Partial Differential Equations IV

Yu.V. Egorov M.A. Shubin (Eds.)

-2 ; :7 I

Partial Differential Equations IV Microlocal Analysis and Hyperbolic Equations

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

7\1

Contents I. Microlocal Analysis Yu.V. Egorov 1 11. Linear Hyperbolic Equations V.Ya. Ivrii 149 Author Index 231 Subject Index 240

.

I Microlocal Analysis Yu.V. Egorov Translated from the Russian by P.C. Sinha

Contents Preface

.......................................................

Chapter 1. Microlocal Properties of Distributions

.................. $ 1. Microlocalization .......................................... 0 2. Wave Front of Distribution . Its Functorial Properties . . . . . . . . . . . 2.1. Definition of the Wave Front ............................ 2.2. Localization of Wave Front .............................. 2.3. Wave Front and Singularities of One-Dimensional

7 8 8 9 9 10

Distributions . ; ........................................ 2.4. Wave Fronts of Pushforwards and Pullbacks of a Distribution ........................................... $ 3. Wave Front and Operations on Distributions . . . . . . . . . . . . . . . . . . 3.1. The Trace of a Distribution . Product of Distributions ........ 3.2. The Wave Front of the Solution of a Differential Equation . . . . 3.3. Wave Fronts and Integral Operators ......................

12 12 12 13 13

Chapter 2. Pseudodifferential Operators

14

.......................... $ 1. Algebra of Pseudodifferential Operators ....................... 1.1. Singular Integral Operators .............................. 1.2. The Symbol ........................................... 1.3. Boundedness of Pseudodifferential Operators . . . . . . . . . . . . . . . 1.4. Composition of Pseudodifferential Operators . . . . . . . . . . . . . . . 1.5. The Formally Adjoint Operator .......................... 1.6. Pseudolocality. Microlocality ............................ 1.7. Elliptic Operators ...................................... 1.8. GArding’s Inequality .................................... 1.9. Extension of the Class of Pseudodifferential Operators . . . . . . .

11

14 14 15 16 16 17 17 17 18 18

I. Microlocal Analysis

Yu.V. Egorov

2

Q 2. Invariance of the Principal Symbol Under Canonical Transformations ........................................... 2.1. Invariance Under the Change of Variables ................. 2.2. The Subprincipal Symbol ............................... 2.3. Canonical Transformations ............................. 2.4. An Inverse Theorem ................................... Q 3. Canonical Forms of the Symbol ............................. 3.1. Simple Characteristic Points ............................. 3.2. Double Characteristics ................................. 3.3. The Complex-Valued Symbol ............................ 3.4. The Canonical Form of the Symbol in a Neighbourhood of the Boundary ............................................ Q 4. Various Classes of Pseudodifferential Operators ................ 4.1. The LF. Classes ....................................... 4.2. The L$ Classes ...................................... 4.3. The Weyl Operators ................................... 0 5. Complex Powers of Elliptic Operators ........................ 5.1. The Definition of Complex Powers ....................... 5.2. The Construction of the Symbol for the Operator A' ........ 5.3. The Construction of the Kernel of the Operator A" . . . . . . . . . . 5.4. The l,-Function of an Elliptic Operator .................... 5.5. The Asymptotics of the Spectral Function and Eigenvalues ... 5.6. Complex Powers of an Elliptic Operator with Boundary Conditions ........................................... Q 6. Pseudodifferential Operators in IR"and Quantization . . . . . . . . . . . 6.1. The Analogy Between the Microlocal Analysis and the Quantization .......................................... 6.2. Pseudodifferential Operators in IR" .......................

.+,

Chapter 3. Fourier Integral Operators ...........................

Q 1. The Parametrix of the Cauchy Problem for Hyperbolic Equations ................................................ 1.1. The Cauchy Problem for the Wave Equation . . . . . . . . . . . . . . . 1.2. The Cauchy Problem for the Hyperbolic Equation of an Arbitrary Order ....................................... 1.3. The Method of Stationary Phase ......................... $ 2. The Maslov Canonical Operator ............................. 2.1. The Maslov Index ..................................... 2.2. Pre-canonical Operator ................................. 2.3. The Canonical Operator ................................ 2.4. Some Applications ..................................... Q 3. Fourier Integral Operators .................................. 3.1. The Oscillatory Integrals ................................ 3.2. The Local Definition of the Fourier Integral Operator .......

18 18 19 20 20 21 21 23 23 23 24 24 26 29 32 32 33 35 36 37 38 39 39 41 43 43 43

44 45 46 46 47 49 49 50 50 52

3.3. The Equivalence of Phase Functions ....................... 3.4. The Connection with the Lagrange Manifold ............... 3.5. The Global Definition of the Fourier Distribution . . . . . . . . . . . 3.6. The Global Fourier Integral Operators .................... $4. The Calculus of Fourier Integral Operators ..................... 4.1. The Adjoint Operator ................................... 4.2. The Composition of Fourier Integral Operators ............. 4.3. The Boundedness in L, .................................. $ 5. The Image of the Wave Front Under the Action of a Fourier Integral Operator .......................................... 5.1. The Singularities of Fourier Integrals ...................... 5.2. The Wave Front of the Fourier Integral .................... 5.3. The Action of the Fourier Integral Operator on Wave Fronts ................................................ $6. Fourier Integral Operators with Complex Phase Functions ....... 6.1. The Complex Phase .................................... 6.2. Almost Analytic Continuation ............................ 6.3. The Formula for Stationary Complex Phase ................ 6.4. The Lagrange Manifold ................................. 6.5. The Equivalence of Phase Functions ....................... 6.6. The Principal Symbol ................................... 6.7. Fourier Integral Operators with Complex Phase Functions ... 6.8. Some Applications ......................................

3

52 53 55 56 58 58 59 61 62 62 63 64 65 65 65 66 67 68 69 71 72

Chapter 4. The Propagation of Singularities

....................... 73 $ 1. The Regularity of the Solution at Non-characteristic Points ....... 73 1.1. The Microlocal Smoothness .............................. 73 1.2. The Smoothness of Solution at a Non-characteristic Point .... $ 2. Theorems on Removable Singularities ......................... 2.1. Removable Singularities in the Right-Hand Sides of Equations ............................................. 2.2. Removable Singularities in Boundary Conditions . . . . . . . . . . . . 0 3. The Propagation of Singularities for Solutions of Equations of Real Principal Type ............................................. 3.1. The Definition and an Example ........................... 3.2. A Theorem of Hormander ............................... 3.3. Local Solvability ....................................... 3.4. Semiglobal Solvability ................................... $4. The Propagation of Singularities for Principal Type Equations with a Complex Symbol ......................................... 4.1. An Example ........................................... 4.2. The Fixed Singularity ................................... 4.3. A Special Case ......................................... 4.4. The Propagation of Singularities in the Case of a Complex Symbol of the General Form .............................

73 74 74 75 76 76 76 77 77 78 78 79 79 80

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I. Microlocal Analysis

Yu.V. Egorov

$ 5. Multiple Characteristics ..................................... 5.1. Non-involutive Double Characteristics .................... 5.2. The Levi Condition .................................... 5.3. Operators Having Characteristics of Constant Multiplicity ... 5.4. Operators with Involutive Multiple Characteristics . . . . . . . . . . 5.5. The Schrodinger Operator ...............................

80

Chapter 5. Solvability of (Pseud0)DifferentialEquations

83

$1. Examples 1.1. Lewy’s Example ....................................... 1.2. Mizohata’s Equation ....; .............................. 1.3. Other Examples ....................................... $ 2. Necessary Conditions for Local Solvability .................... 2.1. Hormander’s Theorem .................................. 2.2. The Zero of Finite Order ................................ 2.3. The Zero of Infinite Order ............................... 2.4. Multiple Characteristics ................................. $ 3. Sufficient Conditions for Local Solvability ..................... 3.1. Operators of Real Principal Type ......................... 3.2. Operators of Principal Type ............................. 3.3. Operators with Multiple Characteristics ...................

83 83 84 85 85 85 86 87 87 87 87

Chapter 6. Smoothness of Solutions of Differential Equations . . . . . . . .

90

$ 1. Hypoelliptic Operators ..................................... 1.1. Definition and Examples ............................... 1.2. Hypoelliptic Differential Operators with Constant Coefficients .......................................... 1.3. The Gevrey Classes ................................... 1.4. Partially Hypoelliptic Operators ........................ 1.5. Hypoelliptic Equations in Convolutions . . . . . . . . . . . . . . . . . . 1.6. Hypoelliptic Operators of Constant Strength . . . . . . . . . . . . . . 1.7. Hypoelliptic Differential Operators with Variable Coefficients .......................................... 1.8. Pseudodifferential Hypoelliptic Operators . . . . . . . . . . . . . . . . 1.9. Degenerate Elliptic Operators ........................... 1.10. Partial Hypoellipticity of Degenerate Elliptic Operators . . . . . 1.11. Double Characteristics ................................. 1.12. Hypoelliptic Operators on the Real Line .................. $ 2. Subelliptic Operators ....................................... 2.1. Definition and Simplest Properties ........................ 2.2. Estimates for First-Order Differential Operators with Polynomial Coefficients ................................. 2.3. Algebraic Conditions ...................................

90 90

............ .................................................

80 81 82 82 83

88 90

91 91 92 93 93 94 95 96 97 98 99 100 100 102 103

5

$ 3. Hypoelliptic Differential Operators of Second Order . . . . . . . . . . . . . 3.1. The Sum of the Squares ................................. 3.2. A Necessary Condition for Hypoellipticity . . . . . . . . . . . . . . . . . 3.3. Operators with a Non-negative Quadratic Form ............ $ 4. Analytic Hypoellipticity ..................................... 4.1. Elliptic Operators ...................................... 4.2. The Analytic Wave Front ............................... 4.3. Analytic Pseudodifferential Operators ..................... 4.4. Necessary Conditions for Analytic Hypoellipticity ........... 4.5. Differential Equation of the Second Order ................. 4.6. The Gevrey Classes .................................... 4.7. Generalized Analytic Hypoellipticity ......................

105 105 106 107 108 108 108 109 110 112 114 115

Chapter 7. Transformation of Boundary-Value Problems

115

............ $1. The Transmission Property .................................. 1.1. Operators in a Half-Space ............................... 1.2. The Transmission Property .............................. 1.3. Application to the Study of Lacunae ...................... $ 2. Distributions on a Manifold with Boundary .................... 2.1. The Distribution Spaces ................................. 2.2. Contracted Cotangent Bundle ............................ $ 3. Completely Characteristic Operators .......................... 3.1. Pseudodifferential Operators and their Kernels ............. 3.2. The Transmission Property .............................. 3.3. Completely Characteristic Operators ...................... 3.4. The Boundary Wave Front .............................. $ 4. Canonical Boundary Transformation ......................... 4.1. The Generating Function ............................... 4.2. The Operator of Principal Type .......................... 4.3. The Differential Operator of Second Order . . . . . . . . . . . ..-. . . . $ 5. Fourier Integral Operators ..................................

115 115 116 119 120 120 121 122 122 123 124 124 125 125 126 126 127

5.1. The Generating Function of the Canonical Boundary Transformation ........................................ 5.2. The Fourier Integral Operator ...........................

127 128

.....................................

129

Chapter 8. Hyperfunctions

$ 1. Analytic Functionals ....................................... 1.1. Definition and the Basic Properties ....................... 1.2. Operations on Analytic Functionals ...................... 0 2. The Space of Hyperfunctions ................................ 2.1. Definition and the Basic Properties ....................... 2.2. The Analytic Wave Front of a Hyperfunction ............... 2.3. Boundary Values of a Hyperfunction ......................

129 129 130 130 130 131 131

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Yu.V. Egorov

Q 3. Solutions of Differential Equations ............................ 132 132 3.1. The Cauchy Problem ................................... 3.2. The Analytic Wave Front ............................... 133 Q 4. Sheaf of Microfunctions ..................................... 135 135 4.1. Traces of Holomorphic Functions ........................ 4.2. The Definition of a Sheaf of Microfunctions ................ 135 4.3. Pseudodifferential Operators ............................. 136 4.4. Fourier Integral Operators .............................. 138

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139

Preface The microlocal analysis is the local analysis in cotangent bundle space. The remarkable progress made in the theory of linear partial differential equations over the past two decades is essentially due to the extensive applicaton of the microlocalization idea. The Hamiltonian systems, canonical transformations, Lagrange manifolds and other concepts, used in theoretical mechanics for examining processes in the phase space, have in recent years become the central objects of the theory of differential equations. For example, the evolution of singularities of solutions of a differential equation is described most naturally in terms of Lagrange manifolds and Hamiltonian systems, the solvability conditions are formulated in terms of the behaviour of integral curves of the Hamiltonian system whose Hamiltonian function serves as the characteristic form, the class of pseudodifferential equations arises in a natural way from that of differential equations under the action of canonical transformations, the class of subelliptic operators is defined by means of the Poisson brackets, etc. The difficulty faced in the microlocal analysis is connected with the principle of uncertainty which does not permit us to localize a function in any neighbourhood of a point of the cotangent space. This paper presents a survey of the most interesting results, from our point of view, of the microlocal analysis achieved over the recent years. Unfortunately, due to lack of space many significant results could not be included. Also incomplete is the list of the literature cited; more complete lists can be found in Egorov [1984], Shubin [1978], Hormander [1963, 1983, 19851, Taylor [1981] and Trkves [1982]. The author expresses his thanks to V.Ya. Ivrii for his useful critical comments.

’

I. Microlocal Analysis

Chapter 1 Microlocal Properties of Distributions

0 1.

Microlocalization

The study of the singularities of solutions constitutes one of the most important problems in the theory of differential equations. In this theory, just as in other mathematical disciplines, one often examines functions modulo smooth ones, so that the points where a given function is infinitely differentiable may be neglected. This approach reflects physical realities: singular points correspond to those phenomena which are most interesting from the point of view of each physical theory. In investigating physical processes that take place in a bounded space an extensive use is made of the principle of locality. Its essence lies in that by knowing the state of the process at a given moment of time in a fixed region 52 of the physical space one may determine, by means of physical laws, the course of the process in a region 51', lying strictly inside 52, for a future time interval. During this time interval the effect of processes taking place outside 52 will have no influence on phenomena in 51' because the effect is propagated with a fintie velocity. We can introduce a more general principle, the principle of microlocality, by examining the phenomenon in a bounded region of the phase space. If we know the state of the process in this region at a certain moment of time, we can describe this process for future close points lying strictly inside the region. In physical terms, this means that the change in the impulse too takes place with a finite velocity because the acting forces are finite. The above-mentioned principles are reflected in mathematical physics in the investigation of singularities of solutions of differential equations. Namely, local properties of such solutions are those properties which remain unaltered when the solutions are multiplied by smooth functions with a small support. Microlocal properties of a solution refer naturally to those properties which do not change on ''multiplication'' of the solution by a smooth function having support in a small neighbourhood of the given point in the phase space. However, this operation is much more complicated. In fact, it consists in multiplying by an ordinary smooth cutoff function with a small support, in applying Fourier transformation, in multiplying successively by a smooth cutoff function of dual coordinates, and in applying inverse Fourier transformation. Instead of Fourier transformation we can also use some other decomposition in plane waves; for example, the Radon transformation. In fact, the microlocal analysis is the local analysis on the cotangent bundle space. A special feature of the microlocal analysis is the fact that localization in the phase space is possible only to a certain extent: the localization of spatial coordinates obstructs that of impulses. In quantum mechanics, this fact is referred to as the Heisenberg uncertainty principle.

9

The last two decades have seen immensely fruitful applications of the microlocality principle to the theory of partial differential equations. Every function (ordinary or generalized) can be regarded as an aggregate of linear differential equations which this functon satisfies. The microlocality principle extends in a natural manner this aggregate to the system of pseudodifferential equations derived from differential equations by transforming the phase space without altering its structure. By applying the microlocality principle we not only obtain a more precise description of singular points of a distribution but we also have a simpler description of the propagation process of these singularities. By this principle we can also extend to distributions the operations defined initially for smooth functions only; for example, the operation of taking trace or the operation of multiplication, etc. Let us explain the idea of microlocalization with the following simple example. Let n > 2 be a natural number and let f be a function in IR" of the form f ( x ) = g(a.x), where a E lR"\O, a * x = a j * x j and g is a function of a single variable. If g(t) has a singularity,for example, if g(t) is not differentiable at t = to, then all the points x lying on the plane a ' x = to are singular points of f. However, f is a smooth function in each direction lying on this plane so that for it singular will be only the direction of the vector a. Radon's theorem enables us to represent each distribution f in g(lR") as an integral of plane waves:

c'&l

f(x) =

s

g,(a * x) da.

lal=l

Therefore at each point x those directions a will be singular for which the distribution g,(t) has a singularity at the point t = X - a .If, instead of Radon's theorem, we apply Fourier transformation, then f can be represented as an integral of plane waves: f(x)=

s

g(a)eia'xda,

where the integration is performed over the whole IR". Now those directions of a become singular for f i n which g(ta) does not decrease, as t + co,rapidly enough. As mentioned earlier, in the modern theory of differential equations the microlocality principle is extensively applied to investigate the singularities of the solution. Many important results achieved in recent years by means of this principle in the theory of boundary-value problems, in the spectral theory, in the theory of functions of several complex variables, and in other branches of mathematics point towards great potentialities of the microlocal analysis.

0 2.

Wave Front of Distribution. Its Functorial Properties

2.1. Definition of the Wave Front. The notion of a singular point of a distribution does not have only one meaning. Depending on the problem under discussion,a singular point may signify a point of discontinuity, or a point where

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I. Microlocal Analysis

Yu.V. Egorov

1. If u E 9'(IR")and cp E Cg(R"),then

the function becomes infinite, or a point where the function has an essential singularity in the sense of the theory of complex variables, etc. For the general theory of distributions, the most natural is the following

WF(cpu) c WF(u). 2. If n: T*(lR")+ IR" is the natural projection, then nWF(u) = sing supp u for every u in 9'(R").

Definition 2.1. A point xo is a non-singular point for a distribution u if there exists a function cp in C;(IR") such that cp(xo)# 0 and cpu E Cm(lR").

2.3. Wave Front and Singularities of One-Dimensional Distributions. Let X and Y be smooth manifolds, and let f:X --+ Y be a smooth map. f is called proper if the set f - ' ( K ) is compact for every compact set K in Y. For a cp E C$( Y ) and a proper map f we set

It follows from the definition that the singular points of a distribution constiUte a closed set. This set is known as the singular support of the distribution u and is denoted by sing supp u. We can easily see that this set is invariant under diffeomorphisms of the space; thus the definition of a singular support can be readily extended to distributions on a smooth manifold. For a smooth manifold X , we denote by T * ( X ) the cotangent bundle space and by T*(X)\O the same space with the zero section removed (see, for example, Amol'd [1974], Egorov [1984]). The following definition and examples are due to Hormander. It should be remarked that in the general theory there are many close concepts that are extensively used, namely, the analytic wave front, Gevrey wave front, oscillation front, etc (see Hormander [1983, 19851, Trkves [1982], and 0 4.2 of Chapter 6).

f*cp(x) = c p ( f ( 4 ) .

Then f*cp E C $ ( X ) and f* is a continuous map from 9 ( Y ) into 9 ( X ) . This enables us to define, by means of duality, the pushforward f,u of every distribution u E 9 ' ( X ) by the formula

(f*u,

l$u(5)1 = ON1 + 151Y)

holds for all 5 E f and all integers N . The wave front of u will be denoted by WF(u). Example 2.1. If the distribution u is a plane wave, that is, if u(x) = g ( a * x ) , where a E IR"\O and g E 9'(lR), then every direction to,non-collinear with the vector a, is non-singular for u. That is, WF(u) may only contain points (xo, to) for which to= ta with t E R\O and a xo E sing supp g . E

cp) = (U,f*cp).

It is obvious that the same construction remains valid for an arbitrary map f (not necessarily proper) when u has a compact support, that is, when u E 8 ( X ) . The pullback f * u of a distribution is defined when f is a submersion of X onto Y. This last condition means that for every point y E Y the set f - ' ( y ) is a smooth submanifold of X and all these submanifolds are diffeomorphic to a fixed k-dimensional smooth manifold. Locally such a map is a projection, and in a suitable local coordinate system it becomes the projection IRk x IR'+Rk. If cp E C ; ( X ) and the support of cp lies in a coordinate neighbourhood, and if x = (x', x"), where x' E IRk, X" E R',then f,cp(x') = f cp(x', x") dx". For any u E 9'(Y ) we now set

Definition 2.2. A point (xo, to)E T*(IR")\Odoes not belong to the waue front of a distribution u in 9'(IR") if there are a function cp in Cg(IR"),with cp(xo)# 0, and a cone f i n IR", with vertex at the origin of coordinates, which contains in its interior the ray (5; 5 = tr,, t > 0 } such that the relations

Example 2.2. Suppose that q Then for the function

11

(f*% cp)

= (u,f*cp).

Iff is both proper and submersion (for example, iff is a diffeomorphism), then f * u and f*u are simultaneously defined for every u E 9'(Y), u E 9 ' ( X ) . Example 2.3. Let n: IR"+ IR' be the projection onto the x,-axis. In this case both n*u, u E 9'(R),and n*u, u E b'(lR"),are defined, and

R"\O, $ E C$(IR"), @(O) = 1 and $(O) > 0.

n*v(x') = 4 x 9 (23 1X2.....xn;

which is continuous in IR", the wave front consists of the ray ((0, tq), t > 0 } (Hormander [1979a]). Summing over q, we obtain from this example a function whose wave front coincides with an arbitrary conical closed subset of T*(IR")\O. 2.2. Localization of Wave Front. It is comparatively easy to establish the following properties of the wave front (see Hormander [1971, 1983, 19853, Egorov [1984]):

~ * u ( x= ) _-

s

u(xl, x 2 , .. .,x,) d x , . . .dx,,.

Using these concepts, we can establish the following Theorem 2.1. Let SZ be a domain in IR",and let u E 9'(Q). A point (xo, to)of T*(Q)\O does not lie in WF(u) i f and only i f there exist afunction cp in Cg(SZ), with cp(xo)# 0, and an E > 0 such that for every smooth function f:supp cp -,IR, with lgrad f ( x o ) - tol< E, thefunctionf,(cpu)(t) is infinitely differentiable on the real line (Guillemin and Sternberg [19773, Egorov [1985a]).

Yu.V. Egorov

12

I. Microlocal Analysis

a.

u E W ( Y )and W F ( u )n N ( Y ) = Then the restriction uI, can be defined as i*u, where i: Y + X is the embedding. Furthermore, if ( y , q) E WF(ul,), then there exists a vector 5 E N ( Y ) such that ( y , q 5 ) E W F ( u )(Hormander [1983,1985]).

2.4. Wave Fronts of Pushforwards and Pullbacks of a Distribution Theorem 2.2. Let f : X WF(f*

+

+

Y be a submersion, and let u E Sl(X). Then

= { ( f ( X I , 44:x E x,(x, Y;?)

E W F ( 4or

Y?;

13

= 01,

This result is an immediate consequence of Theorem 2.3, since N i = N ( Y ) .

where 'f: denotes the transpose matrix of the Jacobi matrix f ; o f f (Hormander [19711).

Theorem 3.2 (Hormander [1983,1985]). Let X be a smooth manifold. Suppose that u1 and u2 E Q'(X), with W F ( u , ) W F ( u 2 )c T*X\O. Then the distribution uluz E W ( X )can be defined as A*(u, @I u2), where A : X + X x X is the diagonal map. Furthermore,

+

Example 2.4. Let rc: IR2 + IR' be the projection onto the x,-axis, that is, rc(xl, x 2 ) = x,. The Jacobi matrix is of the form rc: = (1,O) and

WF(u1,u 2 )

= { ( x ,5 + 49; ( x , 5 ) E WF(u,),or 5 = 0, (x,q) E W F ( u z )or rl = 0; 5 + q # O} Thus WF(rc,u) is contained in the set of projections of those points of the wave front of the distribution u where singular directions are parallel to the x,-axis. The pullback f * u of a distribution was defined above for submersionsf only. Using continuity, we can attempt to define it for the more general case by setting f *u = lim f *uj, where { u j } is a sequence of smooth functions converging to u in W ( Y )and f *uj(x)= uj(f ( x ) ) . As Hormander [1971] has shown, such a path leads to the following

Theorem 2.3. Let f : X

+Y

Here also Theorem 2.3 is applicable, since Nd = { (x, x , 5, - t)}.

Example 3.1. If Y and 2 are transversally intersecting submanifolds of X 2nd if u and v are smooth densities on Y and 2, respectively, then uv is a smooth density on Y n 2. 3.2. The Wave Front of the Solution of a Differential Equation Theorem 3.3. Let u E 9'(R"), and let P(x, 0,) be a differential operator with smooth coefficients. Let P(x, DJu = f E C"(lR"). Then

be a smooth map. Set

N , = {(y, q) E T*Y\Q 3~ E X , y = f(X),'s:q = O}.

W F ( u )c Char P

If u E W ( Y )and W F ( u )nN, = 0, then the distribution f *u can be defined by continuity and the inclusion

W F ( f*u) c { ( x ,5); 3(Y, ?) E WF(u),y = f ( x ) , 5

In the general case, we have W F ( u )c WF(Pu)u Char P. In particular, if P is then WF(Pu)= W F ( u )(Hormander an elliptic operator, that is, if Char P = 0, [1983,1985]). We recall that P is said to be hypoelliptic if sing supp Pu =I sing supp u. If WF(Pu)3 W F ( u )for each u E Sl(X), P is called a microlocally hypoelliptic operator. The question of singularities of the solution of a differential equation will be examined later in Chapter 4.For the present, we note one of the consequences of Theorems 3.1 and 3.3.

= Y'(X)?}

Example 2.5. Let M be a k-dimensional smooth submanifold in IR" and p a smooth function on M . Set u(x) = p 63 6 ( M ) .Then W F ( u )= N(M)\O, where N ( M ) is the conormal bundle of M , N ( M ) c T*(IR").If p(xo) # 0, then (xo,5 ) E W F ( u )if and only if the vector 5 is directed along the normal to M . This result follows from Theorem 2.3 if one considers the submersion IR" + M .

Theorem 3.4 (Hormander [1983, 19853). Suppose that u E W(R"+'), P(t, x, D,, D,)u = f E C"(lR"+'), t E IR, x E IR". If the plane t = 0 is non-characteristic, that is, if P(0, x ; 1,O) # 0, then the traces Dfu(0, x ) E W(IR") are defined for all k 2 0. Moreover,

Wave Front and Operations on Distributions

u E C"(IR; Q'(IR")). - 2

3.1. The Trace of a Distribution.Product of Distributions. Let us demonstrate how to define, using theorems of the preceding sections, some of the operations for distributions that have been initially defined for smooth functions only. Theorem 3.1. Let X be a smooth manifold and Y a smooth submanifold of X . Let N(Y )be the normal bundle of X the normals being tangent to X . Suppose that

{(x, 5 ) E T*(IR")\O, Po(x,5 ) = 0},

where Po is the characteristic form of P.

holds. Moreover, the pullback f * u is unique.

0 3.

=

This result follows from Theorem 3.1, since the points (0, x , z, 0)do not lie in W F ( u )and hence W F ( u )n N ( M o )= where M , is the plane t = 0.

a,

3.3. Wave Fronts and Integral Operators. Let 0,and 0,be domains in IR" and IR", respectively. Suppose that A is a continuous linear operator from 9(Q2) into 9'(Ql) with the Schwartz kernel K belonging to 9'(0, x Q2). Consider

Yu.V. Egorov

14

the sets

M1 = { ( x , 5); 3y E Qz, (x, y, 590) E WF(K)l, M , = { ( y , rl); 3x E 9 1 , (x, y, 0, -44 E WF(K)}.

Theorem 3.5. Suppose that K E W(Q1 x Q,), u E &(QZ), with MI = 0, M , n WF(u) = 0. Then the distribution =

is defined and

s

K ( x , Y ) U ( Y ) dy E WQJ

WF(A4 = { ( x , 5); 3(y, rl), (x, y, 5,

-44 E W W ) , ( y , rl) E WF(u)}. the product K ( x , y ) u ( y ) is defined, in We note that since M , n WF(u) = 0, view of Theorem 3.2. The condition M , = 0signifies that A: C;(Q,) + Cm(Q,), that is, under this mapping smooth functions go over to smooth functions.

Chapter 2 Pseudodifferential Operators

9 1. Algebra of Pseudodifferential Operators 1.1. Singular Integral Operators. In many questions of the theory of partial differential equations there arises the necessity of using singular integral operators. These are operators of the form

where K ( x , z ) is a function having singularity at z = 0 only and K ( x , tz) = t-"K(x, z ) for t > 0. Assuming that

s

K ( x ,Z) dz = 0,

121=1

we can define A correctly in the sense of the principal value, that is, Au(x) = lim e+O

R- m

I. Microlocal Analysis

15

These theorems prove to be crucial in studying smoothness of solutions of elliptic equations. The fundamental solution F ( x , x - y) of an elliptic differential operator L(x, D) of order rn has a singularity of order Ix - ylm-" at x = y. Therefore the mth order derivatives of the solution of the equation Lu = f can be expressed in terms off by means of singular integral operators. Thus D"UE LZp'(0) for la1 < rn if Lu E L,(Q), p > 1, and u E C'"+'(Q) if Lu E C"(Q), O
+

1.2. The Symbol. In his works S.G.Mikhlin had noted that with every operator A of the form (1) one can associate a function a(x, 5 ) defined in the cotangent space. This function, called a symbol, plays a leading role in the calculus of singular integro-differential operators. Different variants of this calculus have been developed in the works of Dynin, Agranovich, Vishik, Eskin, Kohn, Hormander, Nirenberg, Seeley, Unterberger, and Bokobza. At present, the most popular theory of pseudodifferential operators is the one developed first by Kohn and Nirenberg [1965]. By a pseudodifferential operator we mean an operator of the form Au(x) = (2n)-"

s

(2)

a(x, 5)ii(t)ei(x.5) d5.

Here u E C;(IR") and ii(5) = fu(y)e-i(y'e;) dy is the Fourier transform of u. The function a is called the symbol of A. It is assumed that lqD!a(x,

01< C,,@(1+ 151)"-'"'

(3)

for all a, /? E Z;.The constants C,,@may depend on the compact set K to which the point x belongs. When a is a polynomial in 5, A turns out to be a differential operator. In particular, A = I if a = 1. We denote by S" the class of symbols satisfying (3). The number rn is called the order of the operator. When a satisfies (3) for any m E IR,the order of A is taken to be -a;S-" denotes the set of such symbols. A pseudodifferential operator A with its symbol a in S" is called classical if there is a sequence of symbols aj E S m - j ,j = 0,1,2,. ..,such that aj(x, t5) = t"-jaj(x, 5 ) f o r t > 1,151 > 1, and

K ( x , x - Y ) U ( Y ) dye e
It was shown in the works of Giraud, Korn, and Lichtenstein that A is a bounded operator in spaces C", 0 < a c 1, while its boundedness in L , spaces, p > 1, was established in the works of Calderon and Zygmund.

-

aj is for all a,8, 15 I > 1 and all integers N 2 0. In this case the notation a used. The function a, is known as the principal symbol of A, and the class of such symbols is denoted by S;.

Yu.V. Egorov

I. Microlocal Analysis

13. Boundedness of Pseudodifferential Operators. We mention a few basic theorems regarding boundedness of the operators under discussion. 1. The operator (2) maps C:(lR") into C"(lR") and can be extended to an operator A : 8'(lR"+ ) 9'(lR"). 2. If rn < -n, then Au(x) = I K ( x , x - y)u(y)dy, where K ( x , z ) = (271)-"Ia(x, 5)eiz'
1.5. The Formally Adjoint Operator. The operator A* is formally adjoint to the operator A if

16

r

J ~ u ( x ) v ( xdx) =

s

u ( x ) ~ * u ( xdx; ) u, u E c?(R").

(5)

We note that pseudodifferential operators can be defined in the spaces of vector-functions. In this case, their symbols are matrices, and the right side of (5) contains the matrix 'a, the conjugate of a. Other formulae remain valid without any change.

1.6. Pseudolocality. Microlocality. It is known (see Hormander [1963])that differential operators are the only operators that are local linear operators actthat is, are such that supp Pu c supp u for all ing from C?(Rn)into 9'(l"'), u E C:(lR"). Pseudodifferential operators possess a weaker property of pseudolocality, namely, sing supp Au c sing supp u, as well as the property of microlocality, namely, WF(Au) c WF(u). The wave front of a distribution u can be defined by means of pseudodifferential operators as follows: for a distribution u with a compact support in IR" a point (x,, 5,) does not lie in WF(u) if and only if there exists a pseudodifferential operator A with the principal symbol a(x, 5 ) such that a(xo, 5,) # 0 and Au E C"(lR").

a(x, 5 ) = (271)"s K ( x , x - y)ei(Y-")'5dY.

Thus every pseudodifferential operator is the sum of a properly supported operator and a smoothing operator. This situation enables us to define the algebra of pseudodifferential operators up to smoothing operators. 3. Let A be a pseudodifferential operator, and suppose that its symbol vanishes for large values of 1x1. Then A defines a continuous map A : H, + H,-" for each real s.

This formula is a generalization of the classical Leibniz formula.

J

:la1

with 2N > rn + n, which is obtained by integration by parts. This formula shows that K E C" when z # 0 and IK(x, z)l < CIZI-,~,that is, the kernel can have only power singularity. The operator A with the Schwartz kernel K ( x , x - y) is called properly supported if K ( x , z ) = 0 for IzI > p, with p > 0, and K E Cm(lR2").Such an operator maps functions in C:(lR") into functions in C:(lR") and distributions into distributions in &(lR"). in &(Rn) Let a given function h(z) in C"(lR") be such that h(z) = 0 for IzI < p and h(z) = 1 for 1zI > 2p, p > 0. Let A be a pseudodifferential operator with the Schwartz kernel K ( x , x - y). We set K , ( x , y) = h(x - y). K ( x , x - y) and K,(x, y) = [l - h(x - y ) ] K ( x ,x - y) and denote by A, and A, operators with it follows that A , is a the kernels K, and K,, respectively. Since K , E Cm(lRZn), smoothing operator. And A , is a properly supported operator. We note that the symbol of a pseudodifferential operator can be obtained from its kernel by the formula

-1

P

If A is a pseudodifferential operator of order rn, then A* is also a pseudodifferential operator of the same order with the symbol

I z I ~ ~ K (z )x = , (271)-" [ ( - d t ) N a ( x , t ) ] e i z ' td5

1.4. Composition of Pseudodifferential Operators. If A and B are two pseudodifferential operators of order m and m',respectively, of which B is properly supported, then the composition C = A o B is defined. It is a pseudodifferential operator of order rn + rn' with the symbol ilal (4) C(X, 5) +U.+ 4 x , S)D:b(:b(x,5).

17

1.7. Elliptic Operators. An operator A of order rn is called elliptic if b(x, 5)l 2 C , 151" - C,, where C , = const. > 0. Elliptic operators are invertible in the algebra considered up to smoothing operators. More precisely, for each elliptic operator A of order rn and for each compact set K there exists an elliptic operator Q of order -m such that Q A = I T,, AQ = I T,, where T, and T, are operators of order -a on K. This immediately implies that elliptic operators are hypoelliptic, that is, u E C" on each open set where Au E C". An operator A of order m is called elliptic at a point (x,, 5,) of T*(lR")\O if b(x, <)I 2 C , 151" - C , for ( x , 5 ) lying in a conical neighbourhood of (x,, 5,). By a conical neighbourhood we mean, as usual, the set

+

...

{ ( x , 5,; Ix - xoI < a,,

/i It1

+

1501

< a,}.

I. Microlocal Analysis

Yu.V. Egorov

18

x by f(xr) and y by f(y'). We immediately note that

For an operator A, which is elliptic at a given point, we may construct a microlocal pararnetrix. It is an operator Q of order - m such that

x - y = f(Xl) - f ( y ' ) = F(X', y')(x' - y r ) ,

+

A q Q = ~ p l +T2, Q q A = 91 Ti, where q(x, D)is a pseudodifferentialoperator of order zero the symbol of which has a support in a sufficiently small conical neighbourhood of the point (x,, to) and which equals 1 in a smaller conical neighbourhood r of this point; as to symbols of Tl and T2, they vanish in r.

where F(x', y') =

ss

a(x, y, 5)u(y)ei("-y)5 d y d5,

(6)

511 < c a , & y ( l+ 151)"-'"',

are also pseudodifferential operators. The symbol b of such an operator has the following asymptotic representation: b(x, 5 )

-1

-

ax

dt. If 5 is replaced by 'F(x', y r ) - ' , then

ilal

+o;acx,

a a

y,

5)ly=x.

Thus the classes of operators defined by (2) and (6)coincide. The Weyl operators Wu(x) = (2n)-" [Ja('+,

5)u(y)ei(x-y'C dy d t ,

(7)

can also be represented in form (6), where a E S".

. 1

fj2.

+

Definition 2.2. Let X be a smooth manifold and let E and F be smooth complex bundles on X . A continuous linear map A : C:(X, E ) -+ Cm(X,F ) is called a pseudodifferential operator if for every open subset V c X , with the property that El, and FI, are diffeomorphic, respectively, to V x (c"~ and V x Cmz,there exists an rn2 x rn, matrix of pseudodifferential operators Aij on Y such that

where the symbol a satisfies the inequalities lD;D!D;a(x,y,

ay-(yr

Definition 2.1. Let X be a smooth manifold. A continuous linear map A : C z ( X )-,C m ( X )is called a pseudodifferential operator on X if for each coordinate neighbourhood U c X, to which there corresponds a coordinate map cp: U -,w, where w is a domain in IR",the map C:(w) 3 u -+ (q-')*A(q*u)E Cm(w)is an operator of the form (2).

uE

1.9. Extension of the Class of Pseudodifferential Operators. The operators of the form Au(x) = (2n)-"

1

where ~ ( y '= ) u(y). The resulting operator is of the form (1.6), and is therefore a pseudodifferential operator. Its principal symbol is complited in accordance with 0 1.9 and equals a,(f(x'), Yr(xr)-').This implies that the principal symbol of a pseudodifferential operator is invariant on the cotangent space. Thus under a change of variables the class of pseudodifferential operators is preserved. This enables us to define a pseudodifferential operator on a smooth manifold.

for all u in C:(K), C , = C , ( K )(Hormander [1983,1985]). This result can be employed for obtaining a sharper estimate for the norm of the operator in terms of its symbol. Let B(x, D)be a pseudodifferential operator of order rn with the principal symbol b,(x, 5 ) satisfying the condition Ib,(x, 5)1 < M for 151 = 1. Then for any real s we can find a constant C = C(s)such that

< Mllulls + Cllulls-(l/2),

jo

r r

1.8. Girding's Inequality. If A is a classical pseudodifferential operator of order m with the principal symbol a,(x, 5) satisfying the condition Re a,(x, 5 ) 2 Co I 5 I",' C , = const., 15 I > 1, then Re(Au, U) Co IIuII& - C1 IIuII&-1)/2

IIBUlls-m

19

2.2. The Subprincipal Symbol. The invariance of the principal symbol under the change of variables enables us to define various classes of pseudodifferential operators on a manifold that are characterized by the principal symbol, namely, elliptic operators, operators of the principal type, etc. In investigating operators with multiple characteristics a significant role is played by the subprincipal symbol

Invariance of the Principal Symbol Under Canonical Transformations that is defined in an invariant manner on

2.1. Invariance Under the Change of Variables. Let us first examine the question of invariance of a pseudodifferential operator under the change of variables. We express a given pseudodifferential operator A in the form (6)and then replace

z 2

= {(x,

0; a,(% 5 ) = 0, d,,gao(x, 5) = O},

the set of double characteristics of the operator A.

Yu.V. Egorov

I. Microlocal Analysis

2.3. Canonical Transformations. We can examine a wider class of transformations under which pseudodifferential operators are mapped into pseudodifferential operators. Let us set

tor [A, B] = AB - B A is proportional to the Poisson bracket of the principal symbols of the operators A and B. Therefore to every such transformation there corresponds a homogeneous canonical transformation of T* IR",and, hence, a homogeneous generating function S and an operator @ of the form (8). In fact, a far more strong result is valid (see Beals [1977]; Singer was the first to obtain results of this kind). Let X and Y be smooth manifolds with H,(T*X\O, C) = 0. Let L"(X) be the space of classical pseudodifferential operators of order m whose symbols a(x, 5 ) have the asymptotic representation a x u j , where aj(x, t5) = t"-jaj(x, 5 ) for t > 1,151 > 1; thus Lm(X)= Lm(X).Let i be an algebraic isomorphism L"(X) + La(Y) such that i: L"(X) + L"(Y) for all rn E Z.Then there exists an invertible Fourier integral operator A : Cm(Y) -+ Cm(X)such that A . i(P) = P.A . The phase function of this operator defines microlocally a canonical transformation that enables us to compute the principal symbol of the operator i(P).

20

s

@u(x)= (2n)-" b(x, t)ii(<)e's(x*o d5,

(8)

where S is a real-valued smooth function on T*IR" such that S(x, t5) = tS(x, 5 ) for t 2 1 and 151 2 1, and det IIa2S/axia<jll# 0 for 151 2 1. As to the symbol b, we assume that b E So and that Ib(x, <)I 2 Co > 0 for 151 2 1. Theorem 2.1 (Egorov [1969, 19841). To every pseudodifferential operator A of order m there corresponds a pseudodifferential operator A', of the same order, such that @ A - A'@ is a smoothing operator. If ao(x, 5 ) is the principal symbol of A , then the principal symbol of A' is defined by the formula

21

u

5 3.

N

Canonical Forms of the Symbol

This result implies the validity of the assertion of $2.1 if one sets S(x, 5 ) =

f (4 5. *

The transformation (x, 5 ) H ( y , q ) under which

v) 5 = , x ,

y = - as(%rl) all

is called a canonical transformation, and the function S in this case is referred to as its generating function. The canonical transformations play a significant role in classical mechanics (see Arnol'd [19743). In general, a canonical transformation is a transformation of IR2", the space of the points (x, 5 ) = (x,, ...,x,, C,,. . ., I&,),that preserves the Poisson brackets

for every pair of functions f and g in Cm(lRz").Since the variables x and 5 do not enjoy equal status in the theory of differential equations, we confine our attention only to homogeneous canonical transformations. That is, we consider only those transformations whose generating functions satisfy the homogeneity condition, namely, S(x, t r ) = tS(x, 5 ) for t 2 1, 151 2 1. However, in the general theory of differential equation arbitrary canonical transformations are also used (see Egorov [1984], Maslov [1976]). 2.4. An Inverse Theorem. It can be shown that the above-mentioned class of homogeneous canonical transformations turns out to be the maximal admissible class that preserves the algebra of the classical pseudodifferential operators. This becomes immediately clear from the fact that the principal symbol of the opera-

3.1. Simple Characteristic Points. The canonical transformations make the investigation of differential and pseudodifferential equations considerably simple. Let us remark that such an approach is impossible within the framework of the theory of "pure" differential equations. One obtains the class of pseudodifferential operators in a natural manner by applying canonical transformations to differential operators. As we mentioned in $ 1.7, the microlocal study of the equation A(x, D)u = f ( x ) at a non-characteristic point (xo, 5') can be carried out easily after constructing a parametrix in a conical neighbourhood of this point. In the order of complexity, next are the simple characteristic points. Suppose that the principal symbol a, of A is a real-valued function such that ao(xo,5') = 0 and dyao(xO,5') # 0,1 1. Then in a conical neighbourhood of (xo, 5') the symbol of A can be reduced to the form 4(x, 5)5,, where q is the symbol of an elliptic operator of order m - 1. By inverting the operator Q, with symbol q, we can reduce the investigation of the equation Au = f to solving an equation of the form D,u = g. To construct the desired canonical transformation, we present a device for finding its generating function. Since the equation Au = f is equivalent to an equation of the form A'-"Au = A'-"f, where A is an operator with symbol (1 1512)1/2,we can take, without the loss of generality, the order m of A to be 1. With the aid of the rotation x' = Tx, we can achieve that aao(xo,5°)/a<1 # 0. The desired canonical transformation @: (x, 5 ) -+ ( y , q) must be such that a d x , 5 ) = q,. This results in the differential equation

+

Proposition 3.1. If A ( x , D ) and B(x, D ) are pseudodifferential operators of the principal type with the same principal symbol, a, = bo, then there exist elliptic pseudodifferential operators R(x, D ) and S(x, D ) of order zero such that A R - SB is a smoothing operator (see Egorov [1984]).

We supplement this equation by the initial conditions n

s = 1 xjqj

for x , = xy.

j=2

By construction, the plane x 1 = xy is non-characteristic, and hence this Cauchy problem has a unique solution in a certain neighbourhood of (xo, qo), where the vector qo is found from the condition that 8S(xo,?')/ax = 5'. Thus qy = 5: for j = 2,. ..,n, and qy = 0. It is easy to see that S(x, tq) = tS(x, q) for t > 0 and that det IJazS(xo,q0)/8xi8qjll # 0. In a similar fashion, the canonical transformation, resulting in a,(x, 5 ) = q,, can be found also when d p , ( x O , 5') = 0 but d,ao(xO,5') # 0 and the form d,ao(xO,5') is non-collinear to the form todx. In this case, we seek the generating function S = S(y, 5). To be definite, we assume that d,ao(xO, 5') = d x , and 5: = 0, I('[ 2 1. S must satisfy the equation

3.2. Double Characteristics. If a,(x, 5 ) = p2(x, 5), where p is a first-order realvalued symbol of the principal type, then the operator A with the principal symbol a, is microlocally equivalent to every operator of the form D: + C ( x , D). Here C ( x , D ) is a first-order operator whose principal symbol coincides with the subprincipal symbol of A when 5, = 0. In the general case, when the principal symbol a,(x, 5) assume real values, with a,(x, 5 ) = 0, 15'1 2 1, and da,(xO,5') = 0 but d2a,(x0, 5') # 0, the question of reducing the symbol to the canonical form in a neighbourhood of (xo, 5,) is resolved under the assumption that the resonances are absent (see Sternin t19841). 3.3. The Complex-Valued Symbol. When the principal symbol a, of the operator A(x, D ) is a complex-valued function, there arises the question of simultaneous reduction of the pair of functions a = Re a, and B = Im a, to the canonical form. Let us list some of the known results. Two operators A and B are said to be equivalent at a point (xo, 5') if there exist an elliptic pseudodifferential operator Q and a Fourier integral operator @ of the form (7) such that the principal symbols of QA@ and @B coincide in some conical neighbourhood of (xo, 5'). 1. If a,(xo, 5') = 0, to # 0, and if A is a principal type operator with the principal symbol a,, then at the point (xo, 5') A is equivalent to an operator with the symbol itl + b(x, 5'). Here 5' = (t2, ...,ln), 5: = 0, and the real-valued and b E S' (IR"-') for every fixed value smooth function b is independent of of x 1 (see Egorov [1984]). 2. If a,(xo, 5') = 0, to # 0, and the forms dya(xO,to)and d$(xo, 5') are linearly independent, then at the point (xo, 5') the operator A is equivalent to an operator with the symbol it, + t2 a ( x , - xy)151 y(x, g'), where a = const., 5: = (S = 0, y(xo, 5'') = 0, and d,y(xO, 5") = 0 (see Egorov [1984]). 3. If a,(xo, 5') = 0 and {a, p } (xo, 5') # 0, then at the point (xo, 5') the operai x 1 t 2 ,where 5: = 0, tor A is equivalent to an operator with the symbol X: = 0 and 5: # 0 (see Sat0 [1970a]). 4. Suppose that a,(xo, 5') = 0 and let the forms da(xO,
and the initial conditions n

s = jc y j t j + - 5' =2 2151

23

I. Microlocal Analysis

Yu.V. Egorov

22

for y , = 0.

Such a problem has a unique solution in a neighbourhood of ( y o ,5'). Here the vector y o is determined from the condition that xo = 8S(yo, (')/at; thus xi" = yy forj = 2 , . . .,n, and yy = 0. It is easy to see that S ( y , t5) = tS(y, 5 ) for t > 0 and 5°)/8yi85jll # 0. that det Il~?~s(yO, Having constructed the generating function S(x, q) or S(y, 5 ) in a conical neighbourhood of the point in question, we can extend it to a neighbourhood of the form o x IR" so that for 151 >, 1 S becomes a non-degenerate homogeneous function. By applying an operator @ of the form

<,,

s

@u(x)= (2n)-" 6(5)eiScx*s d5,

+

we can reduce the first-order equation Au = f to the form Bu = 9;moreover, the principal symbol b,(y, q ) coincides with q, in the conical neighbourhood of ( y o ,qo). We can then find an elliptic operator Q, of order zero, such that the operator RBQ has the full symbol q, in the conical neighbourhood of (yo, 1'); here R is a parametrix for Q. In conclusion, we mention

+

<,+

+

Definition 3.1. An operator A ( x , D ) is said to be an operator of the principal type if the form d x , S a O (5~) ,is non-proportional to the form todx at each characteristic point (xo, 5') E T*Q.

3.4. The Canonical Form of the Symbol in a Neighbourhood of the Boundary. Let X be a symplectic manifold, and F and G be two hypersurfaces in X defined, respectively, by the equations f = 0 and g = 0. F and G are said to be

It is clear from the above results that the investigation of a principal-type equation Au = f with a real-valued principal symbol leads to the solution of elliptic equatons and of equations of the form D,u = 9. We mention f ,

Y'

24

I. Microlocal Analysis

Yu.V. Egorov

It is clear that, in particular, P ( 5 ) # 0 for all sufficiently large 151. For a parametrix of P(D) one can take an operator Q(D) with the symbol q(<)= h(<)P(()-', where h E C", h(5) = 0 for 151 < a and h(5) = 1 for 151 2 b; here a is so large that P ( 5 ) # 0 for 151 2 a. It can be shown (Hormander [1983, 19851) that Q E L;,',, with some m', p > 0. If A ( x ) is an invertible matrix depending on X, then q(A(x)S)E SF'l-p. 2. Let qa(x)= 1, u E Z", where (pa E Cg(IR") and qa(x)= 0 for Ix - a1 2 1, and lD8qa(x)l < C, for all CI and B. Let a E Srd. Then a function of the form

glancing surfaces at a point p E F n G if at this point df and dg are linearly independent and { f, g} ( p ) = 0 but { f, { f,g > ( p ) # 0, { g , { f,g } } ( P ) # 0. Melrose [1976] has proved that by means of a canonical transformation every pair of glancing surfaces can be reduced to the form f = x 1 and g = 5: - x 1 - t2, and to the form f = x 1 and g = 5: - x 15: - 5 2 5 3 if one restricts to only those canonical transformations that are homogeneous in 5. Let A be a strictly hyperbolic operator of second order in the domain 51 = { x E IR", x 1 > 0 } ,n 2 3, and let a(x, 5 ) be its principal symbol. Let p be a point in T*(dQ)\O such that a(p) = 0 and {a, x l } ( p )= 0. Melrose's theorem enables us to reduce A to its canonical form in a neighbourhood of p , with the plane x1 = 0 preserved. If {a, {a, x l } >( p ) < 0, then A is equivalent to an operator with the symbol 5: x 1 5 i 5,5,,. If {a, {a, x l } }( p ) > 0 (in this case p is referred to as a point of diffraction),then A is microlocally equivalent to the operator 0:- xlD," - D,D,,. This enables us to construct microlocally a parametrix for the boundary-value problem for A in the form of an Airy-Fourier integral operator which in the local coordinates is of the form

>

+

2

2 q a ( t I ~ I - p ) ~ , ( x I ~ a I b ) a 5") (xa,,

a. B

belongs to the class SFd; here 5" is an arbitrary point in supp qa(<151-p) and x,, is an arbitrary point in supp ~p,(x15'1~). This function may be regarded as an approximation to the symbol a(x, 5). Similar properties are also possessed by the function

+

where a[;!(., 5 ) = (iDs)d(iDx)ya(x, 5). The calculus developed above for operators of the class L" can be extended to those belonging to the class L;,d with p > 6. This last requirement is imposed so that the asymptotic series corresponding to the symbols of the product or of the adjoint operator are series in descending order of 151. By a change of variables x = q ( y ) ,the symbol a(x, 5 ) goes over to a ( q ( y ) ,'q'(y)-lq) + .... Thus the class Sfd is preserved if 1 - p < 6 < p . Under these conditions, a class of operators L;,,(X) is defined on a smooth manifold X.This condition is sufficientfor an elliptic operator to have a parametrix. The operators from the class LpO,dare bounded in L , if 6 < p . For an operator A, with symbol a E to be compact, it is necessary and sufficient that a(x, 5 ) -,0 as 5 -,CQ,and the convergence is uniform if x lies in a compact subset (see Kohn and Nirenberg [1965], Hormander [1983,1985]). Calderon and Vaillancourt have proved that the operator A with symbol in SPqp is bounded in L , when p < 1. More precisely, they showed that the operator A defined by

where S , S1,S , are the phase functions and A(s) the Airy functions

.

rm

0 4. Various Classes of Pseudodifferential Operators 4.1. The L;, classes. We have considered hitherto pseudodifferential operators of the form (2) from the class L" with the symbols belonging to s", that is, with symbols satisfying the inequalities (3). The most important property of L" is that there exists a parametrix for an elliptic operator in L". If L-" = L", the elliptic operators are invertible in the classes L"/L-", and no other operator does have an inverse in this class. From this point of view more interesting are L;,d, the spaces of operators of the form (2)with symbols belonging to the classes SFd. These classes are characterized by the estimates

n

lD;D,8a(x, <)I

Examples. 1. Let P ( D ) be a hypoelliptic differential operator of order m,with constant coefficients.It is known (Hormander [19631) that such operators are characterized by the relations = 0, lim IDaP(5).P(5)-'I IFI+m

~ u ( x=) (2.n)-"

Vu z 0.

jj

e'(X-Y).r

4x7 Y , 5 M Y ) dY d5

< 6 < 1, m < ( p - 6 ) n and if lD:D,BD;a(x, y, 5)l < C(1 + for 0 < IyI < 2[n/2] + 2,O < Iu + 81 < 2mj, where mi is the smallest value of the is bounded in L , if 0 < p

< C",,(l + 15l)"-pl"l+@',

where p 2 0 and 6 2 0. As in (3), the constants C,,, may depend on R for 1x1 < R.

25

15l)"+dl"+Bl-Plyl

-

integer r for which r(1 - 6) 2 5n/4 (see Calderon and Vaillancourt [1971]). Hormander has established that the operator A with symbol a and having compact support with respect to x and y is bounded in L, when p > 0, m < 0 and 2m < ( p - 6)n. He also demonstrated that this result fails to be valid if m > 0 or if 2m > ( p - 6)n (see Hormander [1983, 19851). Chin-Hung-Chin has shown that A may not be bounded if 6 = p = 1 and m = 0.

26

I. Microlocal Analysis

Yu.V. Egorov

Properties close to the above mentioned ones are possessed by operators whose symbols belong to Sl;,p,b, the classes introduced by Kumano-go and Taniguchi [1976]. A symbol a E Sl;,p,bif p;D,Ba(x,

<)I

< Cu,,l(x,

+ lxIP(1 + Ill),

<)I < A,,,l(x, <)l+al81-'al, 1(x + y, 5 ) Q A,(1 + lylP1(x, 8,

p;D,Bn(x,

<),I5 - ?I

(A4) R(x, 0) Q C(l

2 0; 0 < 6 < 1; To

tl

2 0.

+

\ 1/2

+ ~ ~ ~ ) m ( x ) +- ~151)]1, u ~1.[ h ( l

l ~ ; ~ , B a ( xt ,) Q ~~,,(1

4.2. The L;,,, Classes. In their works Beals and Fefferman [1975,1977] proposed a wide-ranging extension of the classes of symbols of pseudodifferential operators. The calculus devel~pedby them enables us to invert operators with symbols of the form

+ ipl(X)X:kS2 + pz(x), Pl(0) # 0, p1: IR" R'; + pl(X)X:k 0; P,(O) > 0. x1t: + iPl(X)51 + P2(X)52 + p3(x), +

51

5:

This was not possible in other variants of the calculus of pseudodifferential operators. The definition of the classes of symbols in this calculus starts by describing the weight functions. The continuous positive functions cp and @ defined on the space of variables (x, 5 ) E IRz" constitute a pair of weight functions if there are positive constants c, C and 6 such that (A,) CP G C ; (A,) @CP 2

C;

where R = Q,/cp;

c,

t p 2 ,

15 - ?I Q cR(x, <)6+1'2.

The pair @, cp is said to be localizable if for every compact set K c IR" a constant E > 0 can be found for which @ 2 ~ ( + 1 I
+

(01) IA(x, 0- 4 Y ,

Let us also mention the class of operators of variable order, introduced by Unterberger and Bokobza [19641. Symbols belonging to this class are homogeneous functions in 5 but m(x), the degree of homogeneity, depends on x, and, moreover,

< cpb, 5)cp(Y, rl)-l < c,

G CQ,(X,5);

+ [XI)',

if Ix - YI Q cR(x, 5)%(y,

The classes close to the above mentioned ones were also studied by Vishik and Grushin [1969] and by Grushin [1971a, 19723. Within the framework of this calculus, one may construct, for example, a parametrix for the operator - A lxlZkin the whole space. For this construction, it is convenient to use a weighted Sobolev space, for example, the space with a norm of the form /

if Ix - Yl Q ccp(x,

(A,) c Q R(x, 5)R(Y,v)-' Q

<)"+al@I-pla!

Here 1is a weight function that satisfies the following conditions: 1 < 1(x, 5 ) < AO(1

(A3) c Q @(x, 5 ) @ ( Y , 4-l Q c , c

21

(0,) c(@cp)-"

+

?)I Q c,

if Ix - Yl Q ccp(x,

0,

15 - rll Q @(x, 5); < en@-Kcp-K< c(@cp)"

for some real numbers k, K and m.

Definition. A function a E Si,,, if

Examples. 1. If @ = (1

+ 151)p,

cp = (1

+ 151)-a,

1= mln 151,

we obtain the Hormander classes SFb. 2. The Kumano-go-Taniguchi classes Sl;,p,bare obtained if @ = ,'A cp = 1-' and ,u = m In 1. 3. If we take @ = 1 151, cp = [l ln(1 I<[)]-', and 1 = p(x) ln(1 [
+

+

u H a(x, D)u(x) = (2n)-"

+

s

+

a(x, 5)6(5)eixcd5

whose symbols a belong to Si,,, constitute a class of operators LL-,,,. These operators map continuously the Schwartz space S into itself. Let us mention the basic facts regarding the calculus of such operators.

IlIIIIHIIII11111111111111111111111111111111111II

28

I. Microlocal Analysis

Yu.V. Egorov

29

-

I. If A E Li,, and B E L$,,, then AB E Lz;. The symbol a o b of the operator AB has an asymptotic expansion a

0

b

-c

ilal

zD;a(x,S)DNx,5)

in the sense that

with Ixk

- ykl

6 c R k ( x , t)dRk(y,V)-1'2?

Example. Let a(x, 5 ) = t1 + ix?kpl(x,5 ) p 1 is a real-valued function and where

I t k - V k l < c&(% 5)d""2).

+ po(x, 5), where p1

E S',

po

E So,

511 2 c15'1 for 15'1 2 c, 5' = ( 5 2 , ..., tn). where 6 = 1/(2k + 1) We set g(x, 5 ) = It1[+ x ? ~ I ~ ' I + ( 1 + IPl(X,

Q1

for each natural number N . 11. If A E Li,,, then the formally adjoint operator A* belongs to Li,, and its symbol a* admits an asymptotic expansion lldl

= gWk(l +

Qjj=1+151,

cpj=l

= g1'2k(l

15))-1'2k,

for j = 2 , 3 ,..., n.

< C(IIAUIIY, + II~IIYJ,

ECW). holds. This implies the subelliptic estimate, because e+"2 c ( l [19771).

Aj+1

j ,

-1

where cj and dj are positive constants. Let aj E Saj. Then there exists a symbol aj, that is, a E Sao such that a a-

+

,

The operator A with the symbol a has a right parametrix and a left parametrix belonging to the class Sp,wherep = In g . In particular, the estimate IlUll'

111. Suppose that Aj E O(@, cp) for j = 0, 1, . . .,with

< Aj + cj, Lj - Aj+1 < dj(Aj-1 - A j ) + ~

151)d(2k-1)/2k

1 aj E Sd".

4.3. The Weyl Operators. Hormander [19791 has introduced more general classes of pseudodifferential operators. He proposed that they be expressed in a form generalizing the Weyl's form: U(X,

jCN

IV. If A E Lo,then A : L, + L, is a continuous operator. V. Let A E O(@, cp). Put H A = Hi,,= { A u ; u E L2, A E L-'1).

Then H o = L2 and S c Ha c S'. Moreover, these embeddings are continuous as topological spaces. and the image sets are everywhere dense; also (Ha)'= -+ H a is continuous. If A E L', p E O(@, cp) and A E O(@, cp), then the map A : HA++" There exists an operator A E L' such that the operator A : Ha++" + Ha performs a topological isomorphism. In particular, H A is a Hilbert space. If a E S' and e-p@acpaD;Dta(x,5 ) + 0 as 1x1 151 -+ 00 for all a and 8, then the operator a(x, D):Ha++' + Ha is compact. In particular, the embedding H i c Ha is compact if p - A + +a as 151 + 00 uniformly for x lying in a and consists of neighbourhood of a compact set K . Here Hi is a subspace in H+" functions having supports in K . Beak has also examined more general spaces where @ and cp are the vectors @ = (O1, ..., @,,), cp = (cpl, ..., cp,,), and Qj and cpj are weight functions of the variables xj and t j .What is more, Gj(x, 5 ) d c@k(x, 5)' for a l l j and k, and if 71(x,5 ) = mink @ k ( x , 5 ) ( P k ( x , 51, then

+

@j(x, t)Gj(y, ~ 1 - lG CCX(X,5 ) + Z ( Y , l)Ic, cpj(x, t ) q j ( Y , v)-' G CCn(x, t) + n ( ~ ?)I" ,

+ 151)' (see Beals

jjat+,

D)u(x)= ( 2 ~ ) - "

<)ei(x-y)%(y)dy de.

(9)

If a E S", then such an operator differs from the usual operator with symbol a by an operator of order m - 1. However, the operators of the form (9) have the important property that they remain invariant under linear symplectic coordinate transformatons: U-'a(x, D)U = (a.x)(x,D). Here x: IR2"+ W2" is a transformation that preserves the form o = A dxj and the operator U is determined uniquely by x to within the constant multiplier. Let V be a finite-dimensional vector space and let g x ( y ) be a positive definite quadratic form in this space for each x E V. For a function u : o + CC belonging to the class C k ( o ) ,where o is a neighbourhood of x in V, we define the horm of its kth differential by the formula G

The metric g is said to vary slowly if there exist positive constants c and C such that gx(Y)

< c =s x + , ( t )< Cgx(t),

Vt E

v.

A function rn, defined on V and assuming positive real values, is said to be g-continuous if there are positive constants c and C such that g,(y) 6 c => m(x)/C

< rn(x + y ) G Crn(x).

I. Microlocal Analysis

Yu.V. Egorov

30

If g varies slowly and m is g-continuous, one can introduce the class S(m, g) of functions u that are smooth on V and for which sup lulf(x)/m(x)< co

for each k

E

Z,.

Clearly, Cg(V) c S(m, 9). Moreover, uu E S(mm’, g) if u E S(m, g) and u E S(m’, 9). If IuI > cm, then u-l E S(l/m, 9). Let B be a quadratic form, with real coefficients, on V’, the dual space of V. The differential operator B(D) is defined by the formula B(D)eixt= B(<)eix6on functions belonging to C2(V).We set

31

form u, regarded as a quadratic form on W 0 W. This yields a(x, 5 ) = eia(Dx.D,;D,J,)/Z

5)az(Y, rl)I(y,r/)=(,,<). Here it is assumed that the hypotheses of Proposition 4.1 hold, that is,

s:: d Cg::,(l + g::l(w,

- W)IN?

m(wl) < cm(w)(l + gGl(w, - w ) ) ~ , w, w1 E W. Moreover, if a, E S(m,, g) and a, E S(m,, g), then a E S(m,m,, g), and

g,B(x) = SUP (x, 5)2/g(m. t

Suppose that the Riemannian metric g varies slowly in V and that m is g-continuous. In this case, g and m are said to be compatible with B if there are constants C and N such that the inequalities

holds for all y, t

E

Examples. 1. The space Szdof symbols a satisfying the inequalities lDpI,Ba(x,<)I

< q P ( 1+ 151)-l@’+alPI

sy(t) < Cgx(t)C1 + s,”(x - Y)lN,

is obtained as S(m, g) if one sets

m ( y ) < Cm(x)C1 + g,”(x - Y)lN

gx.t(Y,rl) = lYI2(1 + 151)2d+ 11IZ(1+ 151)-zp, m(x, 5 ) = (1 + 151)”Moreover, g varies slowly if and only if p < 1.Since

V.

Proposition 4.1. If g and m are Compatible with B and if g,(y) < g:(y) for all x and y, then the continuous function eiB%(x) is defined for u E S(m, g), and leiB(D)u(x)I < m(x)Ilull, where

for all N . Here h(x, 5)’ = sup g,,t/gz,r.

g,qt(Y, ‘I)= IYl2(1 + 151)2p + 1112(1+ 151)-26, it follows that the inequality g,,{ < g:,< holds if and only if 6 < p. The compatibility of g with u implies that

11.11 is some seminorm in S(m, g) that depends on C and N.

If u E S, then u = eiB(% = U(1, x) is the value at t problem

au

-=

at

iB(D)U,

U=u

=

1 of the solution to the

at t = O .

that is, 6 < 1. Thus the calculus is developed for 0 < 6 < p < 1 and 6 < 1. These conditions are necessary for the operators belonging to SPqdto be bounded in L, (Hormander [1983,1985]). 2. If we take

5r2

It is easy to see that the function

S,,C(Y, rl) = lY12& + I?IZ% then the operators belonging to S(m,g) become operators of the BealsFefferman class. In this case, the condition @cp 2 1 coincides with the condition g < 9“.The remaining conditions may be reduced to the condition

serves as a Schwartz kernel for the operator (9), and, moreover,

s(

a(x, 5 ) = We set

w = v 0 V’,

K x

WY, rl) < C W , 0, where w , 5 ) = WX, OcP(x, 0 - l

: 3

+ -,x

and

- - e-”< dt.

4x9 5 ; y, rl) = (Y, 5 ) - (x, rl), (x, 5 ) E

w,

5)Ix - YI2 for some 6 > 0.

w,

(Y, ?) E W.

It is immediately clear from the definition that the formally adjoint operator a(x, D)* has the symbol a(x, 5). The symbol a(x, 5 ) of the composotion al(x, D). az(x,D) can be found by applying Proposition 4.1 to the space W 0 W and the

+ 15 - rlI2R-’(y, ?) < CCWY, rl)W,5)-’16

Theorem 4.1 (Hormander [1979]). Let g < g‘, and let the metric g and the ficnction m be compatible with the form u. The operator a(x, D), with the symbol a in S(m, g), is bounded in L, if and only if m is bounded. This operator is compact in L, if and only if m(x) -,0 as x + co.

32

Yu.V. Egorov

I. Microlocal Analysis

Using the calculus developed by him, Hormander [1979] obtained an explicit formula for the index of an elliptic operator that generalizes the results of Atiyah and Singer and of Fedosov [1974].

Assume that the principal part of the symbol a, of A does not .take negative values. Then it can be shown that the operator A - IZ is invertible for those negative I that are sufficiently large in absolute value and that for them the inequality ll(A - IZ)-'11 < CIA[-' holds. We assume further that A - IZ is invertible for all A < 0. Then in the A-plane there exists a contour r which comes from -a from below the real axis, goes around A = 0 along a given curve and then again goes off to -a but now above the real axis, and is such that the spectrum of A lies to the right of r.For Re z < 0, we set

Theorem 4.2 (Hormander [1979]). Assume that the metric g is compatible

< ( 1 + 1x1 + 151)-6 for some 6 > 0. Let a E S ( l ,g), where with CJ and that g(x' g% 5 ) a(x, 5 ) E L(C", C").Let the matrix a-'(x, 5 ) exist and be bounded outside some ball B. Then the operator a(x, D ) is Noether and ~

ind a

=

-(2ni)-"

(n - l ) ! (2n - l)! ~

s

tr(a-' da)'"-'.

dB

Here an orientation in IR'" has been so chosen that the condition d 5 , '.. A dtn A dx, > 0 is satisfied.

A

dx,

A

We also note the following result which generalizes Theorem 2.1. Theorem 4.3 (Hormander [1979]). Suppose that the metric g varies slowly and that m and g are compatible with 0. Suppose that for each point ( x , 5 ) E IR" x IR" there is a value b = b(x, 5 ) < ( 1 + lt;12)1/3such that b(ldx12 + ld5I2(1 + l5l2)-') < gx,&dx,d 5 ) b2(ldxI' + ld5I2(1 + 1512)-'). Suppose that a E S(m, g ) and a = 0 for large 1x1. Let cp E Sy and cp = 0 for large 1x1 + 151. Let A , = @A@*,where @ is the Fourier integral operator @u(x)=

s

~ ( x5)ii(<)eis("*" , d5

with a non-degenerate phase function S that defines the canonical transformation

Then a ,

ES

, : outside each conical neighbourhood of the set

{ ( x , W x , <)/ax):(x, 5 ) E cone SUPP cp} and a ,

o

33

x E S(m, g ) in a conical neighbourhood of the set { ( W x , 5 ) / X , 5): (x, 5 ) E cone SUPP cp}.

Moreover, a , o x can be expressed explicitly in terms of a, and $ b(x, 5) < (1 + for 6 < 2/3, then the principal term of a , o x equals 1(2nn)"cp(x,i)121dets g x , c)l-14z, 0, where (z,i) = x - l k 5).

fj5. Complex Powers of Elliptic Operators 5.1. The Definition of Complex Powers. Let us consider an elliptic pseudodifferential operator A of order m on an n-dimensional closed manifold M . We shall define its power A' for Z E C and show in the sequel how these operators can be used in the theory of index and in the spectral theory of elliptic operators (see Seeley [1967] and Shubin [1978]).

where 1' is defined as a holomorphic function of 1 that equals e' for I E r. Using the Hilbert identity

(A - p ) ( A - I I ) - l ( A - pz)-I

In

= lIl'eiz a r g a

= ( A - AZ)-, - ( A - pz)-I,

we can easily verify that the semigroup property AZAW

= A'+,

holds if Re z < 0 and Re w < 0. In particular, the relation A _ , = (A-')k holds for non-negative integers k. Moreover, for Re z < 0 the operators A , are bounded in each space H,(M). If z E C, we can define A' by the formula A' = A k .A,+, where k is an integer such that k > Re z. It is easy to verify that A' = I , A' = A , A-' is the inverse of A and that A" coincides with A , for Re z < 0. For Re z < k, A' is a bounded linear operator from the space H"(M) into the space HS-mk(M),and depends holomorphically on z E C.The family of operators A' possesses the group property: A'A" = A'+w for all z and w E C. 5.2. The Construction of the Symbol for the Operator A'. The theory of pseudodifferential operators enables us to construct the symbol of A' in an explicit form. We start by constructing the symbol for a parametrix of the operator A - IZ. Let X be a coordinate neighbourhood on M that can be identified with a domain in IR" by fixing a coordinate system. We represent the symbol a(x, 5 ) of A in X in the form of the sum

c aj(x,0, rn

a(x, 5 ) =

where aj(x,5 ) =

1a,(x)<', j

= 0, 1, ..., m.

Cl=i

j=O

It is natural to seek the symbol of the parametrix of A - AZ in the form of a sum of functions that are homogeneous in (5, A'""). We denote these functions by bO_,-j(x, 5, A), j = 0, 1, . . ., where bO_,-i(x, t5, tmA)= t-m-ib!,-i(x,

(,I)

for t > 0 and

151 + 1 1 1# 0.

34

I. Microlocal Analysis

Yu.V. Egorov

35

5.3. The Construction of the Kernel of the OperatorA'. We recall that the Schwartz kernel of the pseudodifferential operator

These functions are determined from the recurrence formulae

Bu(x) = ssei(x-y)%(x, <)u(y)dy d t ,

where b E S", is defined for m < - n by the formula

s

j = 1,2, ...

K(x,y) = ei(x-Y)Sb(x, 5 ) d5.

The homogeneous components b$;*Tj(x, 5 ) of the symbol of A' are now constructed by means of the above homogeneous components of the parametrix in exactly the same way as the powers A' themselves were constructed with the aid of the resolvent ( A - AI)-l. It is clear from the construction that l~!,-~(x,5, A) are holomorphic functions of A on and inside r.For Re z < 0, we now put

A'bO-,-j(~, &A) d l ,

2ni

defining the values of A' as above. In particular, for j gives b$;,O(x,

1

4 ) = 2.ni

sr

A"[a,(x,

The Schwartz kernel K, of the operator A' is defined by this formula for Re z < -n/m and is a holomorphic function of z. It can be shown (Seeley [1967]) that the function K,(x, y), with x # y, can be extended to an entire function of z that vanishes for z = 0 , 1 , 2 , . . . . Moreover, KJx, x) can be extended to a meromorphic function having no more than simple poles at the points z.J = ( j - n)/rn; j = 0, 1,2, . .. . The residues yj(x) at these points equal

j = 0, 1, .. ., =0

the Cauchy formula If zj = I E Z + , then yj(x) = 0 and

5 ) - A]-' dA = ai(x, 5).

It is not difficult to see that the functions b$;fj(x, 5 ) are positive homogeneous in 5 of degree mz - j, and that the semigroup property

1

lal+p+q = j

a (w) 0

(iD<)"b$;.-O,(x,t).Dxbmw*-q(x,t

We note that at the extreme left pole zo = - n/m, the residue yo is given by

(z+w) 0 )U.1l =bm(z+d)-j(X, 5)

and the kernel K Ois regular at x = y, and

holds for Re z < 0 and Re w < 0. For an arbitrary z E (c we can set b$;fj(X,

5)=

1

p+q+lal=j

1 u (.z-k),O 7(iD<)"a%p(x,5)'Dxbm(z-k)-q(x, 5 )

if k E Z is chosen so that Re z < k. It can be verified that these functions b$;,!j do not depend on k, and for Re z < 0 they coincide with the functions defined initially, and that for integers k the relation b$)!j(x,

5 ) = u $ ~ - ~ (5x),

holds. Here and above, U?)(X, 5 ) are the homogeneous components, of degree j , of the symbol of Ak. Having constructed the symbols of A' in a coordinate neighbourhood X , we can construct these symbols on the whole M by means of the decomposition of unity. Any singularity which these symbols may have for 151 = 0 can be removed by multiplying by a smooth cutoff function.

i :It is clear that b-, >. we have

1

1

= (a,

- A)-', and by the recurrence relations defining b-m-j,

where yk, are polynomials in a,, . . .,

and their derivatives of orders

<j .

d

Yu.V. Egorov

I. Microlocal Analysis

5.4. The (-Function of an Elliptic Operator. In the spectral theory a significant role is played by the [-function which is defined for Re z < -n/m by the formula

5.5. The Asymptotics of the Spectral Function and Eigenvalues. We shall mention some of the simple results regarding the asymptotics of the spectral function of a self-adjoint elliptic operator A on a smooth manifold M (see Shubin El9781 and Seeley [1967]). Assume that a,(x, 5 ) > 0 for 5 # 0. Then A is a semibounded operator, and its eigenfunctions cp,, (p2, . . . constitute a complete orthogonal system. We shall assume that these functions have been normalised and the corresponding eigenvalues A,, A,, ... have been arranged in the increasing order. Let E , be the spectral projector of A, that is, let

36

[A@)

=

s,

KZb, x ) dx.

An analytic continuaton of this function yields a meromorphic function that has no more than simple poles at the points zj = ( j- n)/m, j = 0, 1, . . ., excluding the points zj = 1 = 0, 1, . . . . Moreover, it is easy to derive formulae for residues and for values of [ ( l ) with the aid of the above formulae for the kernels. Let us assume that a smooth positive density dx is fixed on M and that A is a self-adjoint operator with respect to this density. Then lA(z)= A;, and the series converges absolutely if Re z < -n/m and uniformly in z for each E > 0 if Re z < -n/m - E. The following arguments are due to Atiyah and Bott. Let A be a normal operator that has a complete set of eigenfunctions $ j corresponding to the eigenvalues Aj. Then the operator A" has a kernel K,(x, y) which is such that K,z(x, X ) = C ~;I$j(x)12,

jM

K,(x, x ) dx =

Taking now A = I LL*, we obtain K,(x, x ) explicitly in terms of the symbol of L. The same is true for I L*L. Thus we can obtain an explicit analytical formula for the index of L. Let us note, in particular, the connection with the classical Riemann zeta function [ ( z ) = n-". To do this, we take for M the unit circle and for A the P, where P is the projection operator onto the subspace of operator -D: constant functions. The eigenvalues of A are n2, n = 1,2,. ..,and their multiplicity is 2 if n > 1 and 3 if n = 1. Hence

We observe that

r

where N ( t )denotes the number of eigenvalues of A not exceeding t. It is evident that N (t)is a decreasing function and that [A(Z)

c:=l +

m

1 1; = 1 + 21;( -22).

:1

=

Assume that, as t -,+a,

N ( t ) = ct"

+

=

1 (u, V j h j .

aja

The kernel of this operator is known as the spectral function of A, and is of the form

where Re 6 < Re a. Then

c1(z)- [,(z) = dim ker L - dim ker L* = ind L.

iA(Z)

=

12;.

If $ is an eigenfunction of the operator L*L corresponding to the eigenvalue 1 > 0, then L$ is an eigenfunction of LL* corresponding to the same eigenvalue. Thus L maps the eigensubspace E,(L*L) in a one-to-one manner into E,(LL*) and dim E,(LL*) 2 dim E,(L*L). Similarly, we find that dim E,(L*L) 2 dim E,(LL*). Thus for L*L and LL* the non-zero eigenvalues and their multiplicities coincide. It is clear that if ll(z) is the trace of ( I + L*L)z and [,(z) the trace of ( I + LL*)z, then

+

Etu

31

L(Z) = c

sp

t" dt"

t ZdN(t).

+ O(tS),

+ f ( z ) = - _ _+ f ( z ) CCI

Z+Ci

and f is a holomorphic functon for Re z < -Re 8. Thus if we know the poles of CA(z)and the residues at these points, we can find the asymptotics of N(t). Similarly, the formula

implies that

j=1

The symbol of A is 5,. Therefore b-, = (5, - A)-' and b - 2 - j = 0 for j > 0. These facts and the above formulae show that yj(x) = 0 forj > 0 and that

Thus when s = 1, [(s) has a pole with the residue 1. What is more, [(O)

=

-3.

t" de(x, x, t).

K,(x, x ) = L

O

Moreover, if e(x, x, t ) = c0(x)tao+ O ( t S o ) and Re Po < Re a,, then

38

I. Microlocal Analysis

Yu.V. Egorov

where the function fo is holomorphic for Re z < -Re b. Since we know the poles of K , ( x , x ) and the residues at these points, we can find, as above, the asymptotics of e ( x , x , t). Thus we obtain the formulae

'

39

If A , has a complete system of orthonormal eigenfunctions qj,we can put

The trace of this operator is given by r

r

r

r

C A;lqj(x)12 dx = 11;

Trace K , ( x , x ) d x =

If we represent A; in the form

5.6. Complex Powers of an Elliptic Operator with Boundary Conditions. The above constructions can be extended to the far more complicated case of a boundary-value problem for a system of elliptic equatons (Seeley [19691). Let A be an elliptic system of smooth differential operators in a domain G and let B be a system of differential operators on the boundary 8G. Let us consider an unbounded operator A , defined in LJG), 1 < p < co, on the space of functions satisfying the conditions Bu = 0 on 8G. We assume that (1) the matrix Ao(x, 5 ) - AZ is invertible for A < 0, (5, A) # 0, and ( 2 ) the boundary-value problem Ao(X', 0; t', D,)u(t) = Au(t),

t > 0;

Bj0(x',t', D,)u(O) = c j , j = 1, . . . , k, u( +a) =0

has a unique solution with arbitrary constants cj E C, A < 0, (<', A) # 0. Here (x', t ) are local coordinates in a neighbourhood of some boundary point and t >,O in G. These assumptions guarantee the existence of the resolvent R , = ( A , - AZ)-l for A < 0 (see Seeley [1969]). What is more, if IAI is large, then ( A , - AZ)-'

where Op(c)f(x)= (2n)-"

ss

1Op(C-,-j) + C Op'(d-,,,-j),

eicx-Y)eC(x, < ) f ( y )d y d t ,

1;

Op'(d)f(x', t ) = ( 2 ~ ) ' - "

N

ds s R n - 1 1 R n - I

t, 5', s)f(y', s) dy' dt', ei(X'-y')t;'d(x',

C - , = [Ao(x, 5 ) - AZ]-', and where C - m - j ( x ,5, A) are homogeneous in (5, Allm) of degree - m - j while d - , -j(x', t, (', s, A) are homogeneous in (5, A'/'", t-', s-') of degree 1 - m - j . The functions C - , - j and d-m-j can be computed explicitly in terms of the coefficients of A and B.

we can extend the function K , ( x , x ) for x $ 8 G to a meromorphic function having no more than simple poles at the points zj = ( j- n)/m, j = 0, 1, ..., z # Z,.The residue yj(x) of this function at zj depends only on c - , - ~ . If zj = ( j - n)/m = I E Z,,the value of K , ( x , x ) can be expressed explicitly in terms of C-,-j.

The function JG K,(x, x ) dx has similar singularities. We can show that the residues at zj = ( j - n)/m equal r

r

JG y j ( x ) d x + J

dj(x')dx', dG

where dj(x')are determined explicitly by the functions d - , - j + l . The same is true also for the values of this function at the points zj = 1 E Z , . In recent years Ivrii [1980b, 19821, Vasil'ev [1984, 19863 and Melrose have obtained more precise results regarding the asymptotics, as t --* co,of N ( t ) in the form N ( t ) = ~ , t " /+~a, t("-l)lrn+ O(t("-l)im ) for boundary-value problems for elliptic equations. These works are based on the study of wave fronts of solutions to the boundary-value problems for suitable hyperbolic equations.

0 6. Pseudodifferential Operators in R"and Quantization 6.1. The Analogy Between the Microlocal Analysis and the Quantization. Let us recall what constitutes the process of quantization of a non-relativistic classical dynamical system with n degrees of freedom. The state of such a system can be uniquely determined by a set of values of the generalized coordinates (xl, . .., x,) and the generalized impulses (t,, . . . , l"), where = m j i j .In classical mechanics, the quantities under observation are the values of the functions F(x, 5). The result of observation of F in a state (xo, to)is the value F(xo, lo). In the classical theory, a significant role is played by the function n

1

6

Yu.V. Egorov

I. Microlocal Analysis

which is referred to as the total energy of the system or as the Hamiltonian. The function V is the potential of the field where the particles move. By the Newton's Second law, mixi = -aV/ax,, and this implies that

2. In a certain sense, lim,,,A = a(x, 5), that is, the original classical system must be derivable, as h + 0, from the quantum mechanical system thus constructed. These conditions do not define the quantization operation uniquely. We shall not discuss here various ways of quantization, but only note the analogy with the microlocal analysis where with the function a ( x , 5 ) is associated a pseudodifferential operator a ( x , D ) in such a manner that the Poisson bracket {a, b} of the two functions a and b becomes the principal symbol of the commutator [ A , BI. The theory of pseudodifferential operators meets the needs of quantum mechanics specially well if to a real symbol there corresponds (even if formally) a self-adjoint operator and if the transition from the coordinate representation to the impulse representation does not present difficulties, that is, if the variables x and 5 play analogous roles.

40

Evidently, the evolution with respect to time of each observable F ( x , 5 ) is described by the equation

The last expression is known as the Poisson bracket of the functions H and F, and is denoted by { H , F } . It follows from this equation that all the laws of classical mechanics are invariant under canonical transformations that preserve the values of the Poisson bracket for any pair of functions. In the quantum theory, the observable quantities are the self-adjoint operators in a Hilbert space X. The state of the system is determined by a wave function $ in 2. The role of coordinates here is played by the multiplication operator tj + xj$ while those of impulses by the differentiation operator $ + Dj+. The result of measuring the quantity A in the state $ is a random quantity having mathematical expectation ( A $ , $). The Hamilton operator H determines the evolution of the state of the system in accordance with the Schrodinger equation d* - = iH$. dt

6.2. Pseudodifferential Operators in IR". We remark that the quantization rules do not define uniquely the operator that corresponds to the function a ( x , t) even in the simplest case of differential operators. Thus with the function x j t j one can associate the following operators:

+

The last of these is to be preferred because it is formally self-adjoint. Therefore with the function a ( x , 5 ) of the general form it is natural to associate the Weyl operator in the form (9). The function a is not necessarily homogeneous in 5, and, in general, the variables x and 5 may enjoy equal status. This is especially convenient in the investigation of some problems of quantum mechanics where the transition from the coordinate representation to the impulse representation is natural.

In the equivalent Heisenberg theory, the state of the system $ remains unchanged while every observable A is subjected to the equation dA -= i ( H A - A H ) = i [ H , A ] . dt

Therefore all the laws of quantum mechanics remain invariant under unitary transformations U : X --t X. By quantization we mean the transition from a classical mechanical system to a quantum mechanical system that is carried out according to the following of") complex-valued functions u(x) whose rules. We take for X the space T2(IR squares are integrable with respect to the Lebesgue measure. We associate with each coordinate x j the operator of multiplication by xi and with each impulse the operator of differentiation, namely, u + hD,u. The quantity h is known as the Planck constant. The problem of quantization consists in associating with each The following two real-valued function a ( x , 5 ) a self-adjoint operator A in S. conditions must be satisfied. 1. The operator A depends on the parameter h, and the commutator [ A , B ] is of the form ihC + O(h), where C is an operator associated with the Poisson bracket {a, b);

41

Definition 6.1. A function a ( x , 5 ) belongs to the class T"' if the following conditions hold: 1. a E c"(IR: x IR;). 2. For any multi-indices CI and /? lDp,8acx, 01 < G p ( 1

+ IXl)"(l + 151)"',

( x , 5 ) E IR" x lR".

To construct the calculus of pseudodifferential operators in this class, it is natural to introduce a small parameter E and then consider the symbols a(&,x, t) with E > 0. '

Definition 6.2. A function a(&,x , 5) belongs to the class T+"if 1 . a € Cm(IR,+ x n;x IR;). 2. For any integer N > 0 N

a(E,x ,

5)=

Ekak(X, k=O

5 ) + E N + l R N ( E , x , t),

I. Microlocal Analysis

Yu.V. Egorov

42

Chapter 3 Fourier Integral Operators

where ak E T", and

+ IxI)"(l + 151)" for any multi-indices a, /?,and for any integer y > 0. Clearly, ID;D,BD,YRN(E, X , OI G Ca,D,y('

D:a

E

T,",

D:a

E

6 1.

5"a E T,"+lal

xaa E TT+lal,

T,",

if a E T,". These relations enable us to construct an algebra of pseudodifferential operators with symbols belonging to T," (Maslov and Fedoryuk [1976]). *

Examples 1. The Schrodinger equation

a+ = - - Ah2+

ihat

The Parametrix of the Cauchy Problem for Hyperbolic Equations

1.1. The Cauchy Problem for the Wave Equation. Let us first examine the solution to the Cauchy problem for the wave equation a2u

+ V(x)+

2m

-=

at2

describes the motion of a non-relativistic quantum particle of mass rn in a field with potential energy V ( x ) . To this equation there corresponds the symbol where a = const. and cp,

+ -2mI l l 2 + V ( X ) , E2

ET

E

= h,

+ 1x1)'.

belonging to T,' if I V(x)l G C ( l 2. The Helmholtz equation

+ are given functions from Cg(IR"). If we put u(t, r) = u(t, x)e-iXSdx,

s

describes the propagation of waves of frequency k in a medium with refractive index n2(x). To this equation there corresponds the symbol

~ ~ 1 + ( 1n2(x), ~ belonging to T: if In(x)l < C ( l 3. The difference equation u:+' - 2u;

E

v=@,

= k-',

u:-1

h2

=a

+ h),

u:+1

- 2u:

+

u;-1

h2

so that

ehd'd'f(t)= f ( t

= u(hn, hm).

Using

u(t, x ) = (272)-"

+ h),

[sin2(hD,) - a2 sin2(hDx)]u(t,x ) = 0. This equation coincides with the above difference equation if the values of u(t, x) are taken at the points of the mesh. To the given equation there corresponds the symbol -

a' sin2 E ( ,

belonging to T," (Maslov and Fedoryuk [1976]).

E =

s [@(r)

+ (27r)-"

we obtain the equation

sin2 ET

fort=^,

This yields

is associated with the wave equation u,, = a2uxx,where u; the formulae ehd'dxf(x)= f ( x

-dv= $ dt

+ 1x1).

+

for t > O ,

a2Au

then this problem reduces to the Cauchy problem for an ordinary differential equation:

+ k 2 n 2 ( x ) ) u ( x )= 0

(A

43

+

1

f [@(r)-

$(5)] eiat151+ixS d{ 1

$(<)I

e-iartSl+ixS d{.

Thus the solution of the Cauchy problem has been represented as a sum of two integrals that define Fourier integral operators with the phase functions +at151 xr. In general, by a Fourier integral operator we mean an operator that maps a function u into the function

+

h, Au(x) = (2x)-"

s

a ( x , 5)fi(<)eis(x*c) d5.

Yu.V. Egorov

I. Microlocal Analysis

Here a E S"' is the symbol of the operator and S is the phase function. It is usually assumed that the phase function satisfies the non-degeneracy condition

To solve the Cauchy problem on a large segment [0, TI, it is necessary to use the Maslov canonical operator (see 5 2.3 below).

44

1 *1

# 0 and that S ( x , t 5 ) = t S ( x , <) for t > 0. Thus a Fourier integral axiarj operator may be regarded as a generalization of a pseudodifferential operator.

det

1.2. The Cauchy Problem for the Hyperbolic Equation of an Arbitrary Order. Let P ( t , x , D,,0,)be a strictly hyperbolic differential operator of order m, with smooth coefficients, in the strip D = { ( t , x ) : 0 < t < T, x E IR"}. This implies that the roots A,, . . ., A,,, of the characteristic equation Po@, x , A, 5 ) = 0

5 # 0, and, moreover, IAj(t, x , 5 ) - &(t, x , <)I co = const > 0

45

1.3. The Method of Stationary Phase. If Re a 3 0, a # 0, then the Fourier Here, and in what follows, zli2is defined in transform of e-ay2is (2n/a)-112e-C2/2a. such a way that 11/2 = 1. Therefore the Fourier transform of ei X'J~:/', a, E IR\O, coincides with the function p y / 2 lajl-l/2e~i~/4e-1 i v:/Za,

n

where (r = x s g n aj. Similarly, if A is a non-degenerate symmetric matrix, the is given by Fourier transform of ei(Ay,y)/2 (2n)"/2 ldet ~ l - W ~ n i s g n A e/ -4 i W ' v , W 3

are real and distinct if

where sgn A denotes the signature of A. Iff E Cg(IRn),the Parseval theorem gives

s

if ( t , x ) E D,151 = 1, j # k. We consider the Cauchy problem for ( t , x ) E D,

P ( t , x , D,, D,)u = 0 for x

Dju(0, x ) = cpj(x)

E

dy

f(~)~il(AY,Y)/2

ldet A I-1/2eni

= (2nA)-"/2

sgn 4 4

s

f(iYl)e-i(A-'%d/2

dv.

Let us expand the exponential function on the right-hand side of this equation in a Taylor's series. Since

IR",j = 0, 1,. . .,m - 1.

The solution of this problem for a sufficiently small t is given by a formula of the form u(t, x ) = (2n)-"

s

ajk(t,x , 5 ) @ j ( < ) e i s k ( f * x ~ ~d<, )

mfl j,k=O

(1)

it follows that I

where the functions Sk satisfy the eikonal equations

5 at = A,(,,

x,

2)

for t > 0

where

and the initial conditions for t = O .

sk=x*(

In physical terms, Sk signifies the distance covered in time t by a ray of light issuing from the point x in the direction of the vector in a medium the optical properties of which are characterized by the function &. It is not difficult to see that Sk satisfies the non-degeneracy condition if T is sufficiently small, because for t = 0 we have

The coefficients C, here do not depend on A or f,and are of the form

<

det Ila2S,/8xiatjll = 1,

k

= 0, ..., m -

1.

The functions a j k (more precisely, their asymptotic series) can be easily found from the linear transport equations which are obtained by equating to zero the functions bjk in the following equality obtained from (1): PU(t, X )

= (2Z)-"

sc

j m-l ,k=O

bjk(t, X ,

~ ) @ j ( ~ ) e i S k (d5. "Xsr)

These coefficients vanish if lcll is odd. We now consider the integral

s

Z(A) = f(y)ei"s'y)dy, where S E C", f E C: and S has a unique critical point x o in supp f. We shall assume that this point is a point of non-degeneracy, that is, we shall assume that det 8 2 S ( x o ) / d x z # 0. Then by a non-degenerate coordinate transformation S ( x ) can be transformed to the form S = 1 ajy,Z/2in a neighbourhood of x o . Applying

I. Microlocal Analysis

Yu.V. Egorov

46

Definition 2.2. The class a* E H 1 ( M ,Z) is known as the Maslou index.

the above estimate, we have P

where N-1

The principal term of the asymptotics is

5 2.

41

The Maslov Canonical Operator

2.1. The Maslov Index. To construct a Maslov canonical operator or a global Fourier integral operator we need some topological concepts introduced in the works of Maslov [1965a] and Arnol'd [1967]. We consider the space C" of points z = x + it, x E IR", 5 E IR", where the unitary structure is defined by the scalar product

An n-dimensional linear subspace A in C" is called a Lagrange subspace if Im(z', z 2 ) = 0 for any two points z', z 2 E A . For example, x = 0 or 5 = 0 are Lagrange subspaces. It is natural to refer to the Lagrange subspaces as real-similar because every such subspace can be transformed to the real subspace 5 = 0 by an unitary transformaton U . The transformation U is defined to within an orthogonal transformation of a real subspace. Therefore the manifold A (n) of Lagrange subspaces is isomorphic to the factor group U(n)/O(n). Since the determinant of an orthogonal transformation is +1, the quantity (det U)? depends only on A and, since JdetUI = 1, determines a point on S , c C. Thus if we take a closed curve y: S' + A(n) to it there corresponds a mapping S' + S' that is characterized by its index. It is possible to deduce from here that H'(A(n), Z) N Z, and for a, a generator of the group of cohomologies H1(A(n),Z), one can take the value of this index. Definition 2.1. A submanifold M" in C" is called Lagrange if at each of its points the tangent space is Lagrange. Let M be a Lagrange manifold, and let z: M + A(n) be a map that associates a tangent space with each point x E M . This map induces another map r*: H'(A(n), Z) -+ H ' ( M , Z) under which a class a* of one-dimensional cohomologies of M is associated with a.

The Maslov index can also be defined as the index of intersection with a certain two-sided (n - 1)-dimensional cycle on M , namely, the cycle of singularities. Let M be an n-dimensional Lagrange manifold and f : M -,IR" the projection of M onto the plane 5 = 0, that is, f(x, 5 ) = x . The set C of points of M , where the rank of df is less than n, is known as the set of singularities off. This set consists, in general, of an open (n - 1)-dimensional manifold C', where the rank of df equals n - 1, and the boundary C - C', the dimension of which is strictly less than n - 2. Thus C defines an (n - 1)-dimensional cycle in M . In a neighbourhood of any point x E C' the Lagrange manifold M is defined by n equations of the form xk

ti = t i ( t k , x i ) ,

= X k ( t k , xi),

+

where f = 1, ..., k - 1, k 1, . . ., n for a certain k, 1 6 k < n. It turns out that the manifold C1can be made orientable by labelling the side where axk/atk> 0 as positive. The expression "in general" used above signifies that the statement becomes valid after an unitary rotation, no matter how small. If on a Lagrange manifold M in ''generic state" we prescribe a curve y that is transversal to the cycle ,Z and whose end points are not on C,then by the Maslov index we shall mean the index of its intersection with C,that is, the difference between the number of transition points from the negative side to the positive side and the number of transition points in the reverse direction. 2.2. Pre-canonical Operator. Let M be a Lagrange manifold in the phase space IR'" of points ( x , 5). We shall denote by a a subset of the set (1,2, . . ., n) And by la1 the number of elements in a. We set ii = (1,2, ... , n)\a.

Proposition 2.1. For each point A E M there exists an a such that in some neighbourhood of this point the coordinates (xu,5;) may be chosen as the local coordinates for M . The atlas of M where the local coordinates have been chosen in accordance with this proposition is known as canonical and the coordinates themselves are known as the local canonical coordinates. Proposition 2.2. Let (xu,ta)be the local canonical coordinates on M in a neighbourhood of A . Then there exists a smooth scalar function S(x,, ti)such that

x-=-- as a

=-.as

t

a<,-'

a

ax,

in this neighbourhood. S is called a generating function. What is more, s = - x .a t

a

+

J

(x. 5 )

(XO, 50)

tdx

I. Microlocal Analysis

Yu.V. Egorov

48

49

When xa, [ a serve as canonical coordinates on M , we set

and the integral is independent of the path joining the points ( x o , Yo) and ( x , 5 ) because M is a Lagrange manifold. We first assume that M is uniquely projected onto the x-space. We introduce the operator K : C g ( M )+ Cm(R:)

2.3. The Canonical Operator. Let M be a simply connected Lagrange manifold, and let x q j = 1 be the decomposition of unity on M subject to a fixed canonical atlas. By the Maslov canonical operator we mean the operator

by the formula

K,(cp): G ( M )

where S is the generating function and d o is the volume density on M that is defined by the standard density dx by means of projection. Note that K preserves the norm in L,:

+

defined by the formula N

K,(q) = j I K q ( x ) l ' d x = slq12 do.

Suppose now that M is uniquely projected onto the [-space. Then we may assume that the generating function S , depends only on [ and that the equation x = - a S , / a g defines M . In this case, we set

s&

( K , q ) ( x ) = L"I2(271)-"

d'(5)ei"s~(~)cp(x([))ei"'~ d<.

Next, suppose that M is uniquely projected onto both x- and [-spaces. Let us compare the values of the operators K and K 1 so that x = x ( [ ) and [ = [ ( x ) on M. We compute the last integral by the method of the stationary phase (see 9 1.3) noting that on M S(x) =

s

(X.

(XO,

5)

5 dx, to)

Sl(8 = -x5

+

s

(x, 5 )

[ dx = - x <

+ S(X).

,

C Kj(qjq)*

j=1

Here Kj is the operator constructed for the jth neighbourhood in accordance with the above-mentioned rule. The quantity sgn a 2 S / a < i , the signature of a 2 S / a [ i , is not defined uniquely, because the local coordinates can be chosen, in general, in more than one way. However, arguments similar to the above show that the value of the operator K(cp) for functions cp having support in such a coordinate neighbourhood is well defined to within functions which have the norm of the order O(A-') in L,. At any rate, this is true for a simply connected manifold M on which the generating function is chosen in every neighbourhood by means of integration from one common point ( x o , to). In the general case, when M is not simply connected, the canonical operator may be well defined on M (to within O(A-')) if and only if 1. dx = 0 for every closed curve y on M , 2. for every closed curve y on M its Maslov index is zero.

1,t

i

(XO, 5 0 )

2.4. Some Applications

We obtain, with the assumption that la2S,([)/a[21 # 0,

1. The Bohr's Quantization Rule. Let I/ be a real-valued continuous function on R such that V ( x ) + +a as 1x1 + 00. Consider the one-dimensional Schrodinger equation

- e(in/4) sgn ( 82 S t / a t 2 ) ~ q + ( x r(x), ) where r ( x ) is a function such that IIrIJL,= O(A-'). Thus if M is uniquely projected onto the c-space, then we must set

Cm(w

-hZ$'' f

+ V ( X ) $ = E$.

For each fixed h > 0 this operator has a pure discrete spectrum and it consists of simple eigenvalues Eo(h) < E , ( h ) < ..* < E,(h) < . * * ,

and E,(h) where sgn a 2 S , / a [ 2 denotes the signature of the matrix a 2 S , / a [ 2 (that is, the difference of the number of positive and negative eigenvalues).

+

+a as m

+ +00.

Moreover,

Em(h) = Am(h)

+ O(h2),

Yu.V. Egorov

50

I. Microlocal Analysis

for all a, B and for any compact set K c M. These inequalities define the class SF,(M x I R N ) of symbols introduced by Honnander [1971]. The class ST,, is usually denoted by S", and one simply writes S," instead of SEl-p, and

where A,,,is determined from the equation

1''

,/-

dx

= nh(m

X-

+

i)

s-"

2. The Quasi-Classical Asymptotics of the Cauchy Problem for the Schrodinger Equation. Consider the equation

a$

h2 ---+ 2 ax2 a2+

at

V(x)$,

=

ns-.

The function S(x, 0) is known as the phase function. It is assumed that S E Cm(Qx IRN\O), and that S is positive, real-valued and homogeneous in 8 of degree 1 and grad,,, S # 0. S is said to be non-degenerate if the differentials d(dS/ae,) are linearly independent, k = 1, 2, . . . , N , at those points where asiae = 0. The integral (2) in which a E Szdand S is a phase function is referred to as an oscillatory integral.

in which x + , x - , x - < x + , are the roots of the equation V ( x , ) = 1. This is the condition, referred to as the Bohr's quantization formula, in accordance with which the Maslov canonical operator can be constructed on the onedimensional Lagrange manifold 5' + V ( x )= 1,that is, conditions 1 and 2 of 9 2.3 above are satisfied.

ih-=

51

.

Proposition 3.1. There exists an operator

t>O,xEIR,

with the initial condition

$ = q(x)e(i/")f(X) for t

with aj E So, b, E S-' and C E S-', such that

= 0.

Let M be the Lagrange manifold { ( x ,<), t = f ' ( x ) } . Let us consider the classical dynamical system

This system defines a phase flow g,: IR2 --+ R2 by means of the translations along its trajectories. The curves M , = g,M are Lagrange manifold for each t > 0. We t smooth. shall assume that these curves and the manifold MT = U O < r < T Mare It is not difficult to see that M T is a two-sided Lagrange manifold. Maslov [1965a] has shown that N

$(l, X , h) = K

1 (ih),$k -k R N + I (X~, ,h)

j=O

for 0 < t < T, with max IIRN+l(t,x , h)llL2(R) < CNhN+'.Here K is the canonical operator constructed for M T .

t

Fourier Integral Operators

Zs(au) = ~ ~ e i S ( ' * @ a 8)u(x) ( x , dx do,

where 0 E IRN,x

E

(2)

M . We shall assume that there exists a constant Ca,B,K such that la;a,aa(x, 0)l

< C a J K ( l+

(ol)"-plal+d'fl'

=i eiS, S

If rn < - N , then the integral (2) converges absolutely. In this case, Z,(au) = ~ ~ e i s ( x ~ e ) L k8)u(x)) ( a ( x ,dx d e

(3)

for any integer k 2 0. Moreover, Lk(au)E SFikU, where a = min(p, 1 - 6 ) . Assuming that > 0, we can define the integral (2) by the formula (3) if k is so large that m - ka < - N ; thus the integral (3) converges absolutely. The integral (2) can be defined in an alternative way also. If h E C:(IRN), with h(8) = 1 in a neighbourhood of 0 E IRN,we set

Since

3.1. The Oscillatory Integrals. Let M be a domain in IR",and let u E C:(Q). Consider the integral

~

where the operator 'L is the transpose of L, that is,

Z&U)

8 3.

~

=

ss

ZS,&(au) =

h(&B)eiS(x.e)a(x, @ ( x ) dx do,

E

> 0.

jl

eis(x*e)Lk (h(&e)a(x,f M x ) )dx

for any k, we can choose k so that rn - ko < - N and then pass on to limit as 8 + 0. Hence lim ZS,&(au) & +O

exists and is independent of h. It is natural to take this limit as Is(au).

52

I. Microlocal Analysis

Yu.V. Egorov

where

3.2. The Local Definition of the Fourier Integral Operator. Let 9 and 9' be domains in IR" and IR", respectively. Consider the integral Au(x) = s I e i s ( x ~ y ~ e )ya, (8)u(y) x , d y do.

S ( X ,&x,

j

s1

w) = s{seis(x.Y.B'a(x, y,0)w(x,y ) dx d y do,

w E Cg(9 x

a).

Example 3.1. If the function S is linear in 8, then Cs = Y x IR", where Y is a submanifold of codimension N . Example3.2. Let H(<)be a positive homogeneous function of degree one that is smooth for # 0, and let

<

A : #(or) -b 9'(9).

S(X, <) = (x, 0- H ( 5 ) .

Examples of Fourier integral operators have been cited in 0 3.1 in connection with the solution of the Cauchy problem for hyperbolic equations. The simplest variant of a Fourier integral operator is provided by an operator of the form

s

a(x, <)ii(<)eis(x*5) d<,

3.3. The Equivalence of Phase Functions. Consider the oscillatory integral

1

j a ( x , f3)u(x)eis(x,e) dx do,

where S is a phase function and a E SEd. If we perform a change of variables ( x , 0) ( x , &x, el) that preserves fibres in T * ( 9 ) we , obtain Z,(au)

=

ss

eis(x*8)ii(x, 8)u(x)dx d8,

Then the condition aS/a< = 0 is equivalent to the condition that x = H'(<) and A = (H'(<),
This example is of a general nature, as shown by the next theorem.

(5)

which was examined above in 6 2.3. A particular case of the Fourier integral operator is the pseudodifferential operator corresponding to the phase function S(x, <) = ( x , 5) in ( 5 ) or S(x, y , 0) = ( x - y, 0) in (4). By a change of variables, one can transform to a pseudodifferential operator the Fourier integral operator with the phase function S(x, y , 0) if this function is linear in 8 and if aS/ae = 0 implies x = y.

Zs(au) =

@x,e))io@oeI = a(x, e).

It is easy to see that the range of this map does not change when one goes over to the equivalent phase function and that A , the set of points lying in this range, constitutes a conical Lagrange manifold in 5.

It is easy to see that K A is a kernel of A in the sense of Schwartz and that A can be extended to a continuous linear map

Au(x) = (27c)-"

G(X,

where

eis(x,y*e) U(X, Y , e)u(Y)v(x)dx d y d e

makes sense and is an oscillatory integral. Hence A is a linear operator acting This is the operator that is known as a Fourier integral from C g ( 9 )into 9'(9). operator. The kernel of A is the distribution K A E 9'(9 x O r )defined by the oscillatory integral (KA,

e))= S ( X , e),

If the above transformation is diffeomorphic, then s" is a phase function and a" E SE,. In this case S and s" are said to be locally equivalent. Consider the map

(4)

Here u E Cg(s2'), x E 9, S is the phase function on 9 x R' x IR" and a E S E , ( 9 x 9' x IR"), with p > 0,6 < 1. If v E C g ( 9 ) ,the integral (Au, 0) =

53

L

Theorem (Maslov and Fedoryuk [1976]). Let A c T*(X)be a conical Lagrange manifold in <. For each point A,E A we can choose local coordinates xl, . . .,x , such that in some neighbourhood of this point A is defined by means of the phase function S(x, 5) = x * - H ( < ) ,where H is a smooth function for # 0 that is positive and homogeneous of degree one.

<

<

The following theorem of Hormander's answers the question posed above regarding the equivalence of phase functions.

Theorem (Hormander [1971]). Let S and f be non-degenerate functions in conical neighbourhoods of the points (x,, do) E 9 x (IRN\O) and (x,, 6,) E Q x (R'\O), respectively. S and s" are equivalent i f and only i f (I) they define at corresponding points one and the same Lagrange manifold; (11) N = R; (111) the signatures of the matrices a2S(x,, 8,)/a02 and a2s"(x,, 6,)/ae2 coincide. 3.4. The Connection with the Lagrange Manifold. We shall demonstrate that the values of the Fourier integral operators are connected with the Lagrange

I. Microlocal Analysis

Yu.V. Egorov

54

(11) Every distribution A of the form

manifold in an invariant manner. To do this, consider the integral

ss

jj

eis(x,e)a(x, 8)u(x)dx do.

( A , u) = ( 2 , p + 2 ~ ) / 4

~ ( a ) , e = e(a, e)

transforms this integral to the form

where

ss

e i % ~ . ~ ) J)c(T) i i ( ~ , d a d8,

s"(n,8) = s(x(a),e(n, el),

~ D ( Aas/ae)/qx, , O)1-1/2a(x,e) = I D(X,

as"/@/~(n, @ ~ - ~ / ~ g), (z(a,

that is, the density of order 3on A , which is the pushforward of the density a& under the map C 3 ( x , 0 ) + ( x , aslax) E A coincides with the density obtained in the same manner from s" and ii. Thus the integral under consideration is connected in an invariant manner with the Lagrange manifold, at least when the number of independent variables N does not change under the coordinate transformation. That the integral is invariant under transition to another parametrization 8' = (el, . . .,ON,) can be checked by the method of stationary phase. We sum up the above discussion in the following. Theorem (Hormander [1971]). Let S(x, 0 ) and s"(x,6) be non-degenerate phase functions in neighbourhoods of the points ( x o ,0,) E D x IRN and (x,, 8,) E Q x I R' defining elements of a common Lagrange manqold, and let = to.

u E Cg(IRn),

@,/& E S;

(4Q1/2). 3.5. The Global Definition of the Fourier Distribution. Let D be a smooth - ii(x,

+n/4+1-2p

u

manifold and A a closed conical Lagrange submanifold in T*(D)\O. Let U j be a finite local cover of D by coordinate neighbourhoods. Suppose that in the conical neighbourhood 5 of each domain U j x (lRN\O), where N = N ( j ) is an integer, the non-degenerate phase function Sj(x, 0 ) is defined, and that the map

'

( x , 0 ) -+ (x, as/w is a diffeomorphism of the domain { ( x ,0 ) E 5, aSj(x,e)/afl= 0 ) on an open subset U of the manifold A . Consider the distribution A = Aj E W(D, D,,), where

1

( A ~u), = (271)-(n+ZN(j))/4

ss

ei(s,(x.e)-w."4aj(x,O)u(x)dx do,

u E Cm(D),dx is the Lebesgue measure in U j , 8 E I R N ( j ) , and where

It can be shown (Hormander [1971]) that

as(x,, eo)/ax = as"(x,, 6,)/ax

ezial4a(x,e)&

qa)= IDX/D%~~/~U(X),

These formulae show that it is more convenient to consider, in place of distributions u, densities of order 3 that are transformed according to the formulae just mentioned. Let C = { (x, O), dS(x, @)/a0= O}. Let d , be a density on C which is a pullback of the Dirac measure in IRN under the map ( x , 0) + dS(x, O)/ae. Finally, let il,. . ., A,, be local coordinates on C that are extended to a neighbourhood of this manifold. Then dc = I D ( A ~. .,., i, as/ae,, ,, . . . , as/aen)/o(x, e11-1 &Il.. . d ~ , , .

eis(x*e)a(x, O)u(x)dx do,

where a E Sp+(n-ZN)i4, p > 3,and where the conical support of the distribution a P lies in a sufficiently small conical neighbourhood of the point (x_,, O0), can be written in the same form by replacing S by s" and a by (z E S:+(n-2N)/4 . Moreover, the conical support of (z lies in some small neighbourhood of ( x o ,Go), and

A diffeomorphic change of variables =

55

( I ) The difference o = sgn a2S(x,8)/a02 - sgn a's"(x, 6)/ae2for the points 0 and 8 that are connected by the relations a q x , eyae = 0, a q x , eya8 = 0, as(x, eyax = as"(x, @/ax = 5 E T:, is a constant in some neighbourhood of (x,, to)on A.

a, J

1,

= { ( x ,to);t 2 1, (x, 0 ) E K } , is a compact subset of the image set of 5 in IR" x IRN(j)under the

p P+ ( n - Z N ( . i W ( ~ n

RNU)

SUPP aj

in which K above diffeomorphism. We denote the set of such distributions by Ir(Q, A). Let

where the points

ek

and 0, are such that

It is not difficult to see that ojkE Z and that this function is locally constant in Ut n U:, in view of the above-mentioned theorem. We thus obtain an integral cochain that defines the element o E P ( A , Z). Consider the corresponding one-dimensional linear complex bundle L on A that is obtained from this cochain in such a way that the multiplication by oi, i is the imaginary unit, corresponds to o in C.Evidently, L is trivial as a complex vector bundle and is defined by the image of o in H ' ( A , Z4).It is clear that L is a R+-bundle because IR+ acts on C trivially. Therefore the spaces S;(A, DlI20 L) are well defined.

Yu.V. Egorov

56

I. Microlocal Analysis

We can establish the existence of the isomorphism Spm+"/4(A,a,,, 0 L)/s7+"/4+'-2p( A ,

0 L ) + Zpm(Q,

A)/zpm+l-2p(Q,

A).

To do this, we proceed as follows. If s E Spm+n/4(A, QIl2 0 L), then for each j an element sj E s; +n/4( uj", Q1/, 0 L ) is defined such that sj = iajksk

in

U; n U:.

51

The phase function is defined in some open conical set and the map

transforms r smoothly into an open conical subset and the set Cs of zeros of aS/a8 in r onto C n ro.

rcX

x Y x (lRy\O)

roc (T*X\O)

x (T*Y\O)

Proposition 3.2. The differential of the map C + T * ( X )is bijective i f and only if n, = ny and

If cone supp s c Uj", then we can define, as above, the element Aj = Aj(s)E Z:(Q, A), corresponding to the symbol aj E S:+(n-2Nj)/4(Uj), and aj& goes over to sj under the map

This condition defines aj as an element of S : + ( n - 2 N j ) 1 4(Cj).Finally, aj is determined by multiplication by a homogeneous cutoff function in Uj. By the theorem of $3.4 a change in the distribution aj leads to a change in A by an element belonging to Z;+n/4+(1-2p) (0).

3.6. The Global Fourier Integral Operators. Let X and Y be manifolds of dimension n, and n y , respectively.The points of X will be denoted by x, those of Y by y and of X x Y by (x, y). Definition 3.1. The submanifold C c T * ( X x Y)\O is called a canonical homogeneous relation from T*Y to T * X if it is, first, closed and conical, second, is contained in (T*X\O) x (T*Y\O), and, finally, if it is Lagrange with respect to the 1-form ax - u y ,where ux = 5 dx and uy = q dy. Example 3.3. Let n, = n y . Suppose that f is a C"-map from T*Y\O to T*X\O that is positive and homogeneous in q of degree 1. The set {(f(y, q), y , q); ( y , q) E T*Y\O} is a canonical relation if and only if cry = f * g X , that is, if the map f is symplectic. Example 3.4. Let X = Y, and let f be the identity map. Its graph is the set A ; = {(x, 5, x, 0, x E X , 5 # 0}, that is, is a diagonal in (T*X\O) x (T*X\O). Clearly, A ; is a canonical homogeneous relation. Let C be a canonical homogeneous relation from T* Y into T * X . The image of C under a map that transforms the point (x, 5, y , q ) into ( x , 5, y , - q ) is a conical Lagrange submanifold in T*(X x Y)\O with respect to the standard symplectic form cr, + oY.We denote this image by C . With this manifold we can associate, as in $ 3.5, Fourier distributions. They are defined microlocally by integrals of the form

The map Cs + C + T * ( X )induces local coordinates (x, 5 ) in some neighbourhood on C s , and the density dCs= ID(S)l-' dx d5. Definition 3.2. A canonical homogeneous relation C from T*(Y ) into T * ( X ) is called canonical local graph if the projection C + T * ( X )(and hence also the projection C + T*(Y ) )is a local diffeomorphism; thus C is a local graph of the canonical transformation. (Clearly, n, = ny in this case.) The density ,u on a local canonical graph can be defined by means of the standard density in T * ( X )(or T * ( Y ) )and the map C + T * ( X )(or C + T * ( Y ) ) . Evidently, E S;l2(C,Q,,), where n = n, = n y . If L is a linear bundle on C that is obtained from the linear bundle associated with the Lagrange manifold A = C , then the map

&

Spm(C, L ) + Spm+n/2(C,Q1/2 0 L),

defined by means of multiplication by isomorphism

A,is bijective. Hence we obtain the

Spm(C, L)/spm+'-2p(c, L ) + Z7(X x Y,

C)/zpm+'-2p(x

x

r, C).

