DEGRUYTER EXPOSITIONS
IN MATHEMATICS
Vladimir E. Nazaikinskii Victor E. Shatalov Boris Yu. Stern in
6
I DE
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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
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 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 —1. Ifk 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 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 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. Similarly,
—
—i
(
(32)
JR' [4)i(x,
—
for x 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
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
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 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 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
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
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