Subsection 2. The identity (66) now shows that the constructed operator R is a microlocal regularizer for the operator (1) in the neighbourhood U1 of the point a. The construction of a regularizer in a neighbourhood of a singular point a (i.e..
of a point a such that H,,(a) = 0) is quite similar. 1.4 Construction of a semiglobal regularizer
Let K be an arbitrary compact set in
In this subsection, we fix the compact
set K as well as the compact sets K,, K2 such that K C K1, K, c k2. Let W being a natural projection. Denote by Ya the trajectory of the contact distribution 'H with its origin
a E S*Rfl be any point of charsH fl,H(K), with jr : S*Rfl
at the point a. Evidently, in a tubular neighbourhood U of the distribution 1,, with the help of some vector field X = integral curve of X with its origin at a, Ya R charsH C
one can realize so that is an
:
ya(O) = a.
Due to the requirement (iii) of Definition I, there exists such that without loss of generality, we can assume that ü > 0. Since an integral curve of the smooth vector field X111 depends continuously on its originating point, there exists a neighbourhood Ua of the point a and e > 0 such that for any E Ua the curve yp(t) lies outside for t E (t0 — e. t0 + r). We can also assume that there exists a neighbourhood such that the pair C (Ia) satisfies conditions (i)—(iii) of the previous subsection and that for this pair, the construction of a microlocal regularizer described in this subsection is possible. We also suppose
thatUa C K, foranya€ K. Since the set char5llfl,r_I (K) is compact, there exists a finite covering of this set by the open sets of the described type. Similarly to the previous subsection, we construct the (open) Lagrangian manifolds (Ua fl char H) x (—e, !o + e)
x R")
170
III. Applications to differential equations
(we do not distinguish between the open set in and the corresponding hoRfl). From the construction above, it is evident that mogeneous set in fl (t0
—
r,
+ e)) fl
K1
= 0,
—÷ T*RIZ on the second factor. x We shall call the manifold L1 a characteristic strip of the open set U3 C be a partition of unity on char5 H fl Let 1, corresponding to the covering of this set. The functions e1 eN can also be considered as the functions on which are homogeneous of order zero. Let C j=1 N be operators of the type (38) or (41) in accordance with the type of the chart U1 (the operator of the type (38) if the chart U is nonsingular. and of the type (41) if the chart U1 is singular). Then the function is a product which is the solution of the transport equation (63) with the of the function initial data e1 and the function i,l,• depending only on the parameter: along the trajectory of the vector field X113 such that i/i, = 1 in a neighbourhood of the origin. Let us choose a function x1(') which is equal to I for 0 t0 — e and which is is evidently equal to zero for r > + Xi (t) E Since the function determined on the entire La, we see that the function (Xi — is a smooth homogeneous function on with support in (0. :0+e). Denote by the Fourier—Maslov integral operator corresponding to the Lagrangian manifold with the amplitude function (Xi — The following affirmation is a generalization of Proposition 3.
with 212 being the natural projection
Proposition 4. The operator
has the order I
—
m in the space
scale W.
Proof Evidently, it is sufficient to prove the boundedness of any operator of the form if the function x has an arbitrary small homogeneous support on the manifold La. Note that each point of La can be considered as a pair of points lying on a trajectory of the vector field V(H1). We aE (a, fi E shall distinguish between the following four cases. A. Both a and are nonsingular points, that is, H1 0, 0. B. The originating a is nonsingular and the endpoint /3 is singular. C. The originating a is singular and the endpoint /3 is nonsingular. D. Both a and /3 are singular points, that is, = 0, H1 = 0. such that H1 (a) 0, H1 (/3) In case A, there exist numbers i0, Hence, the intersection of the manifold L with the manifold = is y'° = transversal. As was shown in Section 1.6, under such a condition the intersection is a Lagrangian manifold whose determining function L fl = "= = into the determining function can be obtained by substituting y'° = of the manifold L. Furthermore, this intersection is obviously the graph of a homogeneous syinplectic diffeomorphism. Thus, in this case, the operator
1. Equations of principal type
can
be represented in the form
=
f
(69)
are FlOs associated with homogeneous symplectic diffeomorphisms. Similarly to the proof of Proposition 3, we obtain the required inequalities for the operators with the help of formula (69). Cases B, C, and D can be reduced to the case A with the help of the transfor-
where the operators
mation
described o
in the proof of Proposition 3. Namely, the operators P' o o and are of the type A if the operator
of type B, C, and D, respectively. Using the inequalities for the operaand the inequality for the operator in the case A, we obtain tors the required result. We use the notation Then, due to Propo+ = sition 2 and the composition theorem for Fourier integral operators (see Section 2 of Chapter 2), we obtain the formulas is
/
H
up to the operators of order —1, where the are FlOs of order 0 such that the projection of the support of their amplitude functions on the second factor of T*(Rfl x Re?) = x lies outside of the set ,H(K1). Summing the equalities (70) over j, we obtain the relation
\ where 3 =
3,,
a,j
\
is a pseudodifferential operator of order 0 such that
= 1 (since the form a partition of unity on char H), and is an S(x, HO of order 0 such that the projection of the support of its amplitude function on = T*Rfl x T*Rfl lies outside the set ,r(K1). the second factor of T*(Rfl x We put
R=3+t. with '1' =
t (x, —i
being a pseudodifferential operator with the principal sym-
T(x,p)=
1—
S(x. p)
H(x, p)
lii. Applications to differential equations
172
which is a smooth function on
= 0 and dH
since 1 — S(x,
0 on
char H. Evidently, we have
up to the operators of order —I, where the operator satisfies the condition B of Subsection 1.2. Hence, the operator R is a right semiglobal regularizer for the operator (I) over the compact set K. 0
1.5
Functional spaces
In this subsection, we present some facts on the theory of Sobolev spaces used above in the construction of a semiglobal regularizer. We recall here the definitions with K being a compact set in R" with a of the spaces and which consists smooth boundary OK. The space is a subspace of
with their supports in K:
of distributions u E
=
I suppu C K).
{u E
(72)
Evidently, the space is a closed subspace of the space We also consider another closed subspace of HS(Rfl) consisting of distributions
which vanish in the interior K of the compact K:
K)
=OinK}.
(u
(73)
We put
(74) is a space and K) is its closed subspace, the space (74) has Since a natural structure of a Hubert space (see K. Yosida IYo II).
(K) is dense in
Lemma 1. The space
K) for any s.
which is everywhere transversal to OK Let X be a smooth vector field on and which vanishes outside a sufficiently small neighbourhood of OK. We assume that the field X is directed inside K at points of OK. Let g, be a one-parameter group corresponding to the vector field X. Then it is evident that:
(i) g,K
K fort >
0;
(ii) g, tends to the identity mapping as 0 in the Affirmations (i) and (ii) show that the set of distributions u K is dense in
H3
gu = a, and that due to (i),
lime .
such
Actually, it is evident that due to (ii),
that suppu —
E
K for
t < 0. Further.
I. Equations of principal type
if x(x) E
x
C
x(x) > O.f x(x)dx = 1, and XF(X) =
I) such that
II
then the function xr *"
infinitely smooth, has its support in an arbitrary small neighbourhood of supp u * u = u. Therefore, the set for sufficiently small r. and H5 — limE.+o is dense in the set of distributions u E This completes the proof of the lemma.
suppu
K and, hence, in D
Now we shall define the pairing between the spaces do this, we need the following affirmation.
Lemma
2. For anu
and
To
any v E H5(R", K), we have (u, v)
Proof Since the space
f u(x) v(x)dx
is dense in Homp(K) such that
(75)
0.
there exists a sequence = u. Evidently, (un, v) = 0. Since the form (u, v) is continuous with respect to its arguments, we u
—
have
(u,v)= U-. lim The latter equality proves the lemma.
Lemma 2 allows us to define the (u, formula
for u E
VE
(76)
(a, 1)) = (u, v),
with
v
by the
being an arbitrary representative of ii. The form (76) determines the mapping —+
= (a. ii).
a i—k
(77)
PropositIon 5. The mapping (77) is an Let
is
= for any v that u
0 for any 13 E H5(R"). Since the spaces
that
Evidently, for such a we have (u, v) = 0 and are dual, one can see
= 0.
Let us prove that (77) is an epimorphism. Let w E be a continuous linear form on the space This form determines a form w' on the space with the help of the formula
w'(v)
=
with i' being a residue class of v in the quotient space a/ vanishes on K). Due to the duality between
(K). Evidently, the form and
there
III. Applications to differential equations
174
exists a function u E
such that
w'(v) =
(u,
u) = Ju(x)v(x)dx.
= Since w' vanishes on K). one can easily verify that supp u fl and hence, u H5(K). It is now evident that w(ii) = w'(v) = (u, v) = (u, This proves the required affirmation. for any 1' E
i:i>
0
Now we shall prove that the spaces form a space scale in the sense of Proposition 6. To do this, we define the mapping
i:
—+
(78)
for s' >
and let v be any representative s in the following way. Let i3 K)) of the element v in of the residue class i3. The residue class (v mod the quotient space (K) does not depend on the choice of the representative v K). We write of the class = (vmodHx'(RhI, K)}, since K) c
=
K)} E
PropositIon 6. The mapping (78) is a dense embedding. K) fl Proof If i(v) = 0, then we have v E for any representative Further, the v of the residue class i,. This means, in particular, v E
restriction of v to the interior part K of K (in the distribution sense) vanishes. Hence, v E Hs(Rn. K), and i' = 0. We have proved i to be a monomorphism. we note that the set To prove the density of the image of i in is dense in Hence, the set of residue classes of the functions of is dense in One can easily see that such residue classes lie in the image i
This completes the proof.
0
To conclude this section, we prove that the mapping (78) is a compact mapping. For any element i E A, we choose Actually, let A be a bounded set in such that v II a representative v II + 1. The set A of such fl representatives is evidently bounded in H's' (Rn). We note, that the image i (A) of the set A with respect to the mapping (78) is equal to the image of the set A with respect to the mapping -+ Hs(Rn)
(79)
the mapping (79) is a composition of the compact mapping H (R") —' —+ (the natural projection), we see that i (A) is a relatively compact set. Since
Hx(Rn) and the bounded mapping HS(Rfl)
2. Microlocal classification of pseudodifferential operators
175
2. Microlocal classification of pseudodifferential operators 2.1 Statement of the problem First of all, let us introduce the localization in the algebra of pseudodifferential operators in a neighbourhood of a point (x0, p°) E (or, more precisely, in a neighbourhood of the corresponding point (xo, p0) E S*Rhl). We say that two pseudodifferential operators P (x, —i and Q (x, —i are equivalent at the point (xo, p0), P there exists a homogeneous neighbourhood U of the point (x0, p0) such that for any pseudodifferential operator R (x, —i with the operator R (x, —ii) (P (x. —ii) — Q (x. —ii)) is an opa(R) E erator of order —N in the space scale Hs(Rn). Here N is an arbitrary large integer which will be fixed in the rest of this section. We denote (here and below) by a(R) the total symbol of a pseudodifferential operator R and by a,,, (R) the homogeneous component of a(R) of order m. Thus, for a pseudodifferential operator R of order m, the function a,,, (R) is a principal symbol of this operator. An equivalence class (with respect to —) of pseudodifferential operators will be called a germ of a pseudodifferenrial operator at the point (xo, p°). We denote by
if
Psd1xOpo, the
set of germs of pseudodifferential operators at the point (x0. Pu). One
can easily verify that the operations P + P o Q are well-defined on and that ellipticity of a pseudodifferential operator in a neighbourhood of (xe, Pu) does not depend on the choice of its representative in the equivalence class with respect to Evidently, the elliptic germs are invertible elements of the algebra Psd(X,,Po).
Moreover, if G : such that G (xo. p°) =
is a homogeneous symplectic diffeomorphism (x0, p0). A 0 (i.e., the point (xo, p°") is a fixed point for the corresponding contact diffeomorphism g), then the elliptic Founer—Maslov integral operator T8(ço) (i.e., such that p (x0, p°) 0) determines the operator TG(co) : Psd(X
1,0)
A
P=
—* Psd(r,po);
oPa
where (TG(ço)Y' is an inverse operator for up to operators of order —N at the point (x0, p°), that is, for any operator P such that a (P) C U, the operators P a (TG(Q) o
—
1),
Po
a
—
1)
pseudodifferential operators of order —N in the Sobolev space scale H'(R0). We denote by Y the group of transformations of the algebra generated
are
by
(i) multiplication by an elliptic operator P
PSd(XOPO) :
Q
° Q;
Ill. Applications to differential equations
(ii) conjugation with the help of an elliptic Fourier—Maslov integral operator Q. with TG(p) being determined by formula (1). The aim of this section is to describe (under some restrictions of the type of generic position) the orbits of action of the group Y on the algebra Such p°)• a description will be used in the following section for constructing a regularizer for some pseudodifferential operators whose contact vector fields possess fixed points. To conclude this subsection, we shall make two remarks. First, it is evident that all elliptic elements P E form a single orbit of the action of a group Y. Indeed, if P is elliptic, then o P is elliptic for any elliptic pseudodifferential operator Q. and P is elliptic for any elliptic FlO (the latter fact is due to the relation P) G((Tm(P)) if P is a pseudodifferential operator of piith order). Further, if P E Psd( is elliptic, then (as was pointed out above) it is invertible in and the inverse element 1. Q E Psd(,,,O) is also an elliptic germ. Hence, we have Q o P = 1 and P Thus, any elliptic germ P is equivalent to the unit operator 1 with respect to the action of the group )). Due to this fact, we can consider below only the germs p°) = O(ord P = ni). such that PE Secondly, using the multiplication by an elliptic germ, we can reduce any germ to a germ P' E of the first order. We denote by PE the first order (and, more generally, P of order k). So, we have reduced the problem P to the problem of the classification on of the classification on For the latter problem, we shall consider the classification only with respect to a subgroup 5.' of the group )) generated by conjugations of the form (1). Note that has a natural structure of a module over the ring the set TG(W): Q i-÷
2.2 Operators of principal type of (microlocal) In this subsection, we shall show that all the germs P E principal type form a single orbit with respect to the action of the group Y. More precisely. we shall show that for the arbitrary two operators H,. 112 such that X,, (x0. p°) 0, 0, there exists an elliptic FlO TG(W) of (xo, p°) order 0 such that H,
/ x, —i — ôxjJ = \
(
Due to Corollary 11.2.3. we have
H2
I
\
x, —i —
iixj
)
H2)) = G*(ai(H,)). The problem of
classification can be solved in two stages. At the first stage, we find a homogeneous
2. Microlocal classification of pseudodifferential operators
symplectic diffeomorphism g in a way such that
a1(H1)(x, p) =
(a1 (H,)) (x, p)
in a neighbourhood of the point (Xe, p°) (reduction of the principal symbol). At we assume that the principal symbols H1(x, p) and H2(x, p) of the second the operators H1 and H2 coincide in a neighbourhood of the point (Xe, p0). H1(x, p) = H2(x, and construct an invertible (in
\
= H(x.
p)
p).
pseudodifferential operator U
that
\
dx;
This gives us an operator
such
satisfying relation (2). Actually, due to Corollary
11.2.3, we obtain
ai(H1) = as
a consequence of (3). If U
—l
is
H2
I TG(l)) = a1(H,)
the operator of the form (5)
for
ii1,
then we
have H1
—i
= (U' o
=
H2(TG(l) o U).
(7)
(7) coincides with (2), since the operator TG(l)oU is an elliptic Fourier— Maslov integral operator due to the composition theorem for the FIOs (see Section Relation
11.2).
First stage (reduction of the principal symbol). First, we note that since X11 0 at the point (x0. p°), there exists a linear contact diffeomorphism g: S*RU such that g (x0, pO*) = (x0, pOt) and Xy, = X11,. Hence, we can suppose without loss of generality that X11, = X11, at the point a. The local character of our considerations allows us (after possible renumbering of the variables) to use the local chart Pi = I in the neighbourhood of the point a S*Rn. In this case, as was shown in Section 1.5, the Contact space StR" can be considered as a subspace in with the corresponding embedding being determined by the relation = 1. Hence, any homogeneous Hamiltonian function H of order 1 is uniquely determined by its restriction on the space SR": 0,
h
=
called a contact Hamiltonian function. If X is an arbitrary contact can be represented in the form X = for a contact Hamiltonian function The function h
is
vector field on S*Rn, it
h
= Xhja,
III. Applications to differential equations
178
where a = dx' + p2dx2 +
is the form which determines the contact + structure of the space S*R?i. Let h0 and h1 be the contact Hamiltonian functions corresponding to H1 and 112,
respectively (with the help of relation (8)). Due to the conditions above, we have 0*
p )=
p
0*
)
0.
To construct the contact diffeomorphism g such that gt(hi) = h0, we path-lifting method. Let us consider a path
use
the
= h0 + t (h1 — h0) between
the Hamiltonian functions h0 and h1 and search for a set of contact
diffeomorphisms
—÷
S*Rfl satisfying the following conditions:
g1 (X(J,
pO*)
=
(xo,
pO*),
(Xh,) =
(12) (13)
it is evident that we can put go = id. We suppose that g, can be represented as a shift by t along the vector field X,. Since g1 is a contact diffeomorphism, we see that X, must be a contact vector field, and hence, it can be represented in the form X1 = Xj, for some contact Hamiltonian function (depending on the parameter 1). By differentiating relation (13) with respect to t. we obtain the equation [Xh,, X1,] + Xh, = for the vector field Xh,. To transform equation (14), we shall use the following affirmation.
Lemma 1. Let h1 and h2 be contact Hamiltonian functions. Then the formula
a(lHh1, Xn,l) =
Xh1(h2)
—
X1(h,)h2
is valid (here X1 is the contact vector field corresponding to the contact Hamiltonian
function h =
1).
This affirmation is a consequence of a more general one, which is also useful by itself.
Lemma 2. Let
be a homogeneous Hamiltonian function on of order I of order k. The formula and H2 be a homogeneous Hamiltonian function on V(H1)
holds where h, =
=
—k
X1(h1))h2
2. Microlocal classification of pseudodifferential operators
179
Proof of Lemma 2. Evidently, we have
=
HIpS
H2r1 +
Hip,
—
—
j=2
1=2
(17)
Due to the Euler identity, we obtain
=
=
—
—
j=2
j=2
= kh2
H2 —
—
j=2
j=2
By substituting the obtained expressions in formula (17), we see that
v(Hi)
1121p,=i
=I(hI
+
h2 = (Xh —
—
j=2
J
to the formulas for Xh derived in Example 2.5.1. This completes the proof of the lemma. due
Proof of Lemma I. Let Hi and 112 be homogeneous Hamiltonian functions on of order I such that The direct calculations show that = a ([V(h1). V(H2)1) = V(H1)
112.
= l} gives
Restriction of formula (18) on the space
a([Xa, Xh,]) = V(H1)
.
Applying formula (16) to the latter relation with k =
1,
we obtain (15).
Due to Lemma 1, the relation (14) can be rewritten as a relation for the contact Hamiltonian function f,:
Xhjf,)—XI(ht).ft+h, =0. 0 at the point (xO, pO*), we see = Xa0 0 at (xo, To solve equation (16). we that Xh, = (I — I) Xh0 + t choose a submanifold 1' C of codimension I transversal to the vector field and passing through the point(xo, pO*), and put
Since h, = (I
—
Oho + 1h1 and Xh() =
ftIrxR,
°
180
III. Applications to differential equations
One can easily see that problems (19) and (20) have a unique solution and that pO*) = 0, df,(xo. pO*) = 0. Hence, the vector field X1, determines a set of diffeomorphisms subject to conditions (12) and (13). Taking into account formula (9), we see that g(h1) = h0. Second stage (reduction of the operator). At this stage, we can assume that the principal symbols of the considered operators H1 and H2 are equal to each other in a neighbourhood of the point (xe, p°). We denote by H(x. p) the principal symbol
of Hi (and of H2) and assume h to be the corresponding contact Hamiltonian function.
We write R = H1 —
Under our assumptions, R is a pseudodifferential
operator of order 0. Let
k
= k=-,o be a representation of the operator R as a sum of the pseudodifferential operators of order k with the homogeneous symbols Rk(x, p). Relation (5) can be rewritten in the form
H2U-UH2=>URA. We search for an operator U in the form
where the Uk are pseudoclifferential operators of order k with the homogeneous symbols U&(x, p). Using Proposition 11.2.1, we obtain the system of equations for the functions (JL(x. p):
V(H)•Uo+R0U0 V(H)•U..,+R0U_1
=0 =•••
(22)
where the dots on the right-hand side denote homogeneous functions which can be found explicitly, provided that the solutions of the previous equations are known. Let r, u1 be the contact Hamiltonian functions corresponding to R3 (x, p), (x, p). Using Lemma 2, we can see that the restriction of the system (22) on = (p' =
2. Microlocal classification of pseudoditlerential operators
1}
c
gives the system of equations for the functions
j = 0.
181
—
1..
=0
Xh (u0) + r0u()
Xh(u_I)+Lro+XI(h)Ju_l (23)
Xh(uk)+[ro—kXI(h)]uk 0 in a neighbourhood of the point (xO. pO*), the system (23) has a 0. solution up to an arbitrary order such that u0(x0. We have proved the following affirmation. Since Xh
Theorem 1. Let (x0, p°) be a point in
(Rn) and let H1 and 112 be pseudodiffer-
ential operators such that H1(xo. p°) = H2(xo, p°) = 0
(24)
E and the contact distributions!,,, do not vanish at the corresponding point (x0. S*Rn. Then the germs H1. 112 are equivalent with respect to the action
of the gmup )) defined above.
2.3
Operators of subprincipal type
In this subsection, we shall describe the orbits of the group Y
on the algebra PSd(XOpo) for operators which satisfy the following condition (absence of resonances).
a contact Hamiltonian function in a neighbourhood of the point (x0, pO*) such that h(xo. pO*) = 0, Xh(xo, pO*) = 0, and let
Condition A. Let h
be
Xo=Xi(h)(xo.pO*),
A1
the eigenvalues of the linear part of the field Xh at the point (XO, pO*)• We > 3. the say that h satis/les Condition A if for any integers rn1, m1 > 0, be
inequality 2n—2
m3A,
A0
(25)
1=0
holds.
DefinItion 1. The pseudodifferential operator ii is said to be an operator of subprincipal type if the restriction of its principal-type symbol on S*Rfl satisfies Condition A.
Thus, in this section, we shall describe orbits of the group Y containing the operators of subprincipal type.
III. Applications to differential equations
182
The importance of Condition A will be shown below in the proof of Lemma 4. To understand the sense of this condition, we shall briefly recall the properties of the spectrum of the linear part of the contact vector field in its fixed point. First of all, we introduce an interpretation of linear parts of a contact diffeomorphism and a contact vector field as linear operators in provided that the point (x0, p0) is a fixed point for this contact diffeomorphism and for this vector field.
If g SM -÷ S*M is a contact diffeomorphism such that g((xo. p°')) = (x0, pO*), then the matrix of its tangent mapping :
(26)
—÷
coincides with the matrix of its linear part at this point. Namely, if z = (z' are coordinates in a neighbourhood of the point (xO, pO*) such that z = 0 at this point, then the local expression for g is
j
= I,... ,2n
—
I)
g'(z) =
(here
and below, we use the
usual
j=
i-+
= X'
(27)
A,'z1+ O(1z12)
summation convention). The matrix
is the matrix of the linear part of g, and hence, Al = (26), is locally expressed by X
2n — I),
I
i-÷
=
A=
(0), the tangent mapping Z
A!X1
Now let X be a contact vector field with the fixed point (xO, pO*)• One can easily
see that in this case, the commutator [Y, depends only on the value of the vector field Y at the point (xo, pO*)• Hence, the mapping Y t-÷ [Y, XJ induces the mapping X.
(29)
The matrix of the linear mapping (29) coincides with the matrix of the linear part of the vector field X. Actually, we have X
= B/z'
+ O(1z12),
where B = H B/Il is a matrix of the linear part of X at (xO, pO*) with respect to the coordinate system z. Furthermore, for Y = Y' we have
and hence, the matrix of the mapping (29) is equal to B.
2. Microlocal classification of pseudodifferential operators
183
is the one-parameter group corresponding to the We point out here that if contact vector field X, then the matrices A(s) and B of the linear parts of g3 and X satisfy the relation
B= We
(31)
as
are now able to examine the properties of the spectrum of a contact vector
field.
Proposition 1 (V.V. Lychagm, ELy lJ). Let X be a contact vector field such that be its linear part (29). Then X (Xe. pO*) = 0 and (h) is an eigenvaiue of with h being a contact 1-famil(i) the number A0 = tonian function corresponding to X; (ii)for any other eigenvalue A, of there exists an eigenvalue A such that (32) S*Rfl
be a one-parameter group of contact diffeomorphisms corresponding to X, g5((xo, pO*)) = (XO, p°') and let A(s) be a matrix of its linear part. Since g.ç is a contact diffeomorphism, we have
Proof Let g5
:
(33)
for some function on S*Rfl. Denote by r the kernel of the form due to (33), we have
Then
a(X)=0 for any X E
r. and hence, the space F is invariant with respect to the mapping
Further,
g(da) = d(g0!) = If X, Y E
.
a) = dfç
A + fc da.
(34)
r, the relation (34) yields = g(da)(X, Y) =
da
The latter relation shows that the mapping
pO*)da (X, Y).
is conformal-symplectic with respect
to the symplectic structure da on I',
da where
=
=
g5,Y)
= JA0da (X, Y),
p°). This relation can be written in terms of the
matrix A(s):
da (A(s) X, A(s) Y) = tio(s)da (X, Y). We also have
VEI',
(35)
Ill. Applications to differential equations
the vector X1 is transversal to 1' (da (X1) = I due to the definition of the vector field X,, : da (X,,) = h) and since I is a subspace of codimension I. since
Evidently.
p = a (p X1 + i') =
=
a (A(s)
(X1) =
Hence. we have
A(s)
X1
= j1o(s)
X1
+ V.
(36)
V E 1'.
By differentiating the relations (35) and (36) with respect to s at and by taking into account equality (31). we obtain
da(BX.Y)+da(X.BY)=Aoda(X,Y). BX1 = A0X1 + V.
the point s =
X,Y El:
0
(37)
V E 1.
(38)
where A0 = /L0(O). Since BX1 = [X1. Xj due to the definition of the mapping
(29), we have
a([X1, XJ) = a (BX1) = Ao. On the other hand, due to Lemma 1. we obtain
a (IX. X I) = X, (h) — X1 (1) h = X1 (h). Ii is a contact Hamiltonian function corresponding to the contact vector field X. Hence, A0 = X1(h). We use the coordinate system (x1: x2 = I) v": P2 p,,) on SR" = in a neighbourhood of the point (xe, pO*). One can easily see that
where
a
X1—j-:
ía
(39)
dx
a
—; ax"
d
a\ — I
dp2
the matrix of dcx on
apn)
..
form a basis in I
I in this basis is i
-
=
with E being a unit matrix of dimension (n—I) x (ii— 1). The relations (37) and (38)
show that the matrix B of the mapping (29) in the basis has
:
the form
(0 B=1
*
...
* ,
(42)
2. Microlocal classification of pseudodifferential operators
185
where B is a (2n — 2) x (2sf — 2)-matrix subject to the condition
'B.I+IB=AoI.
(43)
Affirmation (i) of the proposition follows from the representation (42) of the matrix
B of the mapping (29). To prove affirmation (ii), we note that, due to (43). the matrix C = B — is a symplectic matrix, that is. (44)
and affirmation (ii) is a subsequence of the following result.
Lemma 3 value
If C is a symplectic ,nairix, then for any eigen-
there exists an eigenvalue
+
such that
= 0.
Pmof of Lemma 3. Let be an eigenvalue of C and let X1 be a corresponding eigenvector. Then we have = $LJXJ.
By multiplying this equality by I, we obtain
=
JAJIX).
Due to (44), the latter relation can be rewritten in the form
= that is, the number —;z1 is an eigenvalue of the matrix 'C and, hence, of the matrix C. The lemma is proved. U Now, if A is an eigenvalue of the matrix B, then
—
an eigenvalue is also an eigenvalue This completes the
=
of the matrix C. Due to Lemma 3, the number = — A, of C, and hence, is an elgenvalue of + = A0 — proof of Proposition I.
is
0
One can now see that the restriction 3 in Condition A is essential, rn = 2. the inequality (25) is not true for arbitrary rn,. For example, since for if and A31 is a pair of eigenvalues with a sum A0 (existing due to Proposition 1). condition (25) fails for rn,1; = I; rn = 0, j i0, J We also point out that ('ondition A cannot be valid for arbitrary large E A0 = 0. Actually. if we put mO = N.m1 = 0 for j 0, we have = Ao = X1 (h) also has another important interpretation. Since X,1 = 0 at (Xe, pO*) and since
ap
(45)
111. Applications to differential equations
186
for some A. we have V(H) = A at the point (Xe. p0). On the other hand, by we obtain for p' I. applying the vector field (45) to the function
V(H).
= A,
p
= X11
= 1. Affirmation (15) of Lemma 1 shows that
since X,, is tangent
= —V(pi)
V(H)
=
.
—X1(h) = —Ao.
Thus. A() can be interpreted as the coefficient of proportionality between the field V(H) and the radial field at the fixed point of XII taken with an opposite sign. In particular. for arbitrary large Gondition A cannot be if the point (x0, p°) is the fired point of the Harnilionian vector field V (H). This affirmation leads to the algebraic unsolvability of the problem of the local
classification of Hamiltonian functions in a neighbourhood of a fixed point of the corresponding Hamiltonian vector field V(H) (see V.V. Lychagin [Ly 1]).
Similarly to the previous subsection, we shall reduce the operator H to the simplest form in two stages. First stage (reduction of the principal swnbol). First, we prove the following affirmation.
Proposition 2. If the contact Hamiltonian functions h0 and h1 satisfr Condition A have equal qiwdratk parts at the point (x0, p°), then there exists a contact diffeomorphisni g such that g ((Xe,
pOX))
= (Xe.
pOX)
g*(hi) = h0.
(46) (47)
Proof Similarly to the previous subsection, we search for a set of contact diffeomorphisms g, with a fixed point (xo. pO*) such that = Xh4). If we realize as a translation along the trajectories of a contact vector field X, with a contact Hamiltonian function f,, then we obtain equation (19) for Sf,:
Xh(f,) —
f, =
— h1.
(48)
We point Out that the difference between equations (48) and (19) is that the vector field X,,, in equation (48) vanishes at the point (Xe, pOX) for all values of:. Since ho and h, have equal quadratic parts, we see that the linear part of the vector field X,,, does not depend on r and is equal to the linear part of both and Xh.
Let (z'
be
a coordinate system in a neighbourhood of the point
(xe, pOX) such that z = 0 at this point. Then the vector field Xh, can be represented
in the form
=
az
+
Y,.
(49)
2. Microlocal classification of pseudodifferential operators
187
where all the coefficients of the vector field Y, are of the order 0 (lzj2). and B = II B/Il is the matrix of the linear part of the vector field X,,. Thus, equation (48) can be rewritten in the form
\
/
azJ
where a(z) = 0 (IzI ), b(z) = 0 (
= Y,f,
+b(z).
The solvability of equation (50) is a
consequence of the following two affirmations.
Lemma 4. If the matrix B satisfies Condition A, then equation (50) has a solution
f, = > in the algebra of the formal power series of z. the condition Re Proposition 3 (V.V. Lychagin, ELy 2]). If the matrix B 0 and b(z) = 0(IzIoc), there exists a solution .ft of equation (50) of order We shall show that the affirmation of Proposition 2 follows from Lemma 4 and Proposition 3; the proof of Lemma 4 is presented below. Due to Lemma 4. there exists a formal solution (51) of equation (50). Denote by f, (z) a smooth function with Taylor series (51). We search for the solution of equation (50) in the form f, = f,(z) + u,(z). Then for u,(z). we obtain the equation of the form (50) with Due to Condition A, there is no eigenvalue A, with ReA = 0. b(z) = Actually, if A, = it, then the number A = —it is also an eigenvalue. Therefore. in this case, we have
N
. A1
+N
+ A0 =
A0
for any number N. This is in contradiction with Condition A. Hence, the obtained equation is solvable due to Proposition 3, and its solution is a smooth function of order O( This completes the proof.
Proof of Lemma 4. Using the linear transformation, we can reduce the matrix B to the upper triangular form B/ = 0 for i <j. Then the diagonal elements B,' of the matrix B are equal to its eigenvalues A, and equation (50) can be written as follows: 2n—I
i=I
a
2n—I
f,Y,f,+a(z)f,+b(z). i.j=1
We shall construct the formal solution of equation (52) by induction. Since b(z) 0( 1z13), the solution of (52) up to the second power of the variables z is f, = 0.
III. Applications to differential equations
188
Suppose that the solution 1(N)
=
up to the power N is already constructed. We search for the solution of (52) up to
the power N +
I
in the form
=
1(N)
= 1(N) + g(!)•
+
By substituting expression (53) for equation (52), we obtain for the function
the equation I2n—I
2is—I
+
I
(54)
/.j=l
is a homogeneous polynomial whose coefficients depend only on the coefficients of the polynomial and, hence, are already known. We denote
where
by
the coefficients of the polynomial F(f"):
=
faZe. I
Now we shall rewrite equation (54) as the system of linear equations for the
unknown coefficients aa of the polynomial g(N+I). To do so, we calculate the action on the left-hand side of equation (54) on the monomial za: of the vector field 2n—I
=
+
(55)
i.j=l
1=1
I >j
where
It is a multiindex (a,,.... ai,,..,) for which at = l,aj = 0 for j
substituting (55) in (54), we obtain a0 aI=N+I
A,a1
—
a,
).o) z0 +
i.j=I
laI=N+I
1=1
I >j
= I
k. By
2. Microlocal classification of pseudodifferential operators
By equating the coefficients at ZU in the latter equality, we obtain the system of equations of Oa: n
aa
A,a1 —
Ao) +
BJ(a1 + l)Ua.
= N + I. (56)
1+1, =
i.j=l
1=1
To investigate the system (56). we introduce the lexicographic order on the set of multiindiccs a with al = N + 1. Namely, we put
(at
<
)
< iff for some number i0, we have a2n_I = fi2n-l. = The system of equations (56) has triangular form with respect to the order (57). This means that the ath equation contains the variables ap for = a (diagonal elements) and for fi < a. Such systems are uniquely solvable iff the diagonal
coefficients do not vanish: 2:s — I
0.
— Ao
The inequalities (58) are valid for all N = 3. 4,... due to Condition A. Thus, 3, and hence, equation (52) has a unique the system (56) is solvable for all N formal solution with vanishing linear and quadratic parts. This proves the lemma.
Remark 1. Similarly, we can prove that the equation
Xu—au=f, where
(59)
the vector field X vanishes at the point : = 0. is solvable if the equality mA1
is not valid for any numbers. If this equality is not valid for
in,
N. then
equation (59) is solvable for any right-hand side f which vanishes up to the order
N-I.
The affirmation of Proposition 2 reduces the problem of the classification of contact Hamiltonian functions in a neighbourhood of a fixed point of their contact vector fields with respect to the group of contact diffeomorphisms to the algebraic problem of the classification of quadratic contact Hamiltonian functions with respect to linear contact transformations. The latter problem has been solved by V.V. Lychagin ELy 21 with the help of results by J. Williamson LWi lJ. Using this result. we obtain the following affirmation.
Theorem 2. Let H be a
Hamiltonian function
and such that XH Cro, pO*) =
0. Assume
on
that h =
HI,,
homogeneous of order I
=i
satisfies
Condition A. Then
[II. Applications to differential equations
190
deterthere exists a homogeneous svmp!ectic transformation G ,nined in a neighbourhood of the point (x0. p°) such that G((xo, p0)) = A (x0, p0) and the function G (H) has the ftrm of the direct sum
G*(H) = Aox'p1 +
p),
where the functions (x. p) are determined by the restriction of the linear part of on the space 1' = Namely, the vector field generates a tenn
(1) each pair of eigen values (A, A0 — A) with A
of the Jordan block corresponding to A; of the suni (60), where k is the (2) each four eigenvalues of the type (A±, Ao — A±), = a ± it generate a term
[a
f(x. p) =
—
+ X*+) Pk+j) + r
+ of the sum (60), k as above; (3) each pair of eigenvalues A —
(a)
f(x, p) =
± i1i generates a term of one of two types
+
Ao
+
PJPk_i+I)
or (b)
f(x. p) =
+
of the sum (60); (4) each cigen value A = (a)
+
generates a term of one of two types
f(x, p) =
x'p1 +
x'pj+i
2. Microlocal classification of pseudodifferential operators
191
or (b)
f(x,p)=
of the sum (60). m1 N0 (the so-called Remark 2. If the condition of the type A is valid for algebraic fixed points), then it is possible to write down the list of the corresponding normal forms which are polynomials of order $ N0 (see V.V. Lychagin and B.Yu. Sternin [LyS 11).
Second stage (reduction of the operator). Similarly to the case of the operator
of principal type, at this stage, we can assume that the principal symbols of the considered operators and H2 are equal to each other in a neighbourhood of the point (Xe, p°). If we apply the procedure described in the previous subsection for the operators of principal type, we shall get equations (23) for the symbols u• of the homogeneous components of the operator U. As in the considered case pO*) = 0, if ro(xo. pO*) 0, we obtain uo(xo. pOS) = 0, which is a conXh(xo. tradition to the ellipticity of the operator U. To overcome this difficulty, we shall construct the operator U of order k (and not of order 0. as above). Then we have
pseudodifferential operators of order k — j. The evident modification of the system (23) is where the Uk_J are
= 0,
Xh(uk) + (rO — k Xi(h))u& Xh(Uk_I) + (r0
— (k — l)Xl(h))uk_l
=
Xh(Uk_j) +
— (k — j)
=
(r0
Now we can choose k in such a way that k
— k X1(h))
.
(x0. p°) =
0,
that is,
= rO(xO. p0) A0
was shown above, following two types: (1) XhU + a u = 0, =b, (as
0). u (x0,
Thus, the equations in the system (61) are of the pOX)
0,
a(xo,p°)=j,
a (x0,
= 0;
integerj >0.
III. Applications to differential equations
192
Similar to the proof of Lemma 4. the conditions of formal solvability of equations
(I) and (2) have the form
aA1 + j
0
aI=N
Proposition 3 allows us to claim that equations (I) and (2) for any N = 1. 2 are solvable in smxflh functions if the linear part of the contact vector field X,, satisfies the following condition.
0 such that E
Condition B. For any integers m1
in,
1, the inequality
2n—2
mlxi holds (here A0, A1 field X,1.A0 = X1(h)).
0
is the spectrum of the linear part of the contact vector
We note that Condition B is a consequence of Condition A. Actually, if there m1 > I, mn1A1 = 0. then we exists a set of numbers m0 such that
have 2it— 2
(Nmno+
l)Ao+
=A().
N + I, that is, Condition A is not valid. Thus, we have derived the following affirmation.
and (N in0 + I) +
Theorem 3. Let H he a first-order
operator with the principal svntho! H (x, p) saiisfving condition A at the point (xO. p°). l'hen the germ of the operator H at the point (xe. p0) belongs to an orbit determined by one of the normal forms. More precisely, there exists an elliptic HO such that H j,ç a normal form (by normal form, we mean here a pseudodifferential operator with a symbol described in Theorem 2).
3. Equations of subprincipal type 3.1 Statement of the problem In this section, we shall consider the equation
ufu=H
3. Equations of subprincipal type
193
with H being a differential operator of order M in the space R" with smooth coefficients, *
H=>
aa(x)
Ia I
(_j
One can easily generalize the subsequent results by considering differential operators on a smooth manifold M (the case M = R" is chosen for simplicity). be the principal symbol of the Let H(x. p) = aa(x) p) = operator (2), and let with aa(x) E
p)
=
p) —
i
I a2Hm(x,p) —
2
axap
+
I
V(H) In v.
—
2
= be its subprincipal symbol. (Here v is the density of some fixed measure on R"). It is well known that the functions H(x, p) and HSUb(X, p) are well-defined on (i.e., they demonstrate appropriate of the space the cotangent bundle behaviour under variable changes). S*Rfl = be a fixed point of the contact distribution IN Let corresponding to the Hamiltonian function H(x, p) (of course, we suppose a0 char H). Evidently, in a small neighbourhood U of the point (x0, p°) E corresponding to the point cr0 E S*RSl. there exists a factorization
H(x, p) = Q(x, p) H1(x, p), where Q(x, p) does not vanish at (xo, p°), and H1(x, p) is a homogeneous function of order I (evidently, Q is a homogeneous function of order m — 1). Then (see Section 1.5) the vector field XH1 determines a contact distribution I,, in any point of U. Denote by (Lin XHJ)a() the linear pan of the vector field X11 at its fixed poiflt a0. We recall (see Section 2) that (Lin X111 can be represented as the linear operator (Lin XII
Tati(S*Rhi).
au
being an arbitrary vector field on a neighbourhood of a0 such that p) are Suppose that H = Q(x, p) H1(x, p) = Q'(x, p) two factonzations of the form (4). then, evidently. H1 = Q. H, with Q being a nonvanishing homogeneous function of order 0 in a neighbourhood of the point with Y E
III. Applications to differential equations
194
a0.
Since ao E char H. (XH
= 0,
we have
X111
up
= QXH;
to the terms which have at least second-order zero at
Y = iP.
Then
= IP,
= (Y(Q).X11;
= QIIP,XHL =
Y.
So, we see that for different factorizations (4), the linear parts of the corresponding vector fields XH, and XH; are identical up to a scalar factor. To avoid this ambiguity, we fix some factorization of the form (4) in a neighbourhood of each fixed 0 in a neighbourhood point a() of the contact distribution 1,,. For example, if
of ao, we put H(x, p) =
(x. p). One can check, however, that all the essential notions below depend only on the residue class of (Lin X11 modulo scalar factors. is different from that in A2??.i (the enumeration of We denote by A1 of the contact field Xy Section 2) the eigenvalues of the linear part (Lin XH at an arbitrary fixed point of the contact distribution 'H. We suppose that these eigenvalues are enumerated is such a way that A1 = X1 (h) (or, in other words, that V(H1) = —A1 and = A1 for j = 2,..., n (see Proposition 2.1); + as above, we use the representation = I of the space S*Rfl in a neighbourhood of a0. We also suppose that the numbers A0 are real and distinct. If the there are more than one fixed point a1 aN, we denote by spectrum Of(LfflXH)aj at the point a1. j = I N. We assume that (x. p) are real-valued functions in a (i) The Hamiltonian functions H(x, p), neighbourhood of their characteristic set char H. The vector field V(H) does not vanish in char H. (ii) Equation (I) belongs to the principal type at any point a e except for aN of fixed points of the contact distribution Xy; these a finite number ai points will be called singular points. (iii) At any singular point a1, the contact vector field X,, satisfies Condition A of Section 2. This means that
m1
(iv) The distribution
0,
3.
has no finite motions. This means that
3. Equations of subprincipal type
195
— first, for any compact set K C S*M, each trajectory of with its origin in K either leaves K or tends to some fixed point of — and, second, there exists no closed system of trajectories of ill passing through any fixed points.
Remark 1. In the case N =
0 (i.e., when the distribution 1,, has no fixed points), the condition (iv) is the usual condition of absence of finite motions (see Section
I).
We need some auxiliary notations to formulate the main result of this section.
As was shown in the previous section, the operator (2) can be reduced to a normal form, = —i with the help of Fourier integral operators in a neighbourhood of each fixed point aj, j = I, . . , N. Since for constructing a normal form of the first type (see .
Theorem 2.2) of the previous section, we can choose one number A from each pair such that = A1. We can assume that for fixed j, all numbers AL in (7) have the sign coinciding with that of the number anyj=1,2 N,
=
A'I
,
=
;
We put for
0
where
=
=
We use the
=
notation Smin
= max (5J + ky), J
0max J
with
being the order of the Fourier integral operator U which reduces the
operator (2) to the normal form (7) (see previous section).
Let K be an arbitrary compact set in R". For any pair (a, s) with o S > Smjn, we consider the operator Li
n
''sn with the domain D a E = lu E ator (10) is a densely-defined closed operator. Let •
s.fT
comp''Re'
u—a I
comp'
Evidently, the oper-
lii. Applications to differential equations
196
be the adjoint operator. We recall that the domain of the operator (1 I) consists of
such that
the functions f E
f>l
(12)
< eli u 11(7
We use the notation
for any function u E
=
=
(1' E
suppu C K).
0.
(13)
Theorem 1 (see I S5 I). Suppose tiuit the eontht,ons (I )—( iv) above are satisfied. Thei: for (lilY S > 5mifl• (1 < (a) tile space Nb,, (K) is finiu'—dinwnsional: sue/i 1/lot f_LN%(7 (K) (i.e.. (f. u) = Ofur OflV (h) for our jimetion f such that i' E N.,.,1 (K )), there exists a function ii E
111 = f
of K. and the inequality
in a
ii
/101(15
for our
e
*
11(7
IL
) with C.,1, independent
of f.
The proof of Theorem I will be carried out in a way similar to that for Theorem I. I. The next subsection deals with the construction of a semiglobal regularizer for the operator H. In the remaining subsections, we carry out the proof of Theorem I with emphasis on the differences between this case and the case of the operators of principal type considered in Section I above.
3.2 Solutions of model equations In this subsection, we consider the model equations of the form (7). For definiteness, we suppose that all numbers A, are positive. Consider the operators
(I 1(x) = j )(x)
=
i
J
f
di f(IAX) —,
3. Equations of subprincipal type where tAx
=
197
By the change of the variables t =
these
operators can be rewritten in the form ()
(Doll 1(x) = i j
f(eATx)dr,
f
f(eATx)dr,
J(x) = I
?Tx = (eAItxl is a one-parameter group corresponding to the vector field (7). We note that the corresponding Hamiltonian flow determined by the symbol E7..1 AkxAp* (we omit here the index j) of the operator (7) has the where
form
=
P1 =
j = 1.2....,
e_Alt
Hence, the operator (18) corresponds to the Hamiltonian flow for negative values of r. and the operator (19) corresponds to the Hamiltonian flow for positive values oft. Of course, for A1 <0, the operators (18) and (19) will interchange their roles. We point out that the operator (16) is well-defined only for the functions f(x) such that f(0) = 0 (this condition is nesessary for the solvability of the equation Hu = in smooth functions). We want to consider the operator (16) in the spaces
I
one can easily check that for these .s, the restriction operator 1(x) i-÷ f(0) is well-defined. However, one has to regularize the operator (16) so as to obtain the operator acting in the spaces (21). This regularization can be performed in the following way. We put
for .c > SO =
o PIfl.
R0iji =
(22)
where Piji = 1(x) — f(0). The operator (17) needs no regularization; so we use the notation = The direct calculation gives I o R0(fj = f(x)
ii
—
1(0) = PjfJ;
(23)
= 1(x).
(24)
We note that the operator — I acts from the space (for s > So) to the space for arbitrary s' and, hence, is an operator of infinite negative order. Denoting it by Q. we can rewrite (23) in the form
Ho Rn[f I = f(x) + is an operator of infinite negative order for s > scale Hi!). where
I
+ Q,
So (but
(25)
not on the entire
III. Applications to differential equations
Proposition 1. The operators R0.
defined above are bounded operators in the
spaces R0
s > so =
:
,
s <s1 =
—+
(26)
2Amin .
(27)
mm
preserves the supports, supp
Moreover, the operator
C supp 1.
Proof. First, we consider the operator (26),
Roll]
[f(tAx) — 1(0)]
=
there exists a constant c such
We shall prove that for any function %fr(x) E the inequality that for any f E IL
ii
C
(28)
II
I
IL
holds. Evidently, without loss of generality, we can suppose that 1(0) = 0. Indeed,
let x(x) E
be a smooth function such that x(x) = 1 in the ball KR,
C KR. Then
RO[fJ =
=
f(x) and
f f
— 1(0)1
[f(lAx) — f(0)]
= *(x) f
x(x) [f(x) — f(0)],
II f(x) IL
We write
Illu
ü(p)
=
f
(29)
a Fourier transform of the function u(x). Since u IL c (II u 'IL2 + Denote Ro[fJ Ro[f] ll,, and flu lila by u. To obtain the estimate of II u we shall use the boundedness of the embedding where
is
II
II
Cu(RhJ),
0< a <min(l,s_
with Ca(Rhl) being the HOlder space with the norm
[UJa = sup
u(x)—u(y) Ix —
3. Equations of subprincipal type
Since f(O) =
II
0,
u iiL
we have
=J
*(x)
f
f(IAX)
C1 f *(x) f C2 f
dx
lila
fJ
dx . [ft2
d:}
Ill
C3
If
(32)
since the integral in curly brackets converges. To obtain the estimate for Ill u Il', we note that
f(fAx) (p) =
(33)
Hence (using the condition f(0) = 0), we see that ü(p) is a distribution with the singularity (at the origin) of a type of generalized-homogeneous function of p A,1) of order —JAI. Using the inequality of the type (A1
we have
=J
Jf
f
dp dp
+1
dp
ff +1
dp dp
J
= II [(I
—
u(x)
+ II *(x) lfrI'u(x)
It [(I
—
u(x)
+ Ill u
(34)
= —ii). The first term on the right-hand side of the relation (34) can be estimated similarly to above. The estimate for u for sufficiently large N (here
111. Applications to differential equations
200
the second term is Ilic
f f
{f 1p12511P12dpj
0
the integral over t converges for s > (34), and (35), we obtain the estimate 'i4r k0[f1 since
II
cli .f
IL.
(35)
Collecting the estimates (32).
cf
for s >
The
boundedness of the operator (26) follows now from the relation
= *(x)
J
=
j [f(iXx — f(0)]
f(tAx) —01
di
which is valid for p. E and 1(0 Let us now estimate the operator Suppose that 0 < (we put u = due to formula (33). we have II
f
u
II
k0[f],
=
s
<
In this case
f(tAx) 115
I
+ p12)'
f
(1 + ,Ap12)c IJ(p)12 dPj
di
cli! IL. (36)
If c <0. then similarly to (36), we obtain II
Since s
u IL
=
f
{f
<0, we can omit the factor lu
IL
=
f
1-(IAHI)/2
(1
If(p)12 dp}
di.
in the integrand of the inner integral,
11(1 +
Finally we note that the inclusion supp
the proof of the proposition.
+
p12)' IJ(P)12dP}
di =
II 1115.
C supp f is evident. This completes
0
3. Equations of subprincipal type
of this film of fronts of the operators R0 and
Lemma 1. Let P (x,
[lie of to We shall use the following affirmation.
operator of order 0. Then we have
be a
ok0} (f) = i f
(P0R%}(f) Proof
Suppose
0
(37)
=i
(38)
f(O) = 0. Then due to formula (33), we have
P [f(tAx)] = Hence,
201
1"
j') f} (i'x).
for the composition P o k0, we obtain o R0}
(f) =
i
f
f) (t?.x)
= The proof of formula (38) is just the same.
We denote by r —* +00 and by T
a set of trajectories of V(H) which tend to the fixed point for a set of such trajectories which tend to the fixed point for
—+ —00.
Then there exists a Lemma 2. Let U be a (small) neighbourhood of the set such that if WF(f) C V. then WF(R0 f) C U. neighbourhood V of the set there exists a neighbourhood conversely, for any neighbourhood V of the set
U of the set
CV.
such that :JWF(f) CU, then
Proof We shall consider only the proof for the operator R0 (the proof of this is exactly the same). If U is a neighbourhood of affirmation for the operator then we choose V in such a way that every trajectory y(r) of V(H) which belongs WF(f) C V. then for to U for r = 0 is contained in V for t <0. If f of order 0 such that supp Q(x, p) C V. some pseudodifferential operator Q(x, we have Q(x, fr) f f(mod Ce"). If the pseudodifferential operator P(x, ji) now has its support outside U, we have due to Lemma 1,
P(x, fr) (R0 f)
P(x,
j
modulo
o R0 0 Q(x.
(x.
o Q(x,
fJ (tAx)dT
(39)
the C°° functions. Since the support of the symbol of the operator
P(x, j,) is contained in the set of trajectories which have their initial point
III. Applications to differential equations
202
outside U. we see that the supports of the symbols of the operators P(x, do not intersect each other. Hence, function (39) is infinitely smooth and Q(x, for any P(x, such that supp P(x, p) fl U = 0. This means that WF(R0) C U. This completes the proof of the lemma. 0
3.3
Construction of a semiglobal regularizer
To describe the procedure of constructing a semiglobal regularizer, we shall define the notion of the wave-front set of an operator.
Definition 1. For any operator we define the wave-fmnt set of QWF(Q) as a union of the wave-front sets WF(Qf) for all distributions f
We shall consider here only the wave fronts of order — I; thus, the symbol in the sense of Section 1 of this chapter. WF(Q) will mean Let K be an arbitrary compact set in and U be a neighbourhood of K. All the constructions below are carried out in U (in particular, we consider only those fixed points which are projected onto U). We shall construct a semiglobal rcgularizer via the "step-by-step" procedure. In the first step, we shall construct a regularizer in a neighbourhood of the union of the fixed points (the singular part of a regularizer). Namely, we construct an such that operator
ujok,=i+&. the V1 are some neighbourhoods of the fixed points which are supposed to be sufficiently small: we also denote by V' a system of neighbourhoods of the Vi'. fixed points a1 such that V, The operator R, will be constructed in the form Here
=
N
(40) with
being a regularizer in a neighbourhood of the point a
(j =
1
N),
fl V1 = 0 for any i (we suppose that j). To globalize our regularizer. We choose the direction on each trajectory on the Hamiltonian flow. For definiteness, we specify that the positive direction on each
trajectory is given by the direction of the Hamiltonian vector field V(H). This direction will be called positive.
3. Equations of subprincipal type
203
was shown in Section 2, there exist Fourier integral operators UW and
As
of order o
and
and —k1, respectively, such that the symbols of the compositions o are equal to I in the neighbourhood Vi', and
g ((Jo)
=a
fr1—m o H o
(42)
are given by formula (a is a full symbol), where is the operator (2), and the (7) for each j = 1, 2, ..., N. Recall here that we assume that P1 does not vanish in the neighbourhood Vi'. We put (j(J)
=
(43)
for those fixed points for which the positive direction on the trajectory induces the direction determined by and
=
(44)
for those fixed points for which the positive direction on the trajectory is opposite We can suppose that the symbol of operator to the direction induced by
(42) equals zero outside the neighbourhood V and that the equality (42) takes V1. we place in V1. Thus, modulo operators with a wave front contained in have
= kOU(J)ok
frl-rn
_frrn_10f1
= =
k o
o
=
o
to formula (24). We carried out the calculations for the case (43); the case of the operator (44) is considered exactly in the same way. In the following steps, we shall construct an operator (the nonsingular part of a regularizer) such that due
(45)
where è is a pseudodifferential operator with the symbol e1(x, p) such that
and
1,
(46)
WF(Q)flir'(K)=e.
(47)
is an operator of order 0 such that
If such an operator is constructed, then the operator (48)
III. Applications to differential equations
is a semiglohal regularizer over the compact set K. Actually, we have
Ii 0k = —H (H — I) = ii0R, —(è+Q)(HoR, —1)
=
(49)
where WF(Q') fl ir'(K) = S*M\
It
0, since e(x, p)
1 on WF(H o
—
1)
C
is evident (exactly in the same way as in Section 1) that in subject to condition (45). it is sufficient for any
order to construct the operator
point a E charH\ II 0 e0(x. p)
to construct an operator
= — —
y
such that
+ Qa' I
atthepointa,
0
outside a neighbourhood of the point a,
0, with ea(x. p) being a positive function.
Repeating the considerations of Section I. for each segment of the trajectory y = y [a. (a being the originating point of y, and being its endpoint with respect to the direction choosen above), we can construct an operator R[y] such that
H o RIyJ = + Q, with ?.,, being a pseudodifferential operator of order 0 such that its symbol equals 1 at a and equals 0 outside some neighbourhood of a p) > 0), and WF(Q) is concentrated in a sufficiently small neighbourhood of This operator will be used in the inductive process described below. Let a be a point of char H\ U7=1 Consider the trajectory y which originates from a in the positive direction. Due to the condition (iv) of Subsection 1, there can be two different cases.
Case 1. The trajectory y leads outside the compact set K. In this case, it
is
sufficient to put
= R[y[a, $1],
(52)
with fi being the point on y which lies outside the set (K). Actually, due to the relation (51), the conditions (50) are fulfilled for the operator (52). Case 2. The trajectory y tends (in the positive direction) to some fixed point In this case, the operator kna will be (i.e., y is an incoming trajectorY for constructed with the help of an inductible process described below.
3. Equations of subprincipal type
205
the point of the first level. We consider a set of We call the fixed point in the negative direction (i.e., the set of the trajectories which tend to the point outgoing trajectories). All fixed points for which these trajectories are incoming ones, are called the fixed points of the second level. The third, fourth, etc.. levels are defined in an analogous way. Due to the condition (iv), each fixed point can belong to only one level; it is evident that the set of levels is finite. We denote by Lev(j) the set of the points of the jth level; so we have Lev(l) = j = 0, I We shall construct the operators k inductively such that (53) where
is as above and W F( Q(J) is contained in the union of the set ir - '(R" \ K)
and of some neighbourhood of the set of input trajectories of the points of the (k + I )-th level. The situation is as shown in Fig. 4.
fixed point
Finite number of levels
trajectory
. non - singular point
fixed point
fixed point
Lev (1)
Lev (2)
Figure 4
Now consider the set of outgoing trajectories for points of the last level L(K). We choose a All these trajectories have their endpoints in the set neighbourhood U of the set of endpoints of these trajectories lying in ,r — '(Re \ K). of the set of outgoing trajectories for Evidently, there exists a neighbourhood fixed points of the kth level such that for all trajectories originating from their endpoints lie in U. We note that if the endpoint of a trajectory tends to some point of the outgoing trajectory for some fixed point, then its origin will tend to some point of some incoming trajectory. Hence, there exists a neighbourhood of incoming trajectories for fixed points of the kth level such that all trajectories
III. Applications to differential equations
206
have their endpoints in Analogously, we construct the of the set of outgoing trajectories of fixed points of have their endthe (k — 1)-st level such that all trajectories originating from points in and all trajectories originating from have their endpoints outside the set n'(K). Inductively, we construct the collection of the neighbourhoods for j = 1, 2 k and the neighbourhood U of the initial nonsingular point a such that (i) all trajectories originating from U have their endpoints in have their endpoints in j = I. .. , k; (ii) all trajectories originating from (iii) all trajectories originating from have their endpoints in = originating from
neighbourhoods Ut'.
.
k—I;
have their endpoints outside
(iv) all trajectories originating from (v) all trajectories originating from
•
have their endpoints in U (see Fig. 5).
fixed point
2
out
3Td fixed
point
—2
ucut
point
Figure 5
Now we can formulate the conditions for the operators Q'1' in (53) more exactly. We require that C
(54)
3. Equations of subprincipal type
207
note that the operator can be constructed absolutely analogously to Case 1. Now suppose that the operator j 0 is already constructed. Then we construct the operator Re'' in two stages. At the first stage. we put We
=
o (H o
—
— I).
(55)
Using Lemma 2. we can check (analogously to formula (49)) that
ii where
= è0 +
(56)
C
At the second stage, we construct the operator
ii o
=
such that
+
(57)
U where and the symbol C p) of the (firstorder) pseudodifferential operator is equal to I in Such an operator can easily be constructed with the help of considerations analogous to those of Section
I; this operator is simply a sum of operators R[yj such that the symbols of the corresponding operators è>, (see (51)) form a partition of unity on Now we put —
=
(H a
— I).
(58)
One can now easily see that the operator satisfies the conditions (53) and (54) with j + I instead of j. Now we can see that the operators = satisfy all conditions (50), since = 0 (there are no points of the (k + 1)-st level). The remaining part of the construction of the operator satisfying the relation (45) and, hence, of the operator R is quite analogous to the constructions of Section I and is left to the reader. The estimates of the obtained regularizer directly follow from the estimates of the Fourier integral operators. the estimates of the operators (carried out in Section I), and the estimates of the operators given in Proposition I. We shall formulate the obtained result.
Proposition 2. Under the conditions of Theorem I. (here exists an operator
R:
—*
such that
0 (we suppose, of course. thats > smjn,a
III. Applications to differential equations
208
3.4
ExIstence theorem for equations of subprincipal type
We note here that the proof of the existence theorem (Theorem I) is exactly the same u.s the proof of Theorem I in Section I. We mention here only the main differences between the cases of principal and subprincipal types. The first difference is that in the case when K consists of a single point, the space is not equal to the zero space. in fact, even for the model equation
= f(x), is a one-dimensional space with the generator This example shows also that in contrast with the equations of principal type. the space can contain functions which are not infinitely smooth. The second difference is that we have proved the solvability of equation (I) not for the entire Sobolev scale but only under some restrictions on the indices s, a of the operator namely for s > 0 < Umas. In general. these restrictions are exact, as can be shown in the following example. Let ii be the operator on a two-dimensional sphere S2 corresponding to the vector field X, which has exactly two fixed points N (north pole of the sphere) and S (south pole of the sphere). In the coordinate system given by the stereographic projection. this vector field can be written in the form the space
ax
(61)
with the numbers A and p being positive and incommensurable in a neighbourhood of the point N. In a neighbourhood of the point S. this vector field possesses a
local representation (62) The existence of such a vector field is evident.
Let us show that for the arbitrary smooth right-hand side of the equation
Xu=f.
(63)
the number of conditions which have to be posed on f for equation (63) to have a smooth solution is infinite. For simplicity, we restrict ourselves to the subspace of the right-hand sides which are equal to zero in some neighbourhoods of the points N. S. Suppose there exists a smooth solution of equation (63). Let UN and U5 be some neighbourhoods of points N and S, respectively, such that the vector field X has the form (61) and (62) in these neighbourhoods. Due to (63), the function of the type (A, of order 0. Hence, it ii is a generalized homogeneous function is equal to some constant in these neighbourhoods: a c. Let = a — c. Then
3. Equations of subprincipal type
209
f(t)dt, where we integrate along the trajectories of the vector field X. with i being a parameter along the trajectory. In particular. this representation is u(a) =
valid in a neighbourhood UN, where the function a has to be equal to some constant also. Hence, for smoothness of the solution a of equation (63). it is nesessary that the equality
jf(t)dt = f f(i)dt be valid for any pair of trajectories of X leading from S to N. The latter affirmation completes the proof. However, under some additional conditions, we can prove a theorem analogous
to Theorem I. which is valid in the part of the space scale H We shall call the sign of the number X, the signature of the fixed point. If the signatures of all the fixed points are identical, then (changing the sign of operator H if nesessary) we can use in the regularizer only the operators of the type R0 in the neighbourhoods of all fixed points. This consideration shows the validity of the following theorem.
Theorem 2. Let all conditions of Theorem I be valid and all signatures of the for any compact set K C R", there exists a bounded lived points be identical.
operator
k: Q:
H o R = I + Q, and
= øfors
K)
>
The corresponding existence theorem is also valid. To finish the discussion of Theorem I, we note that we considered equation (I) in the Sobolev spaces H and obtained the regularizer and the existence theorem for c< s> In general, Of course, in the Sobolev space scale. these restrictions are exact. However, the question arises whether one can find a class of spaces such that the regularizer has (m — I )-th order as it takes place for the equations of principal type. We shall not discuss these questions in full generality but we shall write down such a class of spaces for model equations. These spaces are anisotropic with respect to different variables; this feature makes the difference between these spaces and the spaces H'. More exactly, the smoothness of the
elements of such spaces with respect to the variable xk is determined by the corresponding eigenvalue AA of the contact vector field, or, which is the same, by the corresponding coefficient in the normal form (7). We put for f e
ii! ilL = 1(1 +
12/A1
+
If(p)12 dp
III. Applications to differential equations
and define the space H1A(R'7) as a closure of the space with respect to this norm. The following affirmations can be proved analogously to Proposition 1.