Let S be a non-degenerate phase function in a conical neighbourhood U of the point (xo,y o , 0,) from X x Y x (WN\O) to which corresponds a neighbourhood of the point co in C . Suppose that a E Spm+(n-N)/2and a = 0 outside a sufficientlysmall conical neighbourhood of ( x o ,y o , Oo). Consider an operator A with the Schwartz kernel K A defined by the formula (KA,

u) = (271)-(n+N)/2 J J J e i . v x . y . e )

4x7 Y , O ) U ( X , Y ) dx dY do,

where u E CT(X x Y ) . Then the symbol of A coincides with the image of a& on C under the map Cs + A + C . By Proposition 3.2, under this map the density dc, = ID(s)l-' dx d5 corresponds to the density d,, and b(x, y , 0) = a(x, y , e)lD(s)l-1/2E y(x x Y x

(IRN\O)).

I. Microlocal Analysis

Yu.V. Egorov

58

59

We thus have the following result.

Hence (KA,

Theorem 4.1 (Hormander [1971]). Let C be a canonical homogeneous relation ffom T * ( Y )into T * ( X )and let KA E I,"(XxY, C), p > i. Then the kernel KA. of the adjoint operator lies in I,"(YxX, C$), where C, is the inuerse image of C under the map S : T*YxT*X -+ T * X x T * Y that interchanges the factors. If a s mP + ( n x + n ~ ) / 4(C, Q,/, 0 L,) is the principal symbol of A, then the principal symbol of A* is S*ii E S~+(nx+ny)/4(CS, Q1/, 0 Lcs),because there exists a natural isomorphism between the bundles LCsand S*LC1.

u, = (27c)-( n + N ) / 2 [ ~ [ e i S ( x * y ~ e )yb, (O)ID(S)11/2u(x, x, y ) dx d y do,

or, equivalently, Au(x) = (27c)-( n + N ) ' 2 ~ [ e i s ( x ~ y ~ e )y b, O)ID(S)I'/2u(y) (x, d y dO for u E C(: Y ) .With this operator is associated a symbol which is obtained from b by a composition with the inverse of the map

b

4.2. The Composition of Fourier Integral Operators. Let A , and A , be Fourier integral operators with the Schwartz kernels K and K,, respectively. Suppose that K , E I,"'(X x Y, C,),where C, is the canonical relation between T * ( Y )and T*(X), and K , E I,"'(Y x Z, C,), where C, is the canonical relation between T*(Z) and T * ( Y ) . Let A,: C:(Z) -,C g ( Y ) and A,: C g ( Y )-+ Cm(X), where A , is a proper map. Let us show that the operator A , A , has the Schwartz x Z, C), where C is obtained from the composition of C, kernel K E Iz1+m2(X and C,. The direct product C , x C, is a symplectic manifold with respect to the symplectic form a, - ay, ay2- oZ, where Y, and Y, are replicas of the manifold Y. Assume that C, x C, intersects the diagonal A in T * ( X ) x T*(Y,) x T*(Y,) x T*(Z), where the diagonal consists of the elements whose components coincide transversally in T*(Y,) and T*(Y,). This implies that no non-zero vector exists that would be normal to the tangent plane to C, x C, and to d relative to the symplectic form. Or, equivalently, there exists no non-zero vector that would be tangent to C, x C, and to A with zero components in T ( T * ( X ) )and T(T*(Z)).Under this condition, the projection of (C, x C,) n A into T * ( X ) x T * ( Z )is a local manifold, with dimension dim X + dim Z, on which a, - az = 0, because ay,- ay2= 0 on A . We now define the composition of C, and C, as a projection of (C, x C,) n A onto T * ( X ) x T*(Z). It is clear that dim(C, x C,) n A = dim C, x C, - codim A = n, ny. Assume that the projection (C, x C,) n A -+ T * ( X ) x T*(Z)\O is a proper injective map. Let us express the transversality condition in terms of phase functions. Let Sl be a phase function defined in a neighbourhood of the point (x,, yo, 0,) E X x Y x (WN1\O) and S , a phase function defined in a neighbourhood of the point (yo,z,, a,) E Y x Z x ( I R N 2 \ O ) . Assume that these functions are nondegenerate and that they define C, locally by the equation aS,(x, y, O)/aO = 0 and C, by aS,(y, z, a)/aa = 0. Assume further that

,

and which may be regarded as an element of the space ST(C, L). As shown above, this symbol is invariantly defined to within an element from S:+'-2p(C, L). Therefore, it is natural to refer to this as the principal symbol. Using Proposition 3.1, we can show that A: &'(Y)-+ g ( X ) if d,,$ # 0. A becomes a continuous operator from C(: Y ) into C m ( X )if dy,BS # 0.

'

+

6 4. The Calculus of Fourier Integral Operators 4.1. The Adjoint Operator. If u and u are two densities of order 3 on a manifold X and if the set supp u n supp u is compact, then (u, u) = ( u , V ) =

s

uu.

The adjoint to A whose kernel is in IT(X x Y, C ) ,where C is a canonical homogeneous relation from T*(Y ) into T * ( X ) ,is defined by the equation (A% 0) = (u, A*u),

24 E

cg(Y,Q1/2),

lJ E

tax,Q1/2).

If, in local coordinates, the kernel Ka(x, y ) of A is of the form (KA, u> = (27c)-( n x + n y + 2 N ) / 4 [[[eis(x,Y*e)a(x, y , O)u(x,y ) dx d y do, then r r r

(KA,, u )

J J J e-is(x.y.e)a(x, Y , O ~ YX) dx, dy d o

= (271)-(nX+nY+2~)/4

for u E Cg. The function - S is a phase function in X x Y x (IRN\O) and the corresponding canonical relation is defined by the image of the map

+

(x.

as,(x, Y , 0 )

for x = x,,

ax

9

Y, -

as,(x, Y , 0 ) aY

9

Y,

dS,(Y, z, a) aY

9

z,

-

. . ., a = ao,that is, that asl(x0, Y o , Oo)/aY

+ as,(Y,,

zo,~,)/aY= 0.

aZz,

"')

A

I. Microlocal Analysis

Yu.V. Egorov

60

As we stated, transversality implies that there is no non-zero vector (0,0, t, rt, z, 0,O) orthogonal to the tangent plane to C, x C, with respect to the symplectic form ax - cry, + ay, - az. It is easy to see that the equation

+

Here N = N , + N , ny.It is important to observe that a,(x, y , 0 ) x a,(y, z, a) is a symbol in a neighbourhood of points satisfying (6). In fact, there exist positive constants c, and c, such that c1 la1 < 101 < c214 provided that dS,(x, y , e)/ay aS,(y, z, o)/ay = 0. Let x(0, a) be a homogeneous function of degree zero that equals 1 for c1 14/2 < 101 < 2c, 101 and vanishes for c1 101/3 > (01 and for (81> 3c, 101. Then the operator A , A , - B has a C"-kernel if

+

implies that a = b = t = 0 (here d = d,,y,8,a). We now consider the function S(X, z, 4 = S,(X,

61

Bu(x) = ( 2 ~ )(nx- +

y , 0 ) + S,(Y, z,a),

n +2N)/4 ~

ssss

ei(SL(x.Y . 8 ) + S 2 ( y , z,a))

x b ( x , z, y , 8, cr)u(z) dz d y d e do,

where

where 0 = ((lei2

+ l a 1 2 ) 1 / 2 ~e,, a) E

\o.

b(x, z , Y , e,a) = x(e, ~ ) u , ( x Y, , ~ ) u , ( YZ,,

IRN~+NZ+~Y

It is clear that S is homogeneous in o of degree 1 and that the equation dSj13o = 0 implies that

Therefore the transversality condition means precisely that S is a nondegenerate function in a neighbourhood of the point (xo,zo, coo). The corresponding canonical relation is of the form

as,). as, ((.,3, ax aZ ' ae z, --

~

that is, S defines C , o C, locally. If we put

A , ~ (=~(2n)-(nx+"Y+2N1)/4 )

as, -as, -~ + -as, =o,

--

aa

ay

11

ay

p l ( x . Y , @a,(x,

I

Y , W 4 Y ) dY do,

ss

A,u(y) = ( 2 7 1 ) - ( " ~ + " ~ + ~ ~ ~eiS2(y,z,a)a2(y, ) ~ ~ z, a)u(z) dz da, where a,

Sml+(nx+n~-2N1)/4(~ P

y

IRNI),

z

S ~ Z + ( ~ Y + ~ Z - ~ N ~ V ~ ( YIRNZ), P

and where S , and S , are non-degenerate phase functions, a, and a, vanish in a neighbourhood of the sections 8 = 0 and a = 0, respectively, with cone supp a, c r,,cone supp a, c r,,and where S , , S , define the manifolds C, and C2 locally, then A , A , u ( x ) = (2n)-("

+

nz+2N)/4

J JJ

1

ei(sl(x.Y,e)+s2(Y.z,o))

x a,(x, y , e)a,(y, z , a)u(z) dz dy d e da.

0)E

sm ,

with in = m,

+ m, + (nx + n,

- 2N)/4

+ ny.

The principal symbol of the composition A , o A , is obtained by restricting the principal symbols of A , and A , to the diagonal of the direct product. It can be verified that the corresponding density on C, o C2is obtained from the product of densities on C , and C, by means of the projection of C, x C , onto the diagonal.

4

4.3. The Boundedness in L,. Let A be a proper Fourier integral operator with the principal symbol a E S,"(C,L). Then the adjoint operator A* is associated with the homogeneous canonical relation C* obtained from C by interchanging X and Y (or T*X and T*Y). Clearly, C o C* is the graph of the identity map in T*Y\O. Since in this case the transversality condition is always fulfilled, it follows that the operator A*A is defined and is a pseudodifferential Qperator with principal symbol lal'. Therefore A is a bounded operator from proGm,(Y,Q,/,) into L:o,,(x, Ql,) and from L:,,(Y, Q1/2) into L:,,(X, vided that C is a canonical local graph. If the principal symbol of A tends to zero as the point tends to 00 along C over each compact subset in X x K then A is a compact operator (see Hormander [197 11). Suppose that now A is a proper Fourier integral operator with the principal symbol a E Sm(C,L ) and that B and B , are pseudodifferential operators with the principal symbols b E Sm'(X) and b, E Sm'(Y),respectively. Since these operators are associated with the graph of the identity map, it follows that the transversality conditions are fulfilled and the compositions B A and A B , are defined. They are Fourier integral operators with kernels from the class In+m'(Xx Y, C, QlI2). The principal symbol of B A is of the form

b(x, X, WX, Y , e)/ax)a(x,Y , e ~ J d , ) - ~ , and that of A B , of the form

I. Microlocal Analysis

Yu.V. Egorov

62

where

It is clear from this and the preceding fact that if A is a proper Fourier integral operator with kernel from I,"(X x Y, C'), then it is a bounded operator from HEo,,,( Y, 52,/,) into H",,,(X, 52,/,) and from HfJ Y, 52,/,) into Hf0;'"(X, SZ,,,) for all real s.

A(x)= Clearly, A

Definition 4.1. A Fourier integral operator A belonging to Im(X x Y, C, SZ,,,) is said to be elliptic in an open conical subset f " x ryc (T*X\O) x (T*Y\O) If its principal symbol vanishes nowhere on C n (r,x rY).

E

s

a(x, 8)eis(x*e) do,

63

x

E0 '.

Cm(SZ').Therefore if the distribution A is defined by (7), then sing supp A c { x E 52, dS(x, 8)/a8 = 0 for some 8 # 0).

If a E SE,(52 x RN), where a

= 0 in

a conical neighbourhood of the set

c = { ( x ,8); x E 52, e E IR~\{o},as(x, eyae = 01,

Clearly, the ellipticity condition implies the existence of constants K , and K , such that m+(n-N)/Z K2. la(x, Y? 2 K l l 4

then the distribution u HIs(au), defined above, is a C"-function. More precisely, we have

For an elliptic operator A we can construct an approximate inverse operator E in r' x r,. Indeed, AA* is an elliptic pseudodifferential operator of order 2m in r, x ryand A*A is an elliptic operator of order 2m in ryx r,. Therefore there exist parametrices R , and R,:

f c 52 x IRN,and let a E SE,(52 x IRN),cone supp a c T\(SZ x 0). Assume that p > 6 and that either p + 6 = 1 or S is linear in 8. Then the distribution

a

-

R1(A*A) I ,

Proposition 5.1. Let S be a non-degenerate phase function in the cone

u ~ +Is(au) is a C"-function i f a has a zero of infinite order at points of the set

-

c = { ( x ,e) E r,as(x, eyae = 01.

(AA*)R2 I .

if a = 0 at points of C, then there exists a symbol b E Szi*-p(SZ x IR"), with cone supp b c r\(O x 0),such that Is(au) = Is(bu) for u E C"(f2).

If we set El = R , A * ,

-

-

E , = A*R,,

-

Definition 5.1. A real-valued function S defined in X x Y x IRN,which is infinitely smooth for 8 # 0 and which is positive homogeneous in 8 of degree 1, is known as an operator phase function if for each fixed x the differential d,,,S does not vanish or if for each fixed y one has d,,,S # 0 for 8 # 0.

we have E , A I in r, and A E , I in r,, where the sign denotes the equivalence modulo smoothing operators. Moreover, an operator is said to be smoothing in r if its symbol and all the derivatives of the symbol decay, as 181 + 00, faster than any power 181-N,8 E r. Let Y = X . The operator A will be called unitary if it is an elliptic operator of order zero and A*A = I , AA* = I . The following theorem generalises Theorem 2.1 of Chap. 2 (see Trkves [1982]). Theorem 4.2. If A is an unitary Fourier integral operator associated with a local canonical graph C, then the principal symbol of the operator A*PA for any pseudodifferential operator P is locally equal to the pullback of the principal symbol of P with respect to the local symplectomorphism T * X + T * X whose graph is C.

Let

cs= { ( x ,y ) E x

sing supp Au c C, I

I

=

jj

a(x, f3)eis(x+%(x)dx df3 =

s

A(x)u(x)dx,

(7)

sing supp u,

u

E b'(Y ) ,

operators sing supp Au c sing supp u,

The Image of the Wave Front Under the Action of a Fourier Integral Operator

I,(au)

o

, where C, is a relation between Y and X . For instance, for pseudodifferential

I

5.1. The Singularities of Fourier Integrals. Let SZ c IR". Let S ( x , 8) be the phase function defined in SZ x IRN, and let SZ' be the set of those points of 52 where d,S(x, 8) # 0 for 8 # 0. If u E C:(SZ'),then

eyae = 01.

It can be deduced from Proposition 5.1 that

'

8 5.

x Y, 38 E IR"\O, a s ( x , y ,

that is, pseudolocality holds.

5.2. The Wave Front of the Fourier Integral. The fact that microlocalization is possible under the sign of the Fourier integral enables us to make the above results more precise by replacing the singular support by the wave front of the distribution. Proposition 5.2. Let 52 c IR", r be an open cone in SZ x (IRN\O), and let S be the phase function in r. If a E S;,(52 x IRN), p > 0, 6 < 1, and a = 0 in a neighbourhood of the set 8 = 0 and i f cone supp a c r,then WF(A) c { (x, aS/dx);(x, 8 ) E cone supp a, dS(x, 8)/% = 0}, where A is a distribution such that A(u) = I,(au).

Yu.V. Egorov

64

In particular, if a proper pseudodifferential operator P is such that P WF(u) (see $4.3 above), then Pu E C" for all u in 9'(Q). For the general case, we have

Theorem 5.1. Let K

E I;(X,

I. Microlocal Analysis

- 0 in

This theorem follows easily from Theorem 3.5, Chap. 1. In the particular case n = N and S(x, y, 0) = S(x, 0 ) - S(y, e), condition 2 is fulfilled automatically, and the following result holds (Hormander [1971]).

A), a E S;+n/4(A, Qlj2 0 L),and K ( u ) = I,(au) for E S7+"i4+1-2pin A\WF(K).

Theorem 5.4. Let u E &(Q) and let A be an operator of the form

s

u E 9 ( X ) . Then WF(K) c A and a

Au(x) = ( 2 ~ ) - " a(x, &i(<)eis(x.c) d5.

Using Sobolev spaces, we can obtain the following result (see Hormander [19713).

Theorem 5.2. I m ( X ,A ) c H(,,(X) if and only if m + n/4 + s < 0. Furthermore, if K ( u ) = I,(au), K E I;(X, A ) and a f 0 on the set where d,S = 0, then K $ H(,)(X)for 0 < m n/4 s.

+

~

Assume that aS(x, ( ) / a x = 0 implies (aS(x,t)/at,5 ) $ WF(u). Then the distribu-

: tion Au E 9'(0)is defined and

+

5.3. The Action of the Fourier Integral Operator on Wave Fronts. Let A be a Fourier integral operator of the form

ss

Au(x) = (27~)-" Its Schwartz kernel is

65

a(x, y , O)u(y)eiS(x,Y.e) d y do.

(8)

s

K ( X ,y ) = (2n)-" a(x, y, O)eis(x-y,e) do.

To be able to apply Theorem 3.5, Chap. 1, we must know the wave front of this kernel.

Lemma 5.1.

This lemma can be easily established by finding the Fourier transform of the distribution K ( x , y ) and then applying the method of stationary phase. This lemma enables us to find WF(Au) (Hormander [1971]).

Thus WF(Au) is contained in the image set of WF(u) under the canonical transformation ( y , v ) + (x, 5 ) generated by the function S(x, q). Since a canonical transformation is invertible, it follows that the inclusion sign in Theorem 5.4 can be replaced by the equality sign in the case of an elliptic operator A. By way of example, we note that Theorem 5.4 enables us to determine completely the wave front of the solution of the Cauchy problem for strictly hyperbolic equations by means of the wave front of the initial data.

0 6. Fourier Integral Operators with Complex Phase Functions 6.1. The Complex Phase. In examining operators with complex-valued symbols there arise in a natural manner Fourier integral operators with complex phase. These operators are represented microlocally in the form

jj

A ~ (= ~p)-" ) eiS(x,y,W 4% Y, 0

4 Y ) dY do.

Such an integral makes sense if Im S 3 0. Besides this, it is usually assumed that

Theorem 5.3. Let u E Q'(f2) and let A be an operator of the form (8). Assume that the .following two conditions hold:

S ( X ,y , t e ) = t s ( x , y, e)

dS(x, y , 0 ) # 0

for t > 0;

if Im(x, y, 0) = 0.

The complex phase function S is said to be non-degenerate if the differentials d(aS/de,), j = 1, ..., N , are linearly independent over the field C at every point of the set C, where aS/ae = 0. Then the distribution Au E 9 ' ( X ) is defined, and

6.2. Almost Analytic Continuation. Let Q be an open subset in IR" and p(x) a non-negative function in 52 that satisfies a Lipschtiz condition. Definition 6.1. A function f is said to be p-flat on Q if for any compact subset 0 such that If(x)l < clp(x)lN,x E K .

K

c SZ and for any integer N 2 0 there exists a constant C = C ( K , N )

=-

Yu.V. Egorov

I. Microlocal Analysis

Two functions f and g are said to be p-equivalent if the difference f - g is a p-flat function. f is said to be flat on the compact set K c 52 if it is p-flat with p ( x ) = dist(x, K). It is easy to see that iff E Cm(52)and iff is p-flat, then any derivative Oaf is also a p-flat function. In particular, such a function is flat on K if and only if D"f(x) = 0 for x E K and for all IQI.

assuming that Im f 2 0 in supp g. Suppose that in supp g f has a unique critical point x = 0 which is non-degenerate. This last condition implies that the matrix 11d2f(x)/axi13xjllis non-degenerate. Suppose that f depends also on the parameters t = ( t l , . . ., t l ) that vary in a neighbourhood V of the point 0 in IR'. Let f be an almost periodic continuation o f f in the neighbourhood of the point x = 0. Let x = q ( t )be the unique solution of the equation d,f(q(t), t ) = 0 and let s ( t ) = Ila2f(q(t),t)/13xi13xjll.By the hypothesis, R(0)is invertible and Im R(0) is non-negative definite. Hence no eigenvalue of the matrix R(O)/i is real and negative. If the neighbourhood V is sufficiently small, then the matrix [s(t)/i]-1'2makes sense; here t E V and the branch of the function is chosen so that = 1. As in 9 1.3, we can show that

66

Definition 6.2. Let 0 be an open subset in C" and K a closed subset in 0. A function f E C m ( 0 )is said to be almost analytic on K if for j = 1, . . ., n the functions 13,f a r e flat on K . The set of functions that belong to Cm(0)and are almost analytic on K will be denoted by A ( 0 , K). We put OR = 0 n IR". Iff E A ( 0 , OR),f loR = 0, then f is flat on OR. Let z = x + iy be coordinates in C",and let 52 be an open subset in IR".We denote by 6 the subset 52 + iIR" in C".We shall identify Q with 6 n ( y = O}.

Proposition 6.1. Let 0 be an open subset in C" that is contained in 6,. Let N ( 0 , 0,) be the space of Cm-functions on 0 that are flat on OR.The restriction on 0, defines an isomorphism of the space A ( 0 , OR)/N(O, 0,) onto Cm(OR). Thus a function f in Cm(OR)may be regarded as an equivalence class (modulo functions that are flat on 0,) of almost analytic functions on OR.A representative of this equivalence class is known as an almost analytic continuation off in 0. This almost analytic continuation can be constructed in the following manner. Let h E Cm(IR1),where h(t) = 1 for It1 < 1 and h(t) = 0 for It\ > 2. Let f E Cm(O,). We set

4

l(2)

- (3""

f

d e t [ ~ ( t ) / i ] - ' / 2 e i l j ( b ( f )j. f=O )

61

L-'Qj(Ox)g(q(t)),

where Qj are differential operators, with constant coefficients, of order 2j, and Qo(Dx)= I . These operators depend on the choice of almost periodic continuations off and g.

6.4. The Lagrange Manifold. Let M be a real symplectic manifold of dimension 2n, and let fi be its almost analytic continuation in C2".Let A c fi be an almost analytic submanifold containing the real point po E M , and let (x, g) be real symplectic coordinates in a neighbourhood W c IR2" of the point po. Let be an open subset in C2"for which @nIR'" = W . Assume that (2,t ) are almost analytic continuation of the coordinates in @ so that (2,f ) map @ diffeomorphically onto an open subset in C'". Suppose that A is defined in a neighbourhood of po by the equations f = d g ( x ) / X ,2 E C",where g is an almost analytic function such that Im 2 0 in IR".

w

Theorem 6.1. Zf (9,i j ) is another almost analytic continuation of coordinates in l? and A is defined by the equation i j = H ( j j ) in a neighbourhood of the point po, 1, . . . in such a way that this series converges

then A is locally equivalent to the manifold i j = dh(j)/ajj, jj E C", where h is an almost analytic function and Im h 2 0 in IR".

Definition 6.3. Let 0 be an open subset in C",M a smooth submanifold in 0 of codimension 2k, and let K be a closed subset in 0. M is said to be almost analytic on K if any point zo E K has an open neighbourhood U in 0 where there exist k smooth complex functions fl, . . . , fk thar are almost analytic on K n U and are such that M is defined on U by the equations f l( z ) = 0, . . .,f k ( z )= 0 and dfi, . . ., dfk are linearly independent. Furthermore, M is locally equivalent to M , , M M I , if M , is defined in U by means of the functions g l , . . ., g k such that fj g j f o r j = 1, . . . , k, that is, fj - g j are flat functions on K n U .

Definition 6.4. An almost analytic manifold A satisfying the hypotheses of Theorem 6.1 at every real point in some real symplectic coordinate system (x, 5 ) is known as a positive Lagrange manifold.

where tij > 0 are chosen for j in Cm(O,).

= 0,

-

N

6.3. The Formula for Stationary Complex Phase. Let us consider the integral

s

Z(2) = e"f(")g(x)dx,

Definition 6.5. An almost analytic manifold A c fi is known as a strictly positive Lagrange manifold if dim A , = 2n and A R is a submanifold in M , %I,I~ 0 for all local representatives A , of A and for all local almost analytic continuations a, of the symplectic form a on Tp(fi),and, finally, if i-'o(v, V) > 0

-

-

for all v E T&l)\TJA

-

,) p

E

-

A , where Tp(A), is the complexification of Tp(AR).

As in the real case, for a conical almost analytic manifold A c T*X\O, on which o,IAa 0 for all local representatives A , and for all almost analytic continuations a, of the form a, the local coordinates x in a neighbourhood of each point po E A R may be chosen so that A can be represented in the form 2 =

Yu.V. Egorov

I. Microlocal Analysis

d g ( f ) / a z .Here 4 are dual coordinates to x and (2, t )are almost analytic continuation of the coordinates ( x , 5). The function g here is almost analytic and positive homogeneous of degree one.

The phase function S defines a map from S,"(lR"x lRN)into Qi0 according to the following rule. If a E S,"(IR" x IRN) and a = a , for large 0 in a small conical neighbourhood of the ray { ( x , , to,), t > 01, with supp a, lying in this neighbourhood, then the distributions A and A , defined by the equations A(u) = I,(au), A,(u) = I,(a,u) are equivalent at 1,.

68

Definition 6.6. A C"-function S(x, 0) defined in an open conical set

r c IR" x (W\O) is known as a regular phase function of positive type if

1. It has no critical points; 2. S(x, t o ) = tS(x, 0) for t > 0; 3. The differentials d(aS/a0,),. . .,d(aS/a0,) are linearly independent over (c on the set Cs, = { ( x , 0) E r,as/a0.=O}; 4. Im S ( X , 0) 0. Theorem 6.2. Let S be a regular phase function of the positive type that is defined in a conical neighbourhood. Let s"(2,e)be an almost analytic homogeneous continuation of S in a conical neighbourhood in (c" x ((cN\O). Let

cs = { ( 2 , Q E (c" x ((cN\O); as"(;,

e)/ae = O}.

Then the image As of the set Cs under the map

is a local conical positive Lagrange manifold. Furthermore, Ask is the imge of Csg, and when s" is replaced by an equivalent almost analytic continuation, the manifold As is replaced by an equivalent conical positive Lagrange manifold. 6.5. The Equivalence of Phase Functions. Let S & C"(r) be a positive type regular phase function. Suppose that a E S,"(IR"x IRN) and p > 3. Let r be an open conical set and let supp a be contained in a closed conical subset in r. Consider the integral

I,(au) =

1

1eiSix~')a(x, B)u(x)dx do,

u E C;(IR").

As in the real case, we can show that WF(A) c { ( x , dS(x, @/ax), (x, 0) E cone supp a n CSR}, where A is the distribution defined by A(u) = Is(au). Suppose now that S , ( x , o)is another positive type regular phase function defined in an open conical set r c IRnx(IRM\O). Let 1, = (x,, to),with

to= as(x,, 0,)/ax = d S i ( X , , w,)/ax and aqx,,

e,)/ae

= 0,

as,(x,,w,)/ao

= 0.

Two distributions A and B E 9'(IR")are said to be microlocally equivalent at a point 1, if I , q! WF(A - B). Let 9L0be the class of distributions from Q'(IR") that are locally equivalent at I , .

69

Definition 6.7. The phase functions S and S , are equivalent at a point 1, for symbols belonging to S, if the images S,"(IR"x IRN)and S,"(IR"x lRM) under the maps defined by S and S , coincide in gAo, Theorem 6.3. Let S and S , be positive type regular phase functions defined in conical neighbourhoods of the points (x,, 0,) E IR" x (IRN\O) and (x,, coo)E IR" x (IRM\O), respectively. Suppose that A , and A,, are equivalent in a neighbourhood of the point (, = aS(x,, 0,)/ax = aS,(x,, o , ) / d x . Then S and s, are equivalent at (x,, 5,) for symbols belonging to S,, p > 3. Definition 6.8. I,"(X, A), p > 3, is a subspace of distributions A belonging to Q'(X, Q,,J for which 1. WF(A) c A R ; 2. For each point I , E A I Rand for any local coordinates x , , . . ., x , in a neighbourhood of the point .(A,) E X A is of the form A(u) = I,(au), where S is a positive regular phase function that generates A near 1, and a E SF +(" - 2N)'4 (lR"x IRN) has support in a small conical neighbourhood of the point (x,, 0,) E Cs, corresponding to 1,.