Theorem 3. The operator R0: !IcA(R't) is bounded for s>
PropositIon 3. The operator
HA(R) is bounded for s <
References
IA
1)
IAn I) (An 2)
lAr II
Operators to Canonical Forms. J. Alinhac. S.. On the Reduction of Different. Equal. 31 no.2 (1979). 165—182. Anderason. K.G, Analwic Wave Front Sets far Solutions of Linear Differential Equations of Principal 73pe. Trans. Amer. Math. Soc. 177 (1973), 1-27. . Propagation pf Analyzicity for Solutions of Differential Equations of Principal l'tpe. Bull. Amer. Math. Soc. 78 no. 3 (1972). 479—482. Vi.. Normal Forms for Func:ion.s Near Degenerate Critical Points, the Wevi Groups of Ak. 1¾, Ek. and Lagrangian Singularities. Functional Anal. AppI. 6 (1973). 254—272.
lAr 2] jAr 31
(ArGi I)
.Mathematical Methods of Classical Mechanics, vol. 60, Springer. Graduate Texts Math.. 1978. . Lagrange and Legendre Cobordism. Functional Anal. AppI. 14(1981). 167—177. 252—260(1981).
Arnold. V.1.. Givenial'. A.B.. Svmplectic geometry. Itogi Nauki Tekh.. Ser. Sovrem.
Probl. Mat., Fundam. Napravleniya. vol. 4. 1985 [Russian). pp. 7—139. [ArVaGus I) Arnol'd, V.1.. Varchenko. A.N.. Gusejn-Zade. S.M., Singularities of Differentiable Mappings, vol. 82. Monographs Math.. Birkhfluser. Boston. 1985. Duistermaat. JJ.. Oscillatory integrals. Lagrange Immersions and Unfolding of SingularID I) ities. Comm. Pure AppI. Math. 27 no. 2 (1974). 207—281. . Fourier integral Operators. Lecture Notes. Courant Inst. Math. Sciences. New (D 2) York 1973.
IDH II
Duistermaat. ii.. Hormander. L.. Fourier integral Operators II, Acts Math. 128 (1972).
(E I)
Egorov. Yu.V.. On the Solvability of D(fferential Equations with Simple Characteristics.
183—269.
Russian Math. Surveys 26: 2 (1971). 113—130.
- canonical Transformations and
jE 2)
Operators, Trans. Moscow
Math. Soc. 24 (1974). 1—28.
. Linear
IE 3)
Equation.v of Principal
(Russian). Nauka, Moscow.
1984.
[F II IF 2)
(Fo I) [Fo 2) (GSha I)
Fedoryuk. M.V.. The Stationary Phase Method and Pseadodjfferen,ial Operators. Russian Math. Surveys 26: 1(1971), 65—I IS.
, Singularities of Kernels of Fourier Integral Operators and Asymptolics of the Solution of Mixed Prnblems. Russian Math Surveys 32: 6 (1977). 67—120. Fok. V.A.. On the ('anonica! Transformation in Classical and Quantum Mechanics. Russian). Vests. Leningr. Univ. 16(1959). 67. ,Basics of Quantum Mechanics. Nauka. Moscow. 1976 (Russian). Ge1 land, I.M.. Shapiro. Z.Ya.. Homogeneous Functions and Applications. (Russian). Uspekhi Mat. Nauk 10 no. 3 (1955). 3—70.
IGShi I)
Gel'fand. l.M.. Shilov, G.E., Generaliztd Functions. Academic Press. New York. 1964.
jGuSch I)
Guillemin. V.. Schaeffer. D.. Fourier integral Operators from the Radon Transform Point of View. Proc. Symp. Pure Math. 27 no. Part 2 (1975). 207—300.
1967. 1968.
References
212
, On a certain Class of Fuchsian Partial Differential Equations. Duke Math. J.
IGuSch 21
44 no. 1(1977). 157—199. lGuSch
. Maskn Theory and Singularities. M.I.T.. Cambrige. MA. 1972 (mimeographed
31 notes).
(GuSt I
II
H II
Guillemin. V.. Stcmberg. S.. Geometrir Asvniptoiics. Math. Surv. 14 (1977). Hormandcr. L.. The u/u.s of Fourier Integral Operators. Ann. Math. Stud. 70 (1971). 33—57.
IH 21
. TheAnafrsis of Linear Partial D4fferential Operators 1,11.111./V. Springer-Verlag. Berlin—Heidelberg—New York—Tokyo. 1983—1985.
(H 31
. Fourier Integral Operators. Ada Math. 127 (1971). 79—183. Leray. J.. Prnhlhnu' de 1. Bull. Soc. Math. Fr. 85 (1957). 389—429. . Pmhlhme di' Cauchv II. Bull. Soc. Math. Fr. 86 (1958). 75—96. . Prnhlhsuede cauchs Ill. Bull. Soc. Math. Fr. 87(1959). 81—180. . Problè,nede Cauehy IV. Bull. Soc. Math. Fr. 90 (1962). 39-156. . Lectures on hyperbolic Eqiuuions with Variable Coeffwient.s. Institute for Advanced Study. Princeton. 1952. . Hyperbolic Diffi'rrn:ial Equations. Lecture Notes, Institute ror Advanced Study. Princeton. 1951.1952. . Solutions Asvmp:otiq:w.s di's Equations aux l'artielle's. Cony. Intern. Phys. Math.. Rome. 1972. Solutions Asymptoliques ci Grnupe Svsnplec:ique. Springer Lecture Notes 459
IL II IL 21 IL 31 IL 41 IL 5J IL 61
IL 7J IL 81
.
(1975). 73—97. IL 91
. Solutions Asyrnp:oliques ci Physique Matluiniutique. Géométrie Symplectique Ct Physique Mathématique Colloq. mi. (Aix-en-Provence 1974) CNRS no. 237 (1975). 253—275.
IL 10 I
. Analyse Ii:grangienne ci Mecanique Quantique (Notion,s upjx:renteks a relies dr dêveloppe:nent asymplotique ci d'indice de Maslov). R.C.P. 25 (1978).
ILy I(
Lychagin, V.V.. Local Cla.cslficatioa of Nonlinear First Order Partial l)iffrreutial Equa-
I Ly 21
I
Ly 31
ILyS II
M II I M 21
IM 31 IM 41 I MF
I MeSj II I MeSj 21
IMclU
tions. Russian Math. Surveys 30: 1(1975). 105—175. . On Sufficient Orbits of a Group of Contact Diffeornorphisins. Matem. Sbomik 104(146) (1977). 148—270. . Spectral Sequences and Normal Forms Lie Algebras of Vector Fields. Russian Math. Surveys 38: 5 (1983). 152—153. Lychagin. V.V.. Sternin. B. Vu.. (in the Micro/oral Structure Operators. Math. USSR — Sb. 56(1987). 515—527. Maslov, V.P.. Theory of Perturbations and Asrn,pwlir Methods. Dunod. Paris. 1972. . Operational Methods. Mir. Moscow. 1976. . Nonstandard Characteristics in Asymptotic Problems, Russian Math. .Surveys 38:6(1983). 1—42. . Asymptotic Methods for :1w Solution of P.seudodijJeren:ial Equatiosis. Nauka. Moscow. 1987 IRussiani. Maslov. V.P.. Fcdoryuk. M.V.. Se,nicla.csical Approximation in Quantum Mechanir.c. Rei. del Publishing Comp.. Dordrevht. 1981. Melin. A.. Sjöstrand. J.. Fourier Integral Operators with Complex-salued Plia.ss' Functions. Sponger Lecture Notes 459 (1975). 120—223.
. Fourier Integral Operators with Complex Functions and Parametrics for an Interior Boundary Value Probletn. Coinni. in POE 1(4) (1976). 313—400. Melrose. R.B.. Uhlman. G.A.. Lagrangian Intersection and the cauclir Problem. Comm. Pure AppI. Math. 32 no. 4 (1979). 483—519.
References IM1ShS JNOsSSh I
INi 1)
I
213
Mishchenko. A.S.. Shatalov, V.E., Sternin. B.Yu.. Lagrangian Maniftilds and the Masiov Operator. Springer-Verlag. Berlin—Heidelberg 1990. Naznikinskij. V.E.. Oshmyan. V.G.. Sternin. B.Yu.. Shatalov. V.E.. Fourier integral 0prrawr.c and the Canonical Operator. Russ. Math. 36 no. 2 (1981). 93—161. Nirenberg, L.. Pseudod:fferensial Operators. Proc. Symp in Pure Math. 16 (1970), 149- 167.
tRh
Oshima. 1.. Sin gulariiie.c in ('ontac: Geon:etrv and Degenerate Pseudothfferenzial Equations. J. Fac. Sci. Univ. Tokyo 1(1974). 43—83. Radon, J.. Uber Be's:i,nnzung son Funkrionen dutch ihre Integraiwerte kings gewisser MannigfalsigA-elten. Ber. Math.-Phys. KI. Sachs. Akad. Wiss. Leipzig 69(1917), 262—277. de Rham. G.. Varisié différentiables. Hermanu. Paris. 1955.
ISaKKa II
Sato. M.. Kawai. T.. Kashiwara. M.. Mirrnfunciion.c and Pseudodifferen:ial Equations.
10 I I R II
(Shu I Si II
1St II IS
IS 21
IS 3J (S 41
Springer Lecture Notes 287 (1973). 263—529. Shuhin. M.A.. Pseudodifferenthil Operators and Spectral Tlworv. Springer-Verlag. Berlin — Heidelberg 1987. Sjostrund. J.. introduction to Psendod,ffe'rential and Fourier integral Operators. Plenum Press. New York. London, 1977. Steinberg. S.. Lectures in Differential Geometry. Prentice-Hall. Englewood Cliffs. 1964. Sternin. B.Yu.. On the Micro.L,jcal Structure of Differential Operators in the Neighbour. hood of a S:atio,zarr Point, IRussiani, Uspvkhi Mat. Nauk 32 no. 6 (1977). 235—236. . Maslovc Canonical Operator Method. Complex Analysis and its Applications. Kiev. 1978 (Russian I. pp. 47—113. . On the Regulurizazion of Equations ofSuhprincipal 7t'pe. Russian Math. Surveys 37: 2 11982). 261—262. . On Differential Equations of Subprincipal Type. Soviet Math. Dokl. 26 (1982). 22 1—223.
IS SI
,
Differential Equations of Subprincipal
Math, USSR — Sb. 53 (1986).
37-67.