6.6. The Principal Symbol. Let us define the principal symbol for elements belonging to Z,"(X, A). In the case of a real function S , the principal symbol is a section in the tensor product of the linear bundle of density 3 on A and the Maslov linear bundle that gives the transition function of the form i', v E Z. In the complex case, it is not possible to define almost analytic densities of order on complex manifolds in such a way that transition functions are continuous relative to small perturbations. Let us start by examining the linearized situation. Let M be a real symplectic vector space of dimension 2n with the symplectic bilinear form a and let fi be its complexification. The Lagrange plane A c fi is said to be positive (positive definite) if Im a(u, U) 2 0 ( > 0) for all u E A , u # 0. Similarly, one defines negative and negative definite planes. We denote by 9the set of all negative definite Lagrange planes. It is clear that if L E T-, then L is transversal to all positive Lagrange planes and is of the form f = B2,where (x, t) are real linear symplectic coordinates in M and B is a symmetric matrix. Let F be a fixed real Lagrange plane in M and let P be its comlexification. Definition 6.9. Let A c fi be a positive Lagrange plane. A basis e = ( e l , .. .,en)in A is said to be admissible if there exist a basis f = (f,, . . .,f,)in F and a plane L E 9such that, for allj, ej is the projection of& along L.

Yu.V. Egorov

I. Microlocal Analysis

The set of all admissible basis will be denoted by B(A). In particular, B(F")is the set of real basis in F. Note that e as defined by f and L is not unique.

We now define an almost analytic linear Maslov bundle 64 on A as a family ' on A with transition functions sa,,. The of admissible coordinate systems U section f E r ( A ; 64) is then defined by an almost analytic function fa on Ua for which fa fa,, f, for all A and p.

70

-

Proposition 6.2. B ( A ) is a union of two disjoint connected subsets so that two basis e and e', corresponding to given f and L, belong to the same component if and only if their orientations coincide. Furthermore, there exists a unique function s = s,: B(A) x B(A) + C\O such that for each compact set K c B(F) x 64-, s,(e, el) depends continuously on e, e' and i f e, e' E E,(K)'. What is more, if we put e/e' = el A ... A e,/e; A .'. A e:, then s2(e,el) = f e / e ' , where the plus sign corresponds to the case when the basis have the same orientation, and s(e, e')s(e', err)= s(e, e") i f e, e', e" E B(A).

J_

Theorem 6.4. Let A be a closed conical positive Lagrange manifold in T*X\O, where X is an n-dimensional paracompact smooth manifold. Then there exists a "natural" bijective linear map P : Y ~ + ~ ~9) ( A+;zpm(x,A)/I;-,(x, A).

Here Y k ( A ;9) denotes the space of equivalence classes of homogeneous sections of degree k.

-

Suppose now that X is a paracompact smooth n-dimensional manifold and

Let us demonstrate how to construct P locally. Let s E Ym+"/4(A;9) and let supp s lie in a small neighbourhood of the point p o . Choose local coordinates xl, ... , x , and a positive type regular phase function S E Cm(lR"x lRN) which generates A , in these coordinates, in the neighbourhood of p o . Then there exists an almost analytic function a on A which is positive homogeneous of degree m + n/4 - N / 2 , and unique to within equivalence, and which is such that

A c T*(X)\O is a positive closed conical Lagrange manifold. If p E A R, then we can define B(T,(A)) by setting, as above, M = T,(T*X); for F we take here the tangent space to the fibre. For any choice of local coordinates in X there arise natural linear symplectic coordinates in T,(T*X) so that the spaces T , ( T * X )and T,(T*X) can be identified provided that p , p E A and are close enough. Therefore if we consider the section

AR 3P

+

e ( p )E w

s J

4

)

9

Definition 6.10. Let I , , . . ., I, be almost analytic functions on A defined in some complex neighbourhood U', A = (A,,. . .,A,). These functions are referred to as admissible coordinates on A if 1. d l , , . . ., dA, are linearly independent over C at real points. 2. Let (dA,, .. ., dI,) be the dual basis to ( d A l , . . ., dA,) in T,(A)*. Then (dA,, . . ., 62,) belongs to the set B,(T,(A)) locally with respect to p E Ua n A , for some compact set K c B(F) x 9-. If UP, p = (p,,..., pn), is another system of local coordinates, then s(dA, dp) is a continuous function in U'n U p n A , and

(2K)-(n+2N)/4

-

-

aJdS.

j e i s ( x . ~ ~ 0) ~ (dXo ,

belonging to the class I T ( X , A). That this procedure is well defined has been established by Melin and Sjostrand [1976].

6.7. Fourier Integral Operators with Complex Phase Function. Let X and Y be smooth paracompact manifolds. If C is an arbitrary manifold in T * ( X ) x T * ( Y ) = T * ( X x Y ) ,then C' is defined to be { ( x ,<,y, -q); ( x , <,y, q) E C}.

-

Definition 6.11. A manifold C c T*(XxY)\O is said to be a positive canonical relation if C' c T * ( X x Y)\O is a closed conical positive Lagrange manifold and C , c (T*X\O) x (T*Y\O).

+

What is more, sa,a 1, S ~ , , S , , ~ sa,o and sa,, are continuous with respect to small perturbations of A, p under which dA, lip remain locally in B,(TJA)) for some compact set K from B(F) x 2.

-

Consider a as a function on C: and its almost analytic homogeneous continuation a on C" x CNwith support in a small conical neighbourhood of the point (x,,, 0,) corresponding to p o . The element P(s)is now defined as the distribution

then it makes sense to say that e ( p ) locally (with respect to p ) belongs to B,(T,(A)) for some compact set K c B(F) x z-.

Moreover, the derivatives 8/82, are defined by the equation dpj = C(apj/aA,) dA, C(apj/aX,) dX,. It is clear that S ( ~ A dp) , has a unique (to within equivalence) almost analytic continuation in A in some complex neighbourhood of the set Ua n U" n A and this continuation satisfies the condition

71

"

-

Let A E IF(X x Y, A), where p > 3.Let A c T * ( X x Y)\O be a closed conical positive Lagrange manifold. Then A is the kernel of a continuous operator A : Cg( Y, a,,,)+ 9 ' ( X , SZ,,,). We note that WF(A) c A ., If C = A' is a canonical relation, then A : Cr(Y, a,,,)+ C m ( X ,121/2),and A can be extended to a continuous operator from C(Y, a,,,) to W ( X ,SZ,,). Let X , Y, Z be smooth paracompact manifolds of dimension n,, n y , n,, respectively. Suppose that A , E Z,"'(X x Y, /Il),

A , E I,"'(Y x

z, A , ) @ > f)

Yu.V. Egorov

I. Microlocal Analysis

§ 3. The Propagation of Singularities for Solutions of

Theorem 3.2. If u E 9‘(Q), I is a connected piece of the bicharacteristic and I n WF,(Pu) = fa, where P is a real principal type pseudodifferential operator of order m, then either I c WF,+,-,(u) or I n WF,+,-,(u) = fa.

76

Equations of Real Principal Type 3.1. The Definition and an Example. Let us recall that an operator P(x, D)of order m is of real principal type if its characteristic form (principal symbol) Po is real valued and the form dPo(x, 5 ) is not proportional to the form 5 dx. The simplest non-elliptic operator of real principal type is D , . Consider the equation D,u = f ( x ) ,where f E Cm(52). Theorem 1.1 implies that 4 , = 0 if ( x , 5 ) E WF(u). Therefore for each C the restriction of u to the plane x 1 = C is defined. The bicharacteristics of D1are straight lines, parallel to the x,-axis, in R2”.By Theorem 3.3, Chap. 1, (x, 5’) c WF(u),=, if (x, 5 ) E WF(u). But $(D’)cp(x’)u(x)= $(D’)cp(x’)u(C,x ’ ) + $(D’)cp(x’)f ( t , x ’ ) dt. This shows that if (xo, 5’) $ WF(u) and xy = C , then no point of the line segment that is parallel to the x,-axis and lies in T*O\O and passes through (xo, 5’) belongs to WF(u). Hence every segment, lying in T*Q, of the bicharacteristic either does not intersect WF(u) or lies completely in WF(u). The same is true for WF,(u) iff E H,(Q).

3.3. Local Solvability. Suppose that P is a principal type pseudodifferential operator of order m in a domain 52 c lR” with a real-valued principal symbol. The bicharacteristics of P are the integral curves of (2) along which po(x(t),5 ( t ) ) = 0. Theorem 3.3. Suppose that P is a pseudodifferential operator with a real principal symbol p o and suppose that dspo(x,5 ) # 0 in T*Q\O. Then for each point xo E Q we can find a neighbourhood U such that, i f f E H,(B), there exists a function u in HS+,,-,(52)n &’(O) satisfying the equation Pu = f in U . Moreover, IIulls+m-l < cII f 1,) where the constant c depends only on s. Proof. Let U be a neighbourhood of the point xo such that every zero bicharacteristic of the function p o that passes through the point ( x , t),x E U and 5 # 0, also passes through a certain point ( x , , C1), where x 1 E aU and 5 , # 0. Such a neighbourhood always exists, because P is a principal type operator. We note that P* is also a principal type operator and that cp E H,(U) if P*cp E H,-,+,( U )and cp E &’( U ) .Indeed, by Theorem 3.2, the set of points where cp E H, is invariant under the Hamiltonian flux Hpo, and because cp = 0 in a neighbourhood of a U , it follows that every zero bicharacteristic contains points at which cp E C“. Hence cp E H, at every point of T*U. Thus cp E H,(U), by Proposition 1.2. By the Banach open mapping theorem, there exists a constant C , depending only on s, such that

3.2. A Theorem of Hormander. By means of a canonical transformation a real principal type operator can be reduced to the form Q(x, D)D,,where Q is an elliptic operator. This enables us to examine the general equation Pu = f, where P is an mth order pseudodifferential operator of real principal type. Let f E Cm(Q).By bicharacteristics of P we mean integral curves of the system of equations dxj-- aPo(x(t),5(t))

dt

’

a5j

5 = -aPo(x(t), 5(t)), axj

dt

j = 1 , ..., n,

(2)

!

if P o ( W , a t ) ) = 0.

:

Theorem 3.1 (Duistermaat and Hormander [1972]). If u E g’(l.2)and I is a connected piece of the bicharacteristic and if I n WF(Pu) = fa, then either I c WF(u) or I n WF(u) = 0. Proof. We may assume that m = 1, because WF(Pu) and the zero bicharacteristics of P are preserved when P is multiplied by an elliptic operator. As we saw in 0 3.1, there exists a Fourier integral operator A, of order zero, such that P = A*D, A T and AA* = I T,, where T, T, are smoothing operators. Therefore if we set Au = u, then D l v = Af + T2v.Thus the proof reduces to that of the case considered above. Since a transformation involving A associates bicharacteristics of P with straight lines parallel to the x,-axis, it follows that the theorem is true in the general case too. 0 Thus on the complement to WF(Pu), the set WF(u) is invariant under translations along trajectories of the Hamiltonian system (2). In exactly the same way, we can prove

+

+

71

IIcplls

< cIIP*cpII,-,+l?

cp

E

CW).

Therefore - the integral J fq dx, where f E H,(Q), is a continuous linear functional of P*cp E H-,-,,,+,. This implies, by the Hahn-Banach theorem, the existence of a function u E H,+,-, for which

s s

f q dx = ~.P*cpdx,

and I I ~ l l s + m - l Pu=J 0

< CJIf 1.,

cp E CT(U),

By definition, u is a generalized solution of the equation

b

F i ;

3.4. Semiglobal Solvability. By semiglobal solvability we mean solvability in every compact subset. That the semiglobal solvability is not a consequence of the local solvability becomes clear from the following example. Example 3.1. Let 52 be the ring f2 = { ( x ,y), 1 < x 2 + y 2 < 4)

Yu.V. Egorov

I. Microlocal Analysis

in the plane. The equation au/acp = f is always locally solvable, but for it to be solvable in 52 it is necessary that

4.2. The Fixed Singularity. The phenomenon noted in the above example is a feature of a large class of operators. Suppose that P is a proper pseudodifferential operator of order m in a domain 4. Let (xo, to)E T*Q\O be a characteristic point at which Hpo # 0. By multiplying P by i, if necessary, one can always achieve that H R e p o# 0. Then the equation Re p o = 0 defines a smooth surface passing through the point (xo, to)and consisting of bicharacteristics of the function Re p o . Assume that Im po has a zero of finite order k # 0 on each bicharacteristic of Re po that is close to the one passing through (xo, to). This condition signifies, in particular, that HRepocannot have a radial direction because otherwise the function Im p o would vanish identically on the bicharacteristic passing through ( x o , to).

78

loZff fb-9

cp) dcp = 0

for 1 < r < 2. This example demonstrates that the following condition is a natural one. Condition A . Let K << Q. Each zero bicharacteristic passing through the point ( x , t), x E K , 5 # 0, contains points the projection of which onto 4 lies outside K . Theorem 3.4 (Duistermaat and Hormander [1972]). Let P be a principal type pseudodinerentid operator of order m, with principal symbol p o , in a domain Q c IR".Let K << SZ. Assume that condition A holds. Then the set N ( K ) = {cp; cp E Cg(K),P*cp = 0 } has a finite dimension. l f f E H s ( 0 ) and f?jdx = 0 for all cp E N ( K ) , then there exists a function u E HS+,-,(Q) satisfying the equation Pu = f in K which is such that IIUlls+m-1

< cllf

' *

'

11s)

where the constant C does not depend on f.

This theorem can be proved by means of Theorem 3.2 and the closed graph theorem. In doing so, one has to use the fact that each solution of the equation P*cp = 0 in b ' ( K ) belongs to Cg(Q), in view of condition A and Theorem 3.2.

Theorem 4.1 (Sato, Kawai and Kashiwara [1973]). Under the above assumptions, there exists a Fourier integral operator A such that A*A = n 1 - m

,.

4.1. An Example (Hormander [1979]). Consider the equation (D1

+ ix:D,)u = 0,

where T and Tl are operators having symbols of order -a in a neighbourhood of the point (0,qo).

Using the example cited above, we have Corollary 4.1. If k is odd and (HRepo)kIm po(xo, to)< 0, then there exists a distribution u for which Pu E C" and WF(u) = { ( x o ,t t o ) , t > O } .

WF(4 = ((0,trlb), t > O } (see Example 2.2, Chap. 1). Then V decays rapidly outside each conical neighbourhood r of the point &. We set

Lo

e<2x:+1/(k + 1 ) + i(x', C')

4.3. A Special Case. Let P be a principal type pseudodifferential operator of order m with principal symbol p o = a + ib. Assume that the following conditions hold: 1. The Poisson bracket { a , b } vanishes at each characteristic point lying in

T*Q\O.

where k is a positive odd number. Let qo = (0, - 1, q"), q" E IR"-'. Assume that v(x')E C', x' = ( x z , . . . , x,,), is a function with compact support for which

u(x) = (24'-

+ T,

,

0 4. The Propagation of Singularities for Principal Type Equations with a Complex Symbol

79

ij(t')dt',

where rois a neighbourhood of ub such that t2 < - C I ['I in r.Then u E C', and, moreover, u E C" for x 1 # 0. It can be shown that WF(u) = { (0,tqo), t > O } ; thus the solution has a singularity at x = 0 that is not propagated.

2. The fields Ha, H , and

1t j a

are linearly independent at each characteris-

-

atj

tic point belonging to T*Q\O. Under these conditions, there exists a homogeneous canonical transformation x of the neighbourhood of the point ( 0 , ~ ' E) T*IR"\O onto a neighbourhood of the point (xo, to)E T*4\0 and there exist Fourier integral operators F and F , , corresponding to x and x - l , such that the operator F,PF - (Dl iD,) has a symbol of order -a in a conical neighbourhood of (0, qo). Condition 1 implies that {a, b } = aa + p b and [Ha, H , ] = aH, p H , . By Frobenius theorem, the manifold p o = 0 is fibered into two-dimensional manifolds the tangent planes to which are generated by H,, and H,. These manifolds are referred to as bicharacteristics. The fields H,, and H , define on bicharacteristics an analytic lattice in which the solutions of the equation

+

+

Yu.V. Egorov

I. Microlocal Analysis

(Ha+ iH,)v = 0 are analytic functions. For this lattice, we can also define classes

Theorem 5.1. Let u E 9'(0),zo $ WF(Pu), and let (2) hold. Suppose that for each j E { 1, 2) there exists a k E { 1,2} such that Yjk(1k) n WF(u) = 0. Then 20 $ WF(u). Let S, denote the subprincipal symbol of P.

80

of harmonic and superharmonic functions. If u E 9'(0),then the smoothness of u at a point (x,5) E T*0\O can be measured with the aid of the function s,*(x,5) = supp{s; u E H,(x, t)}.This function is lower semicontinuous and positive homogeneous of degree zero. Theorem 4.2 (Duistermaat and Hormander [19721). Assume that conditions 1 and 2 are satisfied. Let u E 9'(0)and P u = f in 0.Suppose that B is a domain on a two-dimensional bicharacteristic and s is a superharmonic function in B such that s < .;s Then the function min(s,*,s + rn - 1) is superharmonic in B. 4.4. The Propagation of Singularities in the Case of a Complex Symbol of the General Form. The most general results in this direction are due to Hormander, Shapira, and Dencker. A majority of these results and the corresponding references can be found in Hormander [1983, 19851. Here we shall cite only one result of Hormander. Theorem 4.3. Let u E 9'(0)and P u = f i n Q. Let I = { t E IR,t , < t < t , } and y ( I ) be a segment of the zero bicharacteristic of the function Re p o . Let Im p o 2 0 in a neighbourhood of ?(I). If y ( I ) n WF(f) = 0and y(t,) $ WF(u), then y(I) n WF(u) = 0. More precisely, iff E H,(y(I))and u E Hs+m-,(y(t,)),then u E H,+,-, at points of y(I). If Im p o < 0 in a neighbourhood of y ( I ) , the theorem remains valid, only one has to take the left end instead of the right end of the segment.

8 5. Multiple Characteristics 5.1. Non-involutive Double Characteristics. Let P = P, P, + Q, where P, and P, are pseudodifferential operators of order rn, and m,, respectively, and the operator Q is of order m + m2 - 1. Assume that the principal symbols p1 and p z are real valued. We take a point zo E T*0\0 at which Pl(Z0) = 0,

PZ(Z0)

= 0,

{PI, PZHZO)

z 0.

(2)

This condition implies that the fields H p I ,Hp2and ta/@ are linearly independent at zo. In particular, an open interval I c IR, 0 E I , can be found such that the bicharacteristics y j of p j passing through zo define an injective map y j : I + cj = { z E T*\O, P j ( Z ) = O},

y j ( o ) = zo.

We note that at those points of the neighbourhood U,,, that lie outside C , n C,, P is a real principal type operator, and for such points the propagation of singularities has been investigated in Q 3. If 1, = { t E I, (- l)kt > 0}, we can define 4 semibicharacteristics at the point 2 as the curves y j k : I , + Zj for which yjk = yjlr, forj, k = 1, 2.

81

Theorem 5.2. Suppose that (2) holds and i S P ( Z O ) / { P l ~PZ} (zo)-

3 $ (0, L 2 , . ..>.

l

Let u E 9'(0), zo $ WF(Pu) and (yl(I)\zo) n WF(u) = 4. Then zo $ WF(u). Simi-

,

larly, i f iSP(ZO)/{Pl? P 2 )

(zo)

+ 4$ (0, - 1, - 2, ...}

and (y2(I)\zo) n WF(u) = 0, then zo $ WF(u).

Corollary 5.1. Assume that (2) holds and that

If

zo $ WF(Pu) and u is a smooth function on certain two semibicharacteristics through z o , then zo E WF(u). (These and more general results can be found in Ivrii [1979,1981], Bove, Lewis and Parenti [1983], Hanges [1979], and Melrase and Sjostrand [1978, 19823.)

5.2. The Levi Condition. When the operator P has multiple characteristics, its properties may significantly depend on the lower terms. In this case, an important role is played by the Levi condition, introduced first in connection with the investigation of the Cauchy problem. This condition signifies that the transport equations are differential equations along bicharacteristics and that their order equals the multiplicity of the corresponding characteristic. Definition 5.1. An operator P E Tm(0) has characteristics of constant multiplicity if its principal symbol p is of the form where rj E IN, p = 4;'. . .q:, and qj are symbols of principal type operators, and the sets q j ' ( 0 ) do not intersect in T*Q\O. Definition 5.2. Suppose that P E 9 " ' ( X ) , its principal symbol is real valued, and suppose that P has characteristics of constant multiplicity. P is said to satisfy the Leui condition Y ( x o , l o )at a point (xo,to)in p-'(O) c T*0\O if e-"qp(ae"q) = O(t"-'J),

t + +a

for each function q ( x ) that satisfies the equation qj(x, d ~ ( x )=) 0 (ifqj(xo,to) = 0) in a neighbourhood of xo and which is such that d q ( x o )= to and for each function a E Cg(0) that has support in a neighbourhood of xo where d q # 0. The operator P is said to satisfy the Levi condition (9) if the condition 9 ( x o ,to) is satisfied for all the points of p-'(0).

82

Yu.V. Egorov

I. Microlocal Analysis

5.5. The Schrodinger Operator. Let 52 be a domain in IR". Let P(x, D)be a pseudodifferential operator in 52 of order m with real principal symbol po(x, 5). Suppose that P has double characteristics so that the set C = { ( x ,5 ) E T*Q\O, po(x, 5 ) = 0) is defined by the equations u,(x, 5 ) = 0, ..., uk(x, 5 ) = 0, where uj(x, t5) = uj(x,t), t > 0, and the forms dul, ..., du,, 5 dx are linearly independent. Assume that C is an involutive manifold, that is, {ui, uj} = 0 on C. Then C is fibered into one-dimensional smooth curves that are integral manifolds for the fields Hu,for each j = 1, .. . , k. We also assume that in a neighbourhood of every point of C,the principal symbol can be expressed in the form Caijuiuj,i, j = 1, ... , k, and that the matrix IIaijll is positive definite. This matrix defines a Riemannian metric g A on C. 1 We assume further that pm-'(x, 5 ) - 2i a'Po(x, 5 ) < on =*

It is clear that (9) is a condition imposed on terms having order

> m - max rj. This condition was introduced by Flaschka and Strang [1971].

5.3. Operators Having Characteristics of Constant Multiplicity. Let us conhaving characteristics of constant multiplicity and sider an operator P E Ym(52) In this case, in a neighbourhood of each satisfying the Levi condition (2). characteristic point (xo,to)the operator P can be expressed in the form x2=oBkQjk+ To for some j , 1 < j < s. Here To is an operator of order zero, Qj is an operator with real principal symbol qj(x, 5 ) E Ssj(sZ),B, E 9 m - k s j ( 5 2 ) . This representation enables us to obtain the following result which is a particular case of a theorem of Tulovskij [1979].

Theorem 5.3. Suppose that the operator P E gm(52) has characteristics of constant multiplicity and that it satisfies the Levi condition (9). Let u E 9'(Q). Then the set WF(u)\WF(Pu) is contained in p-'(0) and is invariant under the translations along bicharacteristics.

axjasj

Theorem 5.5 (Boutet de Monvel [1975]). Suppose that all the conditions listed above are satisfied. If u E B'(52) and Pu E Cm(52),then WF(u) is a union of geodesics, lying on Z, in the metric g A . Conversely, i f (x, 5 ) E C and if y, a geodesic arc in the metric g A , passes through ( x , t),then there exists a distribution u E #(a) such that WF(u) n U = y n U and Pu E Cm(U),where U is a sufficiently small conical neighbourhood of the point ( x , 5).

5.4. Operators with Involutive Multiple Characteristics. We assume now that C c T*X\O is a closed conical manifold of codimension d defined by the equations ql(x, 5 ) = 0,. . . , qd(x,5 ) = 0. Here the functions q j are smooth, real valued,

and are such that: 1. The fields H q l ,. . . , Hqa and lD,are linearly independent at every point of

As an example of an operator satisfying the hypotheses of Theorem 5.5, one i a can cite the Schrodinger operator P = - - a'd. If Pu E C", then in this case at WF(u) consists of straight lines that are parallel to the subspace t = 0, T = 0 and lie in the cone = ... = t, = 0, T < 0 (see Boutet de Monvel [1975]). We remark that several other important results concerning the propagation of singularities for operators with multiple characteristics have been obtained in the works of Bony, Grigis, Ivrii, B. Lascar, R. Lascar, Shapira, Sjostrand, and others. In particular, see Ivrii [1974, 1980a, 19813, Bony and Schapira [1973], Chazarain [1974], Grigis [1976, 19793, B. Lascar [1981], B. Lascar and Sjostrand [1982], Menikoff [1979], Sjostrand [1976], and Uhlmann [1977].

C;

2. { q j , q k } = 0 On C;j , k = 1, ..., d. We may assume that q j are homogeneous in 5 of degree 1. Consider now a proper classical pseudodifferential operator P E Y m + , ( X ) where , m E IN, k E IR. Let its principal symbol pm+, be such that P m + k = 0 on C and P m + k # 0 outside C; P m + k has a zero of order m at each point of C. Take a point p on C, and let a,(t) denote a homogeneous polynomial in t E T,(T*(X))/T,(C) = F, of degree rn that arises in the Taylor expansion of P m + k at the point p. Suppose that 3. ap(t)# 0 for all p E C when 0 # t E F,. The following condition imposed on the lower terms is a natural generalization of the Levi condition. 4. If Qj = qj(x,D), then there exist classical pseudodifferential operators A , of order k such that P = &lSm A,Q;l.. . Q? microlocally. When m = 2, for instance, this condition implies that the subprincipal symbol of P vanishes on C. Theorem 5.4 (Sjostrand [1976]). Assume that the conditions 1-4 hold. Let F = T,(T*X)/T(C), and let F x,F be the product of bundles on C. Let zl(s, t), . . ., zm(s,t ) be the roots of the equation a,(s + zt) = 0; s, t E F,. Assume that for (s, t ) E F x F and for linearly independent s and t the multiplicity of zj is a constant that does not exceed 2. If u E 9 ' ( X ) , Pu E Cm(X),and p E WF(u),then r, c WF(u),where r, is an integral manifold, passing through p, of the fields

Hql'...'Hqd.

83

L

Chapter 5 Solvability of (Pseud0)Differential Equations

8 1. Examples 1.1. Lewy's Example. Consider the unit sphere 52 in C' = { ( z l , z2): 1z21' = l}. If a change of variables is performed by the formula z ; = (zl - l)-', z; = z2(z1- l)-', the surface 52 is defined by the equation

1z,I2

+

1

+ + 2 , + 1z2(' = 0. z1

I. Microlocal Analysis

Yu.V. Egorov

84

The operator

9 = 2-

;(

a

az, - z2

+

1.3. Other Examples. Simple considerations show that the set of smooth functions f for which the equation (1) has a solution (even if generalized)constitutes a set of the first category.

g)

a

is tangent to 52. The operator 9 is the one that is referred to as Hans Lewy's operator. In 1956, Lewy constructed a functionf(x,, x,, x 3 )E C"(lR3) such that the equation

a

We note that for L = - - i x - the operator LL is of the form ax a y

--

has no solution in the class of continuous functions in any subdomain w c lR3.

1.2. Mizohata's Equation. Consider the equation au ax

+.

ax, + a2

__

, while the operator

(5

P = L L L L = -+x*-

-

85

a2 ay2)

2

+-

a2

ay2

real coefficients, and the equation Pu = f has no solution at the origin for set of functions f of second category in Cg(lR2)(Trkves), ts a function f E C"(W2) such that the equation

au = f ( x , y). aY

9u-fu=O

lXk-

If k is even, the substitution t = x k + l / ( k + 1 ) transforms this equation into the equation

as no solution, apart from u = 0, in the rectangle 1x1 < r, lyl < r'. Kannai [1971] has shown that the equation Au = f, where A = D, -t iX,D;,

which is solvable for any distribution g E 9'. If k is odd, then (1) is, in general, not solvable in any neighbourhood of the origin (see Grushin [1971]). Indeed, let { Q j } be a sequence of disjoint discs converging to the origin, each disc lying in the domain x > 0. Let f = 0 outside Q j ,f E C", f ( x , y ) = f (- x , y), and let f > 0 in Qj. If we now set

u

then (1) implies

av + iXk-av = 0,

-

dx

aY

aw

-

ax

+ iXk-aw = f ( x , y). aY

Clearly, v = F(xk+'/(k + 1 ) + iy), where F is an analytic function. As to the function w, it satisfies w(0, y) = 0 and w = G(xk+'/(k+ 1 ) + iy), where G is an analytic function at the points of the complement set to Qj. It implies that w = 0 in R 2 \ U Qj so that, in particular, w = 0 on aQjforj = 1,2,. . . . But then

as no solution at the origin, and that A is a hypoelliptic operator. Hormander [19631 has established the insolvability of a linear self-adjoint fferential operator of second order, with real coefficients, defined by the

+ (1 + x:)(D;u - D:u) - x , x , D , D ~ u - D , D ~ ( x ~ x , u ) + x,x,D,D3u + DlD3(X,X3U).