ISSh II
Stemin. B. Yu.. Shatalov. V.E.. On a Method of So! s'ing Equations with Simple Character. isue.c. Muth. USSR — Sb. 44 (1983). 23—59.
ISSh 21
. On an Integral Representation and the Associated Transform of Analytic Functions of Seceral Cample.r Variables. Soviet Math. DokI. 37 (1988). 38—41. . Maslos' 's Method in Analytic Theory of Diffi'rentaai Equations. IRussian I. (to be published) Nauka. 1991. . On Leru's Re.cidue Theors'. Springer Lecture Notes 1453 (1990). 109—I 19.
ISSh 31 (SSh 41
ISSh 5j
. The ('ontaci Geometry and Li,,ear Dj/j'erentia/ Equations. Springer Lecture Notes 1108 (1984), 257—277.
ISw II
IT I)
Schwartz. L.. Analyse Mathématique I.!!. Hermann. Paris. 1967. Taylor. M.E.. Pseudodifferential Operators. Princeton University Press. Princeton, New Jersey. 1981.
I Tr II
IUnB II
IV II I VeLi Ii IVo
Treves, F.. introduction to tim! Fourier integral Operators 1.11, Plenum Press. New York and London. 1982. Unterberger. A.. Bokobza. I.. Les Optraieur.s de Caldernn—Zygmund Précisés. C.R.A.S. 259(1964). 1612—1614. Vainberg. B.R.. Asymptotic Methods for Equation.s of Mathematical Physics. Ixdatel'stvo MGI.), Moscow. 1982 IRussianl. Vcrgne. M., Lion. C.. The Well Representation. Maslov indez and Theta Series. Boston. 1980. Voros. A., Semiclassical Approximations. Saclay preprint. D.Ph.. 1974.
References
214 IVo
21
IWe II
. The WKR—Maslov Method for Nonseparable July 1974. Weinstein. A.. Sympleciic Manifolds and their Iligrangian
Saclay preprint D.Ph.. Adv. in Math.
6 (1971). 329—346. (We 21
IWe
IWi II
[Y lj IYo II
. On Maslov .c Quantization Condition. Fourier Integral Operators and Partial Differential Equations, Springer Lecture Notes 459 (1975). 341—372. . Lectures on Symplectic Manifolds. Providence. khodc Island. 1977. pp. 1-48. Williamson. J.. On an Algebraic Pmblem Concerning the Normal Forms qf Linear Dynamkal Systems. Amer. Math. 58 no. I (1936). 141—163. Canonical Operator. Hokkaido Math. J. 4 (1975). 8—38. Yoshikawa, A.. On Yosida, K.. Functional Analysis. Springer-Verlag. Berlin—G6ttingen—Heidelberg, 1965.
Index
Action in the canonical chart 70 Amplitude function 78 Chain, even 5 Chain, odd 5 Commutation formulas 38 Contact diffeomorphism 54 Contact distribution 57 Contact Hamiltonian function 177 Contact product 52 Contact structure 45, 60, 62 Contact vector field 55. 59, 62 Contactization 49 Current 7 Current, inverse image 9
Determining function 68
Determining function, normal form of 77
Duality 19 Equation of principal type (locally) 150 Equation of principal type (microlocally) ISO
Exterior differential 3, 7 Exterior product 7 Form, even 2 Form, odd 2 Fouricr—Maslov integral operator 112
Hamiltonian function of principal type 56
Hamiltonian function, contact 56 Hamiltonian vector field 55. 59. 62 Homogeneous distribution 24 Homogeneous distribution, formally 24 Homogeneous function 13 Homogeneous function, associate 29 Homogeneous function, odd 13 Homogeneous function, positive 13 Homogeneous symplectic diffeomorphism 54
Homothetic transformation 70, 84 Index of the curve 144 Induced mappings (of the phase space) 63
Integral of an even form 5 Integral of an odd form 6 Integral of the form 4 Integration over the fibre 8 Inversion formulas 40 Lagrangian manifold. nondegenerate 65 Legendre submanifold 46 Lemma, classification 73 Lemma on the gradient ideal 81 Lemma, stabilization 96 Leray form 13 Local elements 78
Maslov's canonical operator Ill Gradient ideal 81
Hamiltonian 53 Hamiltonian distribution 57 Hamillonian function 55
Operator of subprincipal type 181 Operator, canonical local 106 Orientable manifold I Orientation, standard 5
Index
216
Oriented mapping 4 Parseval identity 40 Phase
function 78
Pseudodifferential operator 120 Pseudodifferential operator. germ of 175 Pseudoscalar I
Quantized manifold Ill Radial vector field 12, 46. 59. 60, 62 Regularization 26 Regularization operator 26 Regularizer 153 Regularizer, microlocal 151 Regularizer. semiglobal 202 Residue 10
Simplex, even singular 5 Simplex. odd singular 5 Simplex, standard 5
Solvable, locally 151 Solvable. microlocally 151 Solvable, semiglobally 151 Stabilization 71. 87 Stationary phase formula 103 Symbol of the PDO 121 Symbol. subprincipal 193 Symplectic structure 59. 60 Symplectic structure, homogeneous 46 Symplectization 50 Theorem. boundedness 132 Theorem, composition 119 Theorem of Euler 13
Theorem on cocyclicity Ill Theorem on the Parseval identity 40 Theorem of Schwartz 7 Theorem of Stokes 6 Transform, Radon 42 Transformation. Fourier 30 Wave-front set ISO
ii
Walter de Gruyte Berlin. New
Anvarhek M. Meirmanov
The Stefan Problem Translated from the Russian by Marek Niezgôdka and Anna Crowley 1992. IX. 245 pages. 17x24 cm. Cloth ISBN 3-11-011479-8 (de Gruyter Expositions in Mathematics, Vol. 3)
The Stefan problem is one of the most classical free boundary problems of parabolic type. It arises from modelling phase-change phenomena, such as phase transitions between, for instance, liquid and solid states of a material. Since the appearance of Rubinstein's important monograph in 1967 this book provides the first systematic
analysis of Stefan-type problems. The existence of classical solutions for the multidimensional Stefan problem was a long-standing problem. The author's approach to the solution of this problem forms the central part ofthe book. Together
with a complete constructive proof of the classical solvability (local in time), examples of critical developments showing the lack of global-in-time solutions in the general setting are given. A careful analysis of the intrinsic structure ofthe free
boundaries that can have the form of mushy zones is provided. For onedimensional Stefan problems, qualitative properties of global classical solutions are studied, including an analysis of their asymptotic behaviour and periodicity. The role of compatibility conditions is discussed. This book is addressed to advanced students and research mathematicians, in particular applied mathematicians and engineers. Contents: Preliminaries . Classical solution of the multidimensional Stefan problem Existence of the classical solution to the multidimensional Stefan problem on an
arbitrary time interval . Lagrange variables in the multidimensional one-phase Stefan problem Classical solution of the one-dimensional Stefan problem for the homogeneous heat equation Structure of the generalized solution to the onedimensional Stefan problem. Existence of a mushy Time-periodic solutions of the one-dimensional Stefan problem• Approximate approaches to the two-phase Stefan problem. Appendix: I. G. Göiz, A. M. Meirmanov: Modelling of binary alloy crystallization.
Walter de Gruvte Berlin New York! New Series
de Gruyter Series in Nonlinear Analysis and Applications Editors: A. Beaseusasa. Paris; B. Florence; A. Friedman, Minneapolis; L-H. Hoffmann. Munich; M. A. Kz,snoselsldl, Moscow; L Nlrenberg, New York Managing Editors: J. Appell, WUrzburg; V. I hmlksa*hsm. Melbourne, USA This series consists of monographs which cover the whole spectrum of current nonlinear analysis and applications in various fields, such as optimization. control theory, systems theory, mechanics, engineering, and other sciences, One of its main objectives isto make available to the professional community expositions of results and foundations of methods that play an important role in both the theory and applications ofnonlinearanalysis, Contributions which are on the borderline of nonlinear analysis and related fields and which stimulate ftirther research at the crossroads of these areas are particularly welcome. The publications in this series are intended to be self-contained and comprehensive. They should prove of value not only to the specialist but may also serve as a guide for advanced lectures and seminars,
Klaus Deimling, University of Paderborn, Germany
Volume
1
Multivalued Differential Equations 1992. XII, 260 pagea, I7x24c,a. auth. ISBN 3-11-013212-3 Coniants:
Chapter I. MuItla 1 Upper Semicontinuity: Some Basic Notation' Upper Semicontinuity' Properties of Use Multis Other Tests for Usc' Remarks' * 2 Lower Semicontinuity: Lower Semicontinuity Selections Locally Lipschitz Approximations of Usc Multis 'Remarks' 3 Measurability: Measurable Multis ' Measurable Selections' Approximation by Step-Multis' Some Consequences Multis of Two Vari-
ables' Remarks
'*4 Mishmash: Tangency Conditions' Bochner Integrals' Monotone Multis' Accretive
Multis ' Some Basic Facts about Banach Spaces' Remarks. Cliapter2. ExIst*nCe Theory In Flake DimensIonal 5 Upper Semicontinuous Right-Hand Sides: The Usc Case' Counter-Examples . The Carathèodory Case . Some Consequences' Remarks '*6 Lower Semicontinuous Right-Hand Sides: The Lsc Case The Carathéodory Case' Some Consequences' Remarks, Chapter 3. SolutIon Sets' *7 'lbpological Properties of Solution Sets: Elementary Properties' lnvariance'Connectedness in the Usc Case 'Connectedness in the Lsc Case 'Funnels' Remarks '§ 8 Comparison ofSolutions: Preliminaries' Extremal Solutions I 'Extremal Solutions U Related Problems'Gronwall'slemma'Convexification ' Remarks. Chapter 4. ExIstence Theory In InfinIte DImensions '*9 Compactness Conditions: Two Examples' Measures of Noncompactness'The Use Case 'The Lsc Case' Remarks• § 10 Noncompactness Conditions: Baire Category' Extreme Points' Proof of Theorem 10,1 'Upsthitz Conditions'Monotonicity ' Hyperaccretlvity 'Remarks, ChapterS. FIxed Pelnis .nd QualitatIve Theory '§ II Fixed Points: Some (Geo-)metric Results 'Weakly Inward Maps' Set.Contractlons' Degree Theory 'An Example' Remarks' * 12 Boundary Value Problems: A Comparison Result'Sturm/Liouville Problems' Solutions in Closed Sets'Remarks'* 13 Periodic Solutions: Reduction to the Regular Case 'Another Fixed Point Problem 'Examples' Remarks '* l4 Stability and Asymptotic Behavior: Stability ' Stability Tests' Asymptotic Behavior' Perturbations' Remarks, AppendIx: Related Topira' Discontinuous Differential Equations' Implicit Differential Equations' Functional
Differential Equations' Perturbations of Dissipative Right-Hand Sides,