PU = ( x ; - x:)D:u

ve cases, for every neighbourhood 52 of the origin there exists a function f E Cg(52) such that the equation Pu = f has no solution u in Y(52). This implies that there is a function fo E S(lR3)for which the equation Pu = fo has no solution u in g'(52)in any neighbourhood of the origin. Definition 1.1. A pseudodifferential operator P is solvable at a point xo if ere are two neighbourhoods U and V of xo such that U c V and for each ction f in C;(U) there exists a distribution u, with support in V , that satisfies e equation Pu = f in U .

In the case of a differential operator, one can take U = K

u

1IQj

f ( x ,y) dx dy

contradicting the definition off.

=

9 2. Necessary Conditions for Local Solvability

w(cos a + i cos b ) ds = 0, 2.1. Hormander's Theorem. Hormander succeeded in explaining the behaviour of the above examples. He [1963] obtained the following result.

I. Microlocal Analysis

Yu.V. Egorov

86

Theorem 2.1. If (x,,

5,)

E

T*Q\O, po(xo,5,)

=0

and c,(x,, 5,) < 0, where

so that the hypotheses of Theorem 2.1 imply that y(t) has a first-order zero at t = 0. This theorem can be generalized by replacing its hypothesis by the ~ ' ( 0= ) . . . = Y"'-~)(O)= 0,

then the pseudodifferential operator, with principal symbol p,, is insolvable at x,. Corollary 2.1. If P(x, D)is a differential operator, and cl(xo,5,) # 0, then P is insolvable at xo.

po(xo,to)= 0 and

To prove Corollary 2.1,it is enough to observe that the function cl(x, 5) is homogeneous in 5 of degree 2m - 1, and is, hence, odd. The proof of Theorem 2.1 is based on the fact that the solvability of P implies

llcplls, G CIIP*cpllSZ?

cp E

ca4,

with certain real C , s1 and s2. For instance, for the Lewy operator P the adjoint operator is of the form P* cpo = e

a + iZ-.a aZ at

= --

- d( 2 1112 +t2 - 1 4 4 -2ir

a a aZ + iz-at

=-

And the function

k odd

(3)

(see Egorov [1971], Nirenberg and Trhves [1970]). 2.3. The Zero of Infinite Order. The following condition ( Y ) is a necessary condition for the solvability of P in the general case (Hormander [1983, 19851). Condition ( Y ) .As t increases, the function y(t) does not change its sign from minus to plus at t = 0. (Clearly, this condition generalizes condition 3.) 2.4. Multiple Characteristics. Less attention has been paid to the case of ultiple characteristics. The necessary conditions known to us for solvability in he case of a characteristic point (xo,5,) at which dpo(xo,5,) = 0 lead to the fact hat in the neighbourhood of this point there cannot be simple characteristic oints where condition ( Y )is violated. We cite a few results of a different nature.

fZir(~1~)

grad w(0) = to w(0) = 0, po(x,grad w(x))= 0, and Im w(x) > 0 when x # 0. We now take for cp a function of the form N

1

~ ' ~ ' (>00, )

(2)

satisfies the equation P*cpo = 0. Let o = { ( x ,y , t);x 2 + y2 + t2 < l} and let h E Cg(w) be a function such that h = 1 for p 2 E x 2 + y 2 + t 2 < 1/4.Then the function f = P*(hcpo)vanishes for p 2 1 and p G f.It is easy to see that Ilf IIsz < CAsze-d/4.On the other hand, IlhcpoIIs,2 c0;1s~-3/4, where the constant Co > 0 depends on s1 but not on 1.Substituting the function cp = hq, into (2),we obtain an inequality that cannot hold for 12 1, if A, is sufficiently large. This establishes the insolvability of P in o. In the general case, when P is a pseudodifferential operator of an arbitrary order the proof is carried out along the same lines. The hypotheses of the theorem enable us to construct a function w for which

cp(x) = h(x)eidw(x) ;l-jaj(x),

87

orem 2.2 (Egorov and Rangelov [1977]). Let po(xo,5,) = 0, x,, 5,) = 0 and (x,, to)E T*Q\O. Suppose that the inequality cl(x, 5) 2 xo12 15 - <, I 2) , m, > 0, holds in some neighbourhood of (x,, 5,). Then udodifferential operator P , with principal symbol p,, is insolvable at xo.

+

We now consider the case P = Q2 + R, where the operators Q and R are of rder m and 2m - 1, respectively. Theorem 2.3 (Popivanov [1975]). Let Q be an operator of real principal type. x,, to)= 0, Re ro(xo,to)< 0 and the function Im ro has a zero of finite odd along the bicharacteristic of qo passing through the point (xo,to),then P is eorem 2.4 (Popivanov [1975]). Let Re qo(x, 5) = a(x, 5) and ,(x, 5) = b(x, 5). Let qo(xo,5,) = 0, 5, # 0, and let k E IN be odd. Assume

aj E C'"(w).

j=O

2.2. The Zero of Finite Order. Let po(x, 5) be the symbol of a principal type operator, a = Re p o and b = Im po. Consider a bicharacteristic of b, that is, an integral curve of the system of equations a 5 k = - ab(x(t),a t ) ), dx,-- ab(x(t),5 ( t ) ) k = l , ..., n, dt 85, at ax, passing through the characteristic point (x,, 5,) when t = 0. Clearly, b(x(t),5 ( t ) ) = 0 in this case. Consider the function y ( t ) = a(x(t),<(t)).We note that C l ( X ( t ) , H t ) ) = Y"t)

Hib(x,,

5,)

=0

for j < k

and

H,kb(x,, 5,) > 0.

en P is insolvable at x,.

6 3.

Sufficient Conditions for Local Solvability

3.1. Operators of Real Principal Type. Theorem 3.2, Chap. 4, which conms the propagation of singularities, immediately yields theorems on local and semiglobal solvability of the equations Pu = f of real principal type if d,p,(x, C;) # 0 in T*Q\o.

Yu.V. Egorov

I. Microlocal Analysis

In fact, P*, the formal adjoint operator to P, is also a real principal type operator. For xo E SZ a neighbourhood o 3 xo can be found such that every = 1, comes out on the bicharacteristic of P* that passes through ( x o ,t),with boundary ao. Therefore, by Theorem 3.2, Chap. 4, * cp E c;(w), cp E &(u), P*cp E Crn(o)

The most general theorem can be obtained for subelliptic operators. These operators can be defined as follows. Let a, = Re p o , a, = Im p o , and

88

CP E &(a), P*CPE Hs(w) * ~p E Hs+m-l(U), where m denotes the order of P. This implies, in view of the closed graph theorem, that there exists a constant C such that IIVlls+m-l

< C(lIP*c~lls+ I I d l s + m - 2 ) ,

and if w is sufficiently small and s

CP E G(o),

+ m 2 1 + 4 2 , then

CP E C:(o)* It follows then that J f?jdx, with f E H...s-m+l, is a bounded linear functional of P*cp in H P s ,that is, there exists an element u E H,such that II(Plls+m-1

d CllP*c~lls,

Hi=C j=l

(a&, ax,

~PEC;(~).

Hence Pu = f i n o. If one considers an arbitrary compact set, instead of a small neighbourhood, then it becomes necessary to impose some additional conditions on the behaviour of the bicharacteristics in u (see 0 3.4, Chap. 4).

3.2. Operators of Principal Type. In his works, Hormander has established that the principal type equation Pu = f is locally solvable if the condition C l ( x o ,to)> 0 holds at each characteristic point (xo,to). This result was further generalized by Nirenberg, Trkves, Beals, Fefferman, and the present author (see Egorov [1984]). The problem of obtaining sufficient conditions was resolved completely in the case of differential operators by Beals and Fefferman [1973]. In this case, the sufficient condition coincides with the following condition, introduced originally by Nirenberg and Trkves: Condition (9). The function y(t) = Re zpo(x(t),t ( t ) ) ,where (x(t),5(t)) is the zero bicharacteristic of the function Im zpo(x, does not change its sign with a change in t ; z E C\O.

r),

Theorem 3.1. If condition (9)holds and dtpo(x, 5 ) # 0, then the differential operator P is locally solvable. Nirenberg and Treves [19701 first established this theorem for operators P having analytic coefficients.For differential operators conditions (9) and (Y)are equivalent. It is not known till now whether or not condition (Y) is sufficient for the local solvability of the principal type pseudodifferential equation with a complexvalued symbol.

a

at,

aai(x, t)

at,

a ),

ax,

i=l,2.

r a = (a1,..., ar), 8 = (bl, ..., /&), we denote by H:H$ the operator llH$l...HYH?. Let k(x, 5 ) be the smallest integer for which H:H$a,(x,

5 ) # 0,

la

+ 81 = k.

f al(x, 5 ) # 0, we set k(x, 5 ) = 0.

Theorem 3.2 (Egorov [1971]). Zf condition (Y) holds for the operator P and p k(x, 5 ) < k for (x, t) E T*K\O, then the equation Pu = f is solvable in K when satisfies a finite number of conditions. More precisely, in this case there are nctions g l , . ..,gN in C;(K) such that, if

s

f i j dx = 0,

I I ~ I I - s ~ C l l f l l - s - m + ~ S~ f @ d x = J @ G d x ,

5)

89

j = 1, . . ., N ;

f E H,(K),

n d'(S2)satisfying the equation Pu = f re exists a function u E H,+m-k(k+l)-l(52) K . I f the diameter of the compact set K is sufficiently small, then the orthonality condition becomes redundant.

We list a few more conditions that are sufficient for the local solvability.

Theorem 3.3 (Egorov [1984]). The operator P is locally solvable at a point xo at each characteristic point ( x , 5 ) condition (Y) and at least one of the following

r),

1. The forms d Re po(x, d Im po(x, 5 ) and 5 dx are linearly independent. 2. There exist a constant N > 0 and a function a(x, t),smooth and positive omogeneous of degree m - 1, such that the inequality

W x , 5 ) + Re a@, t)p0(x, 5 ) + Nlp0(x,5)l2I5t-' 2 0 Ids in some neighbourhood of the point in question. 3. Let a = Re p o , b = Im p o , where a is the symbol of the principal type erator. Let V'b be the derivative of b in the direction of the vector field that is gent to the surface a(x, 5 ) = 0 and orthogonal to the field Ha. There exists a constant K such that the inequality I W x , 5)12 < Klb(x, 01 Ids in some neighbourhood of the point (xo,to)for all the points ( x , t) lying on e a(x, 5 ) = 0, 151 = 1. case, a neighbourhood u c SZ can be found for each real s, s < m - 1 + that the equation Pu = f, with f E Hs-m+l(52), has a solution u E H,(SZ) n &'@) in o.Moreover, llull, < Cell f Ils-m+l, E = diam o,where the constant C is independent o f f or E.

Yu.V. Egorov

I. Microlocal Analysis

3.3. Operators with Multiple Characteristics. We examine only the case P = QZ R, where the orders of Q and R are, respectively, m and 2m - 1.

1.2. Hypoelliptic Differential Operators with Constant Coefficients. A basic feature of such operators is the behaviour of the roots [ of the equation P ( [ ) = 0 as [ + co.Namely, the condition

90

+

Theorem 3.4 (Popivanov [1975]). Let Q be a principal type operator and dcqO(x,CJ) # 0 for CJ # 0. Assume that condition (Y) holds. If ro(x, 5 ) = 0 (mod qo), then for f E b'(SZ)n Hs-zm+z(SZ)there exists a function u E &(Q) n Hs(SZ) satisfying the equation Pu = f i n o;o is a sufficiently small neighbourhood of xo. Moreover, IIulIs G C(o)Ilflls-2m+Z + C1 IIUlls-l.

max( -n/2, - 1 - n/2 + m), we may take C, = 0. Here C(o) + 0 as diam w + 0. When qo is a real-valued function, we can obtain more precise results.

If

s2

Theorem 3.5 (Popivanov [1975]). Let Q be a real principal type operator and dtqO(x,CJ) # 0. The operator P is locally solvable if at least one of the following conditions is satisfied: 1. Re ro(x, CJ) 2 0 if qo(x, CJ) = 0, 5 # 0. In this case the following estimate holds: I I d l m - 1 6 C ( ~ ) l l f ' ~ l l ~ - m CP, E Cg(o). 2. Re ro(x, 5 ) > 0 if qo(x, 5 ) = 0, 5 # 0. In this case,

Chapter 6 Smoothness of Solutions of Differential Equations

5 1.

Hypoelliptic Operators

1.1. Definition and Examples. As we know, every classical or generalized solution of the Laplace equation is actually an infinitely differentiable, and even analytic, function. Petrovskij has shown that all the solutions of the equation P(D)u = 0, where P(D)is a differential operator with constant coefficients, are analytic functions if and only if P ( D ) is an elliptic operator. A complete characterization of the differential operators P(D)for which every solution of the equation P(D)u = 0 is an infinitely differentiable function was given by Hormander. Definition 1.1. A pseudodifferential operator P ( x , D)is said to be hypoelliptic in a domain SZ c lR" if the conditions u E b'(SZ)and P ( x , D)u E Cm(o)imply that u E Cm(o)for every subdomain o c SZ. Definition 1.2. A pseudodifferential operator P ( x , D) is said to be microlocally hypoelliptic in a domain 51 c T*R" if the conditions u E &(R")and WF(Pu) n o # 0imply that W F ( u )n o = 0for each subdomain o c T*SZ.

91

JIm[I + cc as \[I + co on the surface P ( [ ) = 0 (1) is necessary and sufficient for hypoellipticity (Hormander [19631). By means of the Seidenberg - Tarski theorem, one can show that this condition is equivalent to the following condition: 3 y > 0, C > 0: l i l y 6 ClIm {I,

if P ( [ ) = 0, l i l > C. (2) We can cite equivalent conditions in terms of the behaviour of P ( 5 ) for real (. Namely, the conditions (1)and (2)are equivalent to each of the following three

P a ) ( < ) / P ( 5+ )0

as

151 + co, 5 E R", Va # 0;

(3)

3y > 0,

c > 0: lP'"'(CJ)IG clP(<)ll
(4)

'('+')+

1

(5)

P(4)

as 151-fco,

~ E I R " , V~EIR".

We note that elliptic operators are characterized by the fact that for them one 1 in (2) and (4).For other hypoelliptic operators, this quantity is The class of semi-elliptic operators, close to the class of elliptic operators, is defined in the following manner. Let m , , . . ., m, be positive integers, and let ak/mk< 1 for all those a = (a,, . . ., a,) for which a, # 0 in the definition of the operator P(D) = a,D". Let Po(D) = where the summation is performed over those a for = 1. If p o ( o # 0 for 5 # 0, then P(D)is called a semi-elliptic particular, if m , = . . . = m, = m, then such an operator becomes lliptic. For instance, p-parabolic equations, introduced by Petrovskij, are semi-

1

Za,D",

We also note the following theorem (Hormander [1963]). Theorem 1.1. Let Q(5) be a polynomial, with real coefficients, of degree m. Let

2 2 be an integer. Suppose that R ( 5 ) is a positive definite homogeneous polyree 2km - 2(k - 1). Then the operator P(D)= Q(D)2k+ R(D) is 1.3. The Gevrey Classes. In proving sufficiency of the algebraic conditions, d above, one usually constructs a fundamental solution of the operator P . s construction depends significantly on condition (2). If we know the exact condition (2), we can obtain estimates for the derivatives of the ution of the equation. Definition 1.3. By a Gevrey class G, = G,(SZ) ( p 2 0) we mean the space of nctions u(x)that are defined in SZ, have derivatives of all orders and satisfy the

1. Microlocal Analysis

Yu.V. Egorov

92

93

We denote by H,,,(R:) the space of functions with the norm

inequalities ID"u(x)l < CB'"'a,"

l ~1112,s= (24-n[(I + 1512)'(1 + l 5 ' l Z Y I W l Z d5,

for some constants C ad B and for all a. Here, as usual, aSa= a!,'. . .a?. Clearly, these inequalities are equivalent to the inequalities

where

5' = (51,. . .) tn-l),

ID"u(x)l < CB'"'(a!),. The class G , consists of analytic functions (in each of the variables x,, The projective Gevrey classes G, = G,(SZ) are also used.

nu.,

IR:

=

( x E IR",x, 2 O } .

Theorem 1.4 (Hormander [1983,19853). Let P ( D ) = 0." + CancmaaDabe a differential operator of order m, with constant coefficients. Let SZ be an open subset in IR; . I f

.. . ,x,).

P(D)u E HF-rn,r,(Q),

Theorem 1.2. If the condition (2) holds for the operator P(D), then every solution u(x) of the equation P(D)u = f,where f E G, and P y = 1, belongs to G,. In particular, if P ( D ) is an elliptic operator, then every solution u of the equation P(D)u = f,with analytic,f,is analytic.

then u E Htg(SZ), where Q < c1,CJ

E Hzr2(Q),

+ z < aj + zj,j = 1,2.

Remark. Theorem 1.4remains valid when a, is a function of x and belongs to

C" .

The quantity P can be found in terms of the polynomial P ( 5 ) by the formula

Corollary. Let the hypotheses of Theorem 1.4 hold, and let Pu E C"(f2). Let for there be two sequences {Q,] and ( z j } such that aj zj -+ co and u E HkfrJ(SZ) all j . Then u E C"(S2).

+

\

(Hormander [1983,1985]).

1.5. Hypoelliptic Equations in Convolutions. A distribution p to be invertible if

1.4. Partially Hypoelliptic Operators. Let us divide the variables x = ( x , , . . ., x,) into two groups: x' = ( x l , .. ., x k )and x" = (xk+,,. . . ,x,). Similarly, a = (a', a"), etc.

u E b',

Dau E 2z(SZ), VN = (a', 0) * u E C"(Q)

IIm cI/ln

Example 1.1. A differential operator P(D)is partially hypoelliptic in x , if and only if P ( D ) = CD,P +

C

k=O

Qk(D')D,k,

C # 0,

p

D' = (01,.. ., Dn-1).

In particular, this condition is satisfied if the plane x , = 0 is non-characteristic. Theorem 1.3 (Hormander [1983, 19851). The operator P(D) is partially hypoelliptic in x" if one of the following equivalent conditions holds: I. IImc1+IRecl+co i f [ + c o a l o n q t h e s u r f a c e P ( c ) = O . 11. There are constants C and y > 0 such that

(1 + 151)'

< C(l + JIm5"1)(1 + 15'0,

P ( 5 ) = 0,

t E IR".

111. lim P'")(t)/P(5)= 0 if a # 0, 15''1+ co. IV. P ( t ) = p,(<")th, where ~ ~ ( 5 is' ' a) hypoelliptic polynomial in X" and PQ(5")/"(t") -+ 0 as 15"l -+ 0, z 0.

za.t=o

* u E C".

+ co

as [ --f co along the surface jl(c) = 0.

(6) Definition 1.5. A distribution p is said to be hypoelliptic if p is invertible and (6) holds.

for any subdomain SZ c IR".

P -1

be said

Theorem 1.5 (Hormander [1983,19851). If each distribution u in .9'(Rn) satisfying the equation p * u = 0 lies in Cm(IRn)), then

Definition 1.4. The operator P ( D ) is said to be partially hypoelliptic in x" if P(D)u E C"(SZ),

p * u E C"

E 6' will

Theorem 1.6 (Hormander [1983,19851). The following conditions regarding equivalent: 1. p is hypoelliptic; 2. u E 9'(IR,), p * u E C"(IR") * u E C"(IR"). 3. There exists a distribution (parametrix) F such that p * F - 6 E C" and

E &"(R")are

ch sing supp F c ch sing supp p. Here ch K denotes the convex hull of the set K .

t 1.6. Hypoelliptic Operators of Constant Strength (Hormander [19633). Let p(x, D) be an mth order differential operator with constant coefficients, and let

Yu.V. Egorov

94

I. Microlocal Analysis

Definition 1.6. An operator P ( x , D) is said to be of constant strength in 52 c IR" if for any point (xo,y o ) E Q the differential operators P(xo, D) and P(yo, D) are such that f%O,

1.7. Hypoelliptic Differential Operators with Variable Coefficients. Let P(x, D) be a differential operator of order m, with smooth coefficients, in a domain Q c IR". Let M j ( x ,5) be functions in Q x IR" such that 1 < M j ( x ,5 ) 6 Cx(l

or -+-<1, a-1 b 2m 2k

5) d cxo,,oaYo, 5).

Theorem 1.7. Let P ( x , D ) be a dfferential operator, with coefficients in Cm(Q), Q c IR", of constant strength in Q. Suppose that P(xo, D), where x,, is some point of Q, is a hypoelliptic operator. Then the operator P(x, D ) is hypoelliptic in Q, What is more, W F ( u )= W F ( P u ) for all u E 9'(Q), that is, P is microlocally hypoellipt ic.

+ l
d > 0,

j = 1, ..., n,

= M?I

... M,B"M;"

I . . .

Theorem 1.9 (Hormander [1983, 19851). Let P be a matrix pseudodifferentiul operator, P E 2 : & 2 ; Ck,Ck), 0 < 6 < p d 1. Suppose that there are (k' x k)matrices A ( x , 5) and B(x, 5) such that the matrix p(x, 5) (the symbol of P ) satisfies the inequality

IA(x,

5)D:D!p(x,

lD:D!Hx, 01 d ca,~,x(l + 151)-d'alMfi-a(X,5)P(X,5) for all tl and fi, x E Q, 5 E IR";the constants Col,B,x are bounded when x varies on bounded subsets of Q. Theorem 1.8 (Hormander [1961]). If a differential operator P ( x , D ) satisfies condition ( H E ) in Q, then it is hypoelliptic in 4. The condition ( H E ) is a natural generalization of condition (4) to operators with variable coefficients. Example 1.2 (Hormander [1961]). The operator P(x, D ) = I

+ IX(Z'(-d)"

satisfies condition (HE), and is, hence, hypoelliptic if v > p. In this case, we may take M j ( x ,5) = M ( x , 5) = (1 + 1512")1/2v(1+ I ~ 1 ~ ~ 1 5 1 ~ " ) - ~ / ~ ~ . Example 1.3 (Hormander [19611). The operator P ( x , D ) = 0:"'+ D;k + ic(x)DyDDb,+ 1 satisfies condition ( H E ) , and is, hence, hypoelliptic if c E C", c(x) E IR and if either a-1' b a b-1 -+-<<, < 1, 2m 2k 2m 2k

-+-

5)*B(x, 511 d ca,@,K(l + 150-p'aJ+dlS',

x

E K,

where K is an arbitrary compact subset in Q, and u, p are any multi-indices. Assume also that IB(x9

IA(x?

M;"".

Definition 1.7. An operator P ( x , D) is said to satisfy the condition ( H E ) in Q if P ( x , D) does not vanish identically in any of the components of 52 and

O
1.8. Pseudodifferential Hypoelliptic Operators. The preceding theorem can be generalized in a natural manner to the case of pseudodifferential operators.

in which the constant C, is bounded when x varies on compact subsets in Q. For a = (al,. .. ,a,) E Z : , fi = (fil,. . ., fin) E Z : , we set MP-a

95

5)-1p(x,

511 + IB(x,

5)-1A(x, <)-'I < CK,

511 d C K 1 5 1 m ' ,

151 3 c K *

XEK,

Then P is hypoelliptic.

A still more general theorem was obtained by Beals [1977]. Let

(1

+ 1512)'/2. Suppose that cp and @ are weight functions, that is,

(111)

@(X?

5) 2 c;

5)/cp(x,0

(IV) @(Y, rl)

-

-

=

d cp(x, 5) < c;

(1) c d @(x,5) d C ( 0 , (11) @(x,O c p ( X ,

(5)

@(Y, rl)/cp(Y, 64, if

@(x,0,cp(Y, n)

-

(5)

-

(?>,

cp(x, 51, if lY - X I d ccp(x, 8

and Irl - 51 d c@(x,5); (V) @(x, 5) 2 c < (5y, (VI) @(x,Ocpk

c > 0, 6 > 0;

8 2 &(5)E,

&

> 0.

Theorem 1.10 (Beals [1977]). Let H and H , be Hilbert spaces and A a pseudodifferential operator, A = a(x: D), with a(x, 5): 52 x IR" --+ 2 ( H , Hl), where 2 ( H , H , ) is the space of bounded linear operators acting from H into H I . Assume that

IID,BDta(x, 5)11

< c", P - J a l c p m -[@I,

where @ and cp are weight functions satisfying the conditions (I)-(VI) on each compact subset of Q (here M and m are real numbers). Let p(x, 5) E 2 ( H , Hl), where a and p have left inverses in each compact set KC Q for sufficiently large

Yu.V. Egorov

96

I. Microlocal Analysis

Theorem 1.12 (Grushin [1972]). Zf the conditions 1 and 2 of the preceding theorem hold for the pseudodifferential operator Y ( y ; D,., D,), then the operator p(x, D ) is hypoelliptic in some neighbourhood of the origin.

15I. Assume that IIp(D!D:a)a-'p-' 11 d C,,@-l"'~p-~~', IlPll

+ l l ~ - l P - l l l d C(5>N

Example 1.6. The operator

for some N and sufficiently large 151. Finally, assume that either p = Z or a is invertible for large 5. Then A is a hypoelliptic operator. 1.9. Degenerate Elliptic Operators. Let N = k + n. Let the coordinates of the point x E IR"be (x', y), where x' E IRk and y E IR". Let m E IN and 6m E IN. We set mo

= {(a,

97

P, 7): la1 + ID1 G m,m + I Y I = lal + (1 + 4lBI).

In IRN we examine the operator

D, >

+ iay'(D2 + D;)l12,

Re a # 0,

is hypoelliptic if r is even or if r is odd and Re a > 0. Example 1.7. As we saw in Example 1.4, the operator 0,"+ y2D2 - Dx is non-hypoelliptic. However, the operator

+

I

0," y2D2 - D,

+ ay2Dx

t

P(Y,D) =

is hypoelliptic if a # 0.

1aaayyyD!,D;

i

mo

Example 1.8. The operator

with constant coefficients. Theorem 1.11 (Grushin [1972]). Assume that the following conditions hold. 1. The operator p(y, 0,) is elliptic when y # 0. 2. The equation p ( y , t', D,)v(y) = 0, where 5' E IRk, 15'1 = 1, has no non-zero solution in S(lR!). Then the operator p ( y , D) is hypoelliptic. Conversely, if condition 1 holds and p ( y , D) is hypoelliptic, then condition 2 is satisfied.

i where k > 1, is partially hypoelliptic: if Pu E C"(V), where V is a neighbour! hood of the origin and u E C" in a neighbourhood of the intersection a V n (Xi X k = xk+l = o}, then U E Cm(l/).

5F

/

1.10. Partial Hypoellipticity of Degenerate Elliptic Operators. Let us consider the operator

'

where t E IR,x

6 Example 1.4. The operator

+

0," y2D:

+ ADx,

is hypoelliptic if and only if A # 2k

Re I = 0,

x

E

IR,

y

E

IR,

+ 1, k E Z.

+ 6m E IN. We assume that

= I - " Y ( t , z,

5)

for all A E IR+,t E IR,z E IR and 5 E IR".

Dy + iay'D,,

Re a # 0,

Theorem 1.13 (Grushin [1972]). Zf the operator dp(t, D,, 0,) is elliptic for t # 0, then it is partially hypoelliptic in IR x IR"(in E+x IR") if and only if the equation Y(t,D,, t ) u ( t )= 0 has only the trivial solution in the class S(IR) (in s(E+)) for all 5 E IR"\O.

is hypoelliptic if and only if r = 2k, k E IN. We set

m = ((01, P, Y): la + PI Q m,m6 2 IyI 2 la1 + (1 + S)/lPI

- m}.

f

Let us consider the operator D) =

IR", ajaE C, 6 2 0, CJ E Z,and CJ dp(A-lt, AT, 1'5)

Example 1.5. The operator

P(X,

E

1qaoy(x,D)Y~DXS,D;, m

where qaPy(x, D ) are classical pseudodifferential operators of order zero and qaoo(x,D ) = qaoo(x)if la1 = m. Let q & , ( x ,5, q) be the leading part of the symbol of the operator qaay.We put g ( Y ; 5, 0,) =

c

4:,y(o;

t, 0)Y'rflD;.

Let us recall that in this case partial hypoellipticity means that u E C" a )and u E C" ( I ; g(i2')) for any open sets I' c I ,

(I' x 0') if dpu E C" (I' x

Example 1.9. The operators

+ iDx + A, + 0,' + ItD, + pDx + v tD,

t'D:

are partially hypoelliptic in IR x IR and in TR+ x lR for all A, p, v in (c.

I. Microlocal Analysis

Yu.V. Egorov

98

for any non-negative integers m,, . . . , mk and any eigenualue ~ ( x5), of the matrix iMm-l(x,5), where Al, . . ., 1 , are the eigenualues of the automorphism cp of the normal space to Char P defined by the relation

We now examine the operator

1

2' =

aja(t,x)t'+tqJIDjDu t X)

j+lalCm

where q E IR,q 1, qm E IN,and [qj] denotes the integral part of q j . Let 9obe the principal part of this operator, that is, Yo= aj,(O, x)t"+qJD{Dx.

p*(cpu, u) = O*(u, u),

N

Theorem 1.14 (Grushin [1972]). If the operators 2' and Yoare elliptic for t # 0 (for t > 0), then 9 is partially hypoelliptic in IR x IR" (in K+ x IR"). Example 1.10. The operator t3Df

+ tZDtDx+ iD,' + At2Dt + pD, + v tD:

and

R,

and

x R for all A, p and v in C. The

A typical example of the operator considered here is the one examined by Grushin:

+ to,' + AD, + pD,, tD: + iD," + ADt.

D:

1.1 1. Double Characteristics. Let us consider on an n-dimensional infinitely differentiable manifold r;;Z a certain d x d system of pseudodifferential operators of the form where I is the identity matrix, p(x, 5) is a homogeneous symbol of degree m and the matrix operator M , is of order < m - 1. Although such an operator P is of a very special form, nevertheless it is of sufficient interest and one comes across this operator in the investigation of, for example, the 8-Neumann problem. We assume that the following conditions hold. 1. The set Char P = ( ( x ,5) E T*Q\O, p(x, 5) = 0 } is a smooth manifold of even codimension 2k and is non-involutive, that is, the restriction of the form o = dx A d( to every space tangential to Char P is non-degenerate; 2. The ratio Ip(x, ()l/d2(x,0, where d is the distance from Char P,is locally bounded; 3. If k = 1, that is, if codim Char P = 2, then the number of rotations of p around Char P is identically zero. Theorem 1.15 (Boutet de Monvel and Trkves [1974]). Assume that conditions 1-3 hold for the operator P. For P to be hypoelliptic with a loss of one deriuatiue, that is, in order that E

&(a),

Pu E H;&D)

3u E

H;:"-'(w),

vo c

n,

s E IR,

+ t2D2 + AD, + p.

Operators having double characteristics have also been studied by Boutet de

: Monvel [1974, 19751, Boutet de Monvel, Grigis and Helffer [1976], Boutet de Monvel and Trhes [1974], Hormander [1975], Menikoff [1977a, 19793, and others.

P(x, D ) = I P ( x , D ) + Mo(x, D),

u

5 ) = 1cjl(x, S)hj(x,t)h,(x,5)

P(X,

is partially hypoelliptic in IR x IR and in same is true for the operators

u E IN.

vu,

Here G*(df,d g ) = Hs(g) and p* is a bilinear form associated with the quadratic form fi defined in the following manner. If Char P is defined by the 2k real equations hj(x,5) = 0, j = 1, . . ., 2k, and det((hj, h,}) # 0 (such functions exist in view of condition l), then

j+lal=rn qjE

99

~

1.12. Hypoelliptic Operators on the Real Line. A complete characterization of those hypoelliptic differential operators whose leading coefficient has zeros of finite order has been obtained by Kannai [1971]. Let L ( x , D) = l L oaj(x)Dj, aj E Cm(IR).Clearly, L is hypoelliptic at all those points where a,(x) # 0. Suppose now that a,(x) = 0 for x = 0 and that for x = 0 a, has a zero of finite order. It is well known (see Wasow [1965]) that in this case there exist m linearly independent solutions ul(x),. . .,u,(x) of the equation 9 u = 0 for which ui(x)= eQ*(x)xPwi(x), where Qi are polynomials in x-''~,and mi

udx) =

1 ui,

j(X)

(log xy',

j=O

for 0 < j

< mi, 1 < i < m.

Ui,

j(x)

N

2

n=O

ui,j,"X"'qi

Here mi, q i are integers, mi 2 0, qi > 0, and the

. asymptotic series for ui, do not, in general, converge even when aj are analytic functions. Let r be the number of functions Qi that vanish identically, that is, Q l ( x ) z . . * = Q,.(x) = 0.

it is necessary and sufficient that

Theorem 1.16. The operator 2 is hypoelliptic in a neighbourhood of the origin a,(O) # 0 and [Re Qi(x)l co,for r < i < m, as x -+ 0.

if and only if

-+

100

Yu.V. Egorov

I. Microlocal Analysis

In some cases it is possible to obtain a more explicit description of the hypoellipticity conditions. Let

Definition 2.1. A pseudodifferential operator P of order m is said to be subelliptic in a domain 52 if there are constants C > 0 and 6 E [0, 1) such that

n

101

rn

P(X,

5 ) = am(x) j =1 (5 - Tj(x)).

The operator 9 is said to satisfy condition (S) if for each root i j that increases without bound as x + 0 the following statement is true: ii(x)/ij(x)+ 1 as x + 0 implies i = j . Theorem 1.17. Suppose that Y satisfies condition (S). For Y t o be hypoelliptic in a neighbourhood U of the origin, it is necessary and sufficient that there exists < C or 1x1 IIm C l + 00 as x + O when a constant C > 0 such that either p ( x , i)= 0 for x E U . The following theorem provides another class of hypoelliptic operators. Theorem 1.18 (Kannai [1971]). Let there exist a neighbourhood U of the origin and a constant C such that ClImil>IReil

asx+O

for x E U provided that p(x, i)= 0. Then there exist constants p and 6 such that 0 Q 6 < p Q 1 and an integer m' such that for any non-negative integers a and B and for any compact subset K c U constants C , ( K ) and C(a, b, K ) can be found with the properties

511 2 c,(K)15Irn'; (b) I P $ ] ( X , 5)l Q C(a, B, K)(1 + 151)-~=+a~lP(x, 5)l (a)

IP(X3

for x E K , 5 E IR and hypoelliptic operator.

151 2 C , ( K ) . If conditions (a) and (b) hold, then Y is a

Example 1.11. Let Y u = x3u' + ( i + x)u. By Theorem 1.16, Y is hypoelliptic. However, condition (b) is not satisfied for this operator. Significant results of the theory of hypoelliptic operators have been obtained by Bronshtejn [1976], Ganzha [1986], Grushin [1972], Popivanov [1976], Beals and Fefferman [1976], Bolley, Camus and Helffer [1976], Bolley [1977], Boutet de Monvel[1974], Folland [1977], Grigis [1976], Grigis and Rothschild [19843, Helffer [1976, 19771, Helffer and Nourrigat [19791, Hormander [1975], Menikoff [1977a, 19793, Metivier [1976, 19801, Parenti and Rodino [1980], Rothschild and Stein [1976, 19791, Trhes [1961], and by others.

Q C(lIP~ll0+

Subelliptic Operators

C(IlP~lls+

IIUlls+rn-a

(7)

IIUlls+rn-l),

E

G(K)*

It is easy to show, with the aid of the averaging operator, that the theorem on smoothness holds for subelliptic operators; namely, if u E &'(a) and Pu E H,(K),

then u E Hs+m-a(K).

This theorem has its generalization in the microlocal terms: If u E 6'(52) and Pu E H,(o), where o is a subdomain in T*Q\O, then u E Hs+m-a(o). In particular, P is microlocally hypoelliptic. If P is subelliptic,then the operator P* is solvable in each compact set K . Namely, there exist functions $, ,. .., $N E Cg(K),satisfying the equation Plc, = 0 in K , such that iff E H,(K), with J f ( x ) m d x = 0, then a function u can be found in the class b'(52)n H,+rn-a(K)that satisfies the equation P*u = f in K . Moreover, IIUII:+rn-a Q cIIfII5. If P is a subelliptic operator on a closed, smooth and compact manifold 52, then the set { u E 9'(52 Pu ),= O}

consists of infinitely smooth functions and is a finite-dimensional linear space. The range of this operator, with the domain of definition gP= H,(Q), is a closed subspace of H,-,(SZ). Example2.1. For even k, the operator D, + ix:D, is subelliptic with the exponent 6 = k/(k + 1). Example 2.2. The oblique derivative problem. We examine in the half-space

+

- {X =(XI,

.**,

xn+1)9 xn+1 > O}

the equation d u = 0,

2.1. Definition and Simplest Properties. The subelliptic pseudodifferential operators constitute an important class of hypoelliptic operators. In its properties, this class is very close to the class of elliptic operators (see Egorov [1975, 19843, Hormander [1979]).

Cg(K),

Such an estimate is valid for 6 = 0 if and only if P is elliptic, that is, if po(x, 5 ) # 0 for 5 # 0. It follows from (7), as is easy to see, that for any real s a constant C = C(s, K ) can be found such that

Rn+1

5 2.

ll~llrn-1)>

for every compact set K c 52.

lil < C or IxllIm il-+ 00

either and

IIUllrn-a

u E HZ(IR?++')

with the boundary condition au ~

ax,

+ a x : - au

axn +I

-f(x) -

for x , + ~=0, a f 0.

Yu.V. Egorov

I. Microlocal Analysis

The Fourier transformation reduces this equation to a pseudodifferential equation

Theorem 2.2. Let P(t, x ) be a polynomial of degree k in two variables (t, x ) E IR2, and let A(t) be a polynomial of degree s < k / 2 whose leading coefficient is 1. The estimate

102

llullo d

where

Av

= (2n)-" I i ( < ) l ( l e i x td t .

The corresponding operator iDl but a is negative. In this case

+ ax:A is subelliptic if k is even or if k is odd

6 = k(k

+ 1)-'

Example 2.3. The 8-Neumann problem. Let 52 be a domain in CC". In 52, let us consider a system of equations au -=

azj

fj(z,Z),

j = 1, . . . , n,

z, = x,

+ iy,.

Take a point A on the boundary 852. Let us introduce a coordinate system in a neighbourhood of this point in which 852 is given by the equation x , = 0 and is situated in x, > 0. When x, = 0, the system takes the form j = 1, . . . , n - 1,

4 . u = g,, where

- - a

c

l au +

ax au

+ AP(t, x ) u

I

A > 0,

,

-

CF(IR2),

holds i f and only i f the following conditions are satisfied: 1. A(t) does not change its sign. 2. B(t, x ) = P(t, x)A(t)-' is a smooth function in lR2 and sgn A ( t ) .arB(t,x ) < 0. 3. There is a constant co > 0 such that

1

i +j
ID:[A(tY'@D,B(t,x)]I 2 co.

If conditions 1-3 are satisfied, then the following estimate holds:

+ IlA(t)au/ax + iAP(t, x)ull0 + AP(t, x)u , u E C2(IRZ).

A~'(~+~+ ) I Ilau/atilo IuII~

d

/Iau

c at + ?A(')-

ax au

lo

2.3. Algebraic Conditions. In order to investigate the inequality (7), we can use microlocalization. Without loss of generality, we may take m = 1.

This system is subelliptic, with 6 = 3,if the matrix C defined by the relation II$(Y)llo d CpdllG(x90(X + YAd-',

is positive definite. The matrix C is referred to as the Levi matrix and 52 is called a strictly pseudoconvex domain if C > 0.

2.2. Estimates for First-Order Differential Operators with Polynomial Coefficients. Such operators serve as models for general subelliptic pseudodifferential operators and are extensively used in investigating the latter (see Egorov [1975, 19841, Hormander [1979]). Theorem 2.1. Let P(t)be a polynomial of degree k in t E IR.The estimate

+ AP(t)uIlo,

uE

0

Theorem 2.3. If a first-order pseudodifferential operator P is subelliptic, with 0 < 6 < 1, in a domain 52, then for every integer k > 0 and every compact subset K c 52 a constant C = C(k, K ) can be found such that the inequality

a

4 = axj + aj(x,y)-a Y n + i

llullo d Cllu'(t)

103

A 2 1,

uE

caw,

holds i f and only i f P(t)does not change the sign from minus to plus as t increases. If P(t)satisfies this condition, then the following estimate holds: Ilu'(t)llo + AIIP(tbJll0+ A1'(k+l)ll~llod c, Ilu'(t) + W t ) u l l o , 1 > 0, u E C$(W).

+ Ad-(k+l)(l-d)

1 la+/Jl
5 + DA-d)$(Y)l10

11 yw$lIoi'""-'~)}

holds for all x E K , 5 E IR",151 = l,A 2 1 and for all $ in C:(IR"). Here we have used the notation

ft&?

5 ) = (iD<)"(iDx)Bf ( x , 5).

Theorem 2.4. Suppose that for every compact subset K << 52 there are constants C = C(K), N = N ( K ) and c0 = cO(K)such that the inequality

I. Microlocal Analysis

Yu.V. Egorov

104

where, with t 2 1,151 2 1, p ,

E

C"O(T*O),

+

PA(y,D)$ = A k T k ( x * T ) p o ( ~ yA-', 5 holds for all x

l 'l

EK,

+ onvk)$

and po(x, t5) = tp,(x,

5),

151 3 i,A 2 1 and for all $ ( y ) E Cg(IR"). Then the estimate

+ IIp*uIlO < c,( llpullo + IIullO),

l/(k+1)

E

Cg(K),

holds for the operator P with principal symbol p,. An analysis of microlocal conditions, similar to the one carried out for conditions of Theorems 2.3 and 2.4, enables us to formulate algebraic conditions that are necessary and sufficient for the operator P to be subelliptic (Egorov [1975]). The first of these conditions is necessary for the inequality (7) to hold for some 6 (not necessarily from [0, 1)). ($) Let a, = Re p , and a, = Im p,. Consider an integral curve of the system

W) = ap,(x(t), t ( t ) ) ,

Theorem 2.7. Let P be a pseudodifferential operator satisfying condition (B).A necessary and sufficient condition for P to be hypoelliptic is that condition ($) holds. For the case of a differential operator, Theorem 2.5 assumes the following form.

Theorem 2.8. For a dgferential operator P to be subelliptic, it is necessary and sufficient that the function k(x, 5 ) assumes at all points of T*O even values not exceeding a certain integer k. In this case, 6 = k(k l)-'. If the function k assumes odd values at a point (x,, lo), the inequality (7)cannot hold in any compact set K containing x , no matter what the value of 6 E IR. In this last case, P is non-hypoelliptic in every neighbourhood of this point.

+

We also remark that Theorem 2.7 implies that among operators satisfying condition (B) we do not have any hypoelliptic operators but not subelliptic operators.

= - a,a,(x(t), t ( t ) ) ,

which is known as a bicharacteristic of the function a2. Put h(t) = al(x(t),<(t)). If a,(x(t), <(t))= 0 and h(t,) < 0, then h(t) < 0 for t p to. Similar condition holds if p , is replaced by ip,. In order to formulate the second condition (B),let us consider the operators

$3. Hypoelliptic Differential Operators of Second Order 3.1. The Sum of the Squares. We consider a second-order operator

s?= For CI

Hy H$ = H"'H!'. . . H";?. We take k(x, 5 ) = 0 if po(x, 5 ) # 0, and if po(x, 5 ) = 0, then k(x, 5) is the smallest integer k for which (B)SUP k(x, 5 ) < k < 00,

5 ) # 0,

la

+ PI = k.

( x , 5 ) E T*O,

x

E

K,

151 = 1. if and only if

Theorem 2.5. The inequality (7) holds with 6 = k/(k + 1) conditions ($) and (B) are satisfied. If po(x0,5,) = 0, (xo, to)E T*O\O and 5,) = 0 for ICI + BI < I, then (7) cannot hold for 6 < l(1 + l)-', H",$a,(x,, xo E K . Thus to every subelliptic operator P there corresponds a natural number k such that 6 = k(k + 1)-l. The subelliptic operators arise when elliptic operators degenerate to order k.

Theorem 2.6. If P is a subelliptic operator, then for every compact subset K c SZ and every real s there exists a constant C = C(s, K ) such that IIP*ulls d C(llPUllS +

IIUlls+m-l),

2x ; + x o + c ,

j=1

(8)

where X,, X , , . . . , X, are first-order differential operators with real coefficients that are smooth in O,c E Cm(0),and where O is a bounded domain in IR". As usual, [ A , B] denotes the commutator AB - BA.

B = (pl, ..., b,), we set

= (cI,, ..., ar),

H",$a,(x,

105

Cm).

Theorem 3.1. Suppose that at every point of O there are n linearly independent operators among X j , , [ X j , ,X,,], [ X j , , [X,,, Xj3]], . . ., where ji = 0, 1, . . ., r. Then the operator 2' is hypoelliptic. If the Lie algebra generated by X,, X , , . .., X , is of constant rank, which is less than n, in a neighbourhood of some point xo E O,then there exists, by Frobenius's theorem, a local transformation of coordinates the application of which results in 2 acting only with respect to the variables x l , .. ., x.-~. If, moreover, the homogeneous equation 2'u = 0 possesses even one non-trivial solution u, then s? cannot be hypoelliptic because the same equation is satisfied by a function u , that equals u for x , < c and equals zero for x , > c, where c is Some constant. The first example of the operator 9satisfying the hypotheses of the theorem was considered in 1935 by Kolmogorov. It was the operator a2

a

a

2' = x + x 7 - T . ax. ay at

Yu.V. Egorov

I. Microlocal Analysis

The proof of Theorem 3.1, given by Hormander [1983, 19853, uses simple geometrical considerations. Let X be a real smooth vector field in T(52).Let us consider a one-parameter group of transformations in 52 that is generated by X. To do this we determine cp as a solution of the equation

Thus, if P(x, D ) is a second-order differential operator with real principal part, then the quadratic form corresponding to this principal part is semi-definite. We observe that (see Beals and Fefferman [1976], Ganzha [1986]) the sign of this quadratic form may change, all the same, from one point x to another point. For instance, the operator

106

a

a2

X : T + -

with the initial condition q(0, x ) = x. Clearly, cp(s, cp(t,x ) ) = cp(t + s, x). We now set

107

ax,

ax,

is hypoelliptic in IR3 (Kannai [1971]).

3

Theorem 3.4. Let

erXu(x)= u(cp(t,x ) )

Y ( xD, ) =

so that d(erxu)/dt= etXXu= Xerxu. Let

where

11

be a differential operator with real C*-smooth coefficients. If x , Y o ( x ,5 ) > 0, where Y o ( x ,<) = 1aij(x)Si5,,x , # 0, 5 # 0 and D,a,,(x) = 0, b,(x) > 0 when x 1 = 0, a l l ( x )= 0,

denotes the norm in Y2.We put I = ( i l , . . ., i k )for 0 < ij < r, and XI

=

then Y is hypoelliptic. Conversely, i f x , Y o ( x ,5 ) > 0 for x , # 0,5 # 0, and Y is hypoelliptic, then a l l ( x ) = 0, D,a,,(x) = 0 for x 1 = 0 and b,(x) 2 0 for x 1 = 0.

[XiI, [Xi2, * * * CXik_l,Xik], ***11*

Let 111 = k.

Theorem 3.2. Let Xj E T(52),0 < sj

3.3. Operators with a Non-negative Quadratic Form. Let Y ( x ,D) be given .by (9). Theorem 3.1 can be extended to such operators that are more general than those of form (8).

< 1,j = 0, 1, . . . , r. Let k

mj = 1 / s j ,

m(l) = I/s(I)

= jC =1

l/sij,

I

= ( i l , ..., ik),

Theorem 3.5 (Olejnik and Radkevich [1969]). Let Y o ( x ,5 ) 2 0 in 52. Let A"(T

and let r~ be a positive number. If t is a sufficiently small positive number, then Ile'm'''x'u - ~ I 6I C,t j=O

r~

but not on

Theorem 3.3. Let P(x, D)be a differential operator, with Cw-coefficients, in a domain 52 c IR". Let its principal symbol p(x, 5 ) be real valued. If (xo,to)E T*52\0 and i f then P cannot be hypoelliptic.

o mz 0,

5 ) = ayo(x, t)/axj,

j=1

3.2. A Necessary Condition for Hypoellipticity. We have shown in 0 1 that a hypoelliptic differential operator with constant coefficients that is not elliptic must have multiple characteristics. This result can be extended to an operator with variable coefficients whose principal symbol is real (see Hormander [19633).

aP(xo,t

A,)(x,

Consider the Lie algebra generated by A,), A,j, and Y,,j = 1, . . ., n, with respect to the operation of computing the Poisson brackets for each pair of functions. Assume that the inequality

In proving this theorem one has to make a significant use of the CampbellHausdorff formula.

to)= 0,

5 ) = aYo(x, t)/a<j,

I ~ I+ ~ ~~t l ,u ~~ , u,~E c;(K),

where C , and C2 are constants in which C , depends only on r and x,,... , x,, so, ...,s,.

P(x,,

(9)

i

1 IAj

lklim

a

l...jk(X,

t)I 2 co > 0,

151= 1,

x

EK,

holds for the brackets Ajl,...,j , = {Xjl,.. ., {Xjk-l,Xi*}, . ..}, where Xi, are Some of the functions AU),A,, or Y,,and K is a compact set in 52. Then 9is hypoelliptic in K and

< C ( K , ~ ) ( l l c p l ~ ~+l l Isl c p l ~ I l J ? E C;(K), cpq, = cp, E ( K )> 0, and y = const. < s + E ( K ) .

Ilcpulls+E(K)

where u E C;(K), cp, cp, A more precise analysis shows that E = m-'.

Example 3.1. The operator ~ ( xD) , = a ( x )+ ~ a/axl, where a E Cm(IRn),a(x) > 0 for x # 0 and a("'(0) = 0 for all a, is hypoelliptic in

Proposition 4.1. For a distribution u in 9'(Q)to be an analytic function in 0,it is necessary and sufficient that for every point x o E Q there exist a

I

SUP j b . Y ! K O ( x , y, z ) ] ~ = f~( -y ~) dy

XE

x

< C1a+81+'a!/?! sup 1 XE

4 lyl
IDyf(x)I (10)

Yu.V. Egorov

I. Microlocal Analysis

Definition 4.4. A continuous linear operator B'(9)+ g(9)is called analytically smoothing if its image is contained in A ( 9 ) .

Definition 4.5. A function u in C"(9) belongs to the class A ( 9 ) if for each subdomain KC 9 there exists a constant 6 ( K )> 0 such that u can be continued analytically to V(u)as a function of x + iy in the domain Qa = { x E K , -6(K) < y j < 6(K),j = 0, 1,. . .,n} and u is bounded in Qd.

110

Proposition 4.2. Let the amplitude a(x, y , 5 ) E C", defined on 9" x 9' x JR", be holomorphic in x and y. Suppose that for each compact set 4"in 9' x 9" there exists a constant C > 0 such that

< Ce-ltl/C,

la(x, y , <)I

5 E R".

V ( X , y ) E A',

111

Definition 4.6. A function u in C"(9) belongs to the class A j ( 9 ) if for each subdomain KC 9 there exists a constant dj(K) such that u admits analytic continuation to y(u) in the variable xi as the function u(xo,. . .,xj-,, x j + iy,, xj+,, . . . , x,) in a domain

Then the operator A, given by Qj.6

dy d t , Au(x) = [ [ a ( x , y, t)ei("-y*r)u(y) is analytically smoothing. Here, and in the sequel, 9"denotes an open neighbourhood of 9 c IR"in C". Theorem 4.4 (Treves [1982]). Let a(x, y , () E C m ( 9 x SZ x IR"). Suppose that this function can be extended to a C"-function of ( x , y , [) that is holomorphic in x , y on the open set

where 6 , > 0. Suppose that the following condition holds: For every compact set A c 9 x 9 and every E > 0 there exist an r > 0, a 6 E (0,6,) and C, C > 0 such that the estimates

w,Y , 511 < C(1 + 151)m, Y , [)I

+,I

la&,

y, [)I

5 JR",

< C eEdlReCI, I

< Ce-IReCl/'

=(X

E

and, moreover, u and au/dx, are bounded in Qj,a for all k # j. If every weak solution u in 9'(9)N of the system of equation 9 u = 0, (12) with analytic coefficients, belongs to A j ( 9 ) ,then for every compact set K c 9 constants d j ( K )and C = C ( K )can be found such that

Iq(u)I< CIIUIIBN.

IIm cl < 6 1Re [I.

Then the kernel r

J

IKo(x,y, z ) = (27~)-" eiZta(x,y , t) d t satisfies (10) and the operator A, defined by the formula y , <)u(y)ei("-y)t dy dr,

Au(x) = [[a(.,

Here BN is a Banach space, containing the solutions of (12), such that the implies the convergence in 9 ( 9 ) N This . is a convergence in the norm 11. consequence of the Baire-Hausdorff theorem. By choosing those solutions of (12) for which (13) is valid, we obtain a class of systems that are not analytically hypoelliptic. Theorem 4.5 (Olejnik and Radkevich [1975]). Let a solution up E 9'(9)of (12) of the form u,(x) = eipxjup(x), p > 1, be such that up E A j ( 9 ) ,where dj(K) is independent of p and the functions I J$(v,)l and laJ$(vp)/dx,l,1 # j , are bounded in Qa, j ( K )by a constant C ( K )that is independent of p. Let B N ( 9 )be a Banach space contained in 9'(9)N and endowed with a stronger topology, and, moreover, IIupIIBN 6

4.4. Necessary Conditions for Analytic Hypoellipticity. Let 52 be a bounded domain in lR"+l,and x = (x,, x i , . . , x,,).

.

eCipr,

where p < 1 and the constants C , and p do not depend on p. Suppose that there is a sequence of points x ( p )E K for which

I J$(Up(Zp,h))l > e-c2p",

+

where Z p , h = (x,(p), ..., xj(p) ih, ..., x,(p)) E Qa,j(K), - 6 , 6 h < 0, and the constants C2 > 0 and 0 < 1 do not depend on p or h. Then the system (12) is not analytically hypoelliptic in the variable xj. Example 4.3. The operator

P

is analytically pseudolocal.

(13)

Qj.8

hold for ( x , y , 5) from the set (11 ) that satisfy the condition dist((x, y), 4 )< r,

K , -6j(K) < Y J < dj(K)},

= 0;

+ t2DyZ + 0,"

is not analytically hypoelliptic in t or y, because the equation Pu = 0 has a solution up(x)= e i w - ~ r 2 / 2 +~ ' Z z - 1 )

Yu.V. Egorov

I. Microlocal Analysis

In this case, llullsl = sup, IuI. However, all the solutions of Pu = 0 are analytic in z (Olejnik and Radkevich [1975]).

Necessary conditions for the analytic hypoellipticity are given in the following theorem.

Example 4.4. The system of differential equations Y u = f,with constant coefficients, has for analytic vector function f solutions that are non-analytic = 0 has a real solution t = (t,, . . ., <,) and in x j if the equation det Yo(t) t j # 0. The leading part of the symbol Yo(<) may be determined in the "Dough-Nirenberg sense": there are integers (s,, . ..,sN) and (t,, ...,t N )such = ll~~(t)11 that the order of the operator Yijdoes not exceed si ti and Yo(<) and the polynomial Yiy is homogeneous of degree si + tj (Olejnik and Radkevich [1972]).

Theorem 4.7 (Popivanov [1976]). Let X ; , k = 1 ak,(x)
112

113

+

at each point x E Q where H ( x ) = 0. Then the operator is analytically hypoelliptic at the origin.

Example 4.5. If the function x o Im aj(x) does not change sign in a neighbourhood of the origin, then the solutions of the equation

The next theorem gives sufficient conditions for the analytic hypoellipticity when n = 2. Theorem 4.8. Let

are non-analytic in x j . This is true, for example, for the Mizohata operators

D,+ itZk+'Dx(Olejnik and Radkevich [1972]).

Example 4.6. Let P1(Dx),P,(D,) and P,(D,) be homogeneous elliptic differential operators, with constant coefficients, of order 2m. The operator pi(Dx)+ IX12pPz(Dy) + IXlZq~p3(D,), where x = ( x , , ..., xk), y = ( y l , ..,, yl), z = (zl, ..., z,) and p and q are nonnegative integers, is analytically hypoelliptic if and only if p = q. It fails tQ be analytically hypoelliptic in y if q < p and in z if q > p (see Olejnik and Radkevich [1972]).

~

1. IIm bI2 < M , IIm al, t Im a(x, y 6

f

j = 1 , . . ., n,

3.

IIm a1

+ it) < 0;

< M41xIZk,

Example 4.7 (Grushin). All the solutions of the equation

+ c(x, y)u = 0 , ,

q . ( x ) = (ajl(x),. . ., ajn(x)), has rank n at every point of Q.

t Im a(x, y

,,

with real analytic coefficients aij, bj and c. In this case, Theorem 4.5 assumes a simpler form.

everywhere in Q. 9 is hypoelliptic in Q if and only if the Lie algebra generated by the system of vector fields (Yo,Y,,, . ., Yn), where

< M , Ixlk, IIm bl < M31xlk,

2. IIm bl

+ it) < 0;

where M . .. , M , are certain constants. If f is an analytic function in 52, then every solution of the equation Y u = f belonging to Q'(52) is analytic in Q (Popivanov [19761).

4.5. Differential Equation of the Second Order. In a domain 51 c IR", we consider an operator 9 'defined by the formula

Theorem 4.6 (Olejnik and Radkevich [19751). Let

Suppose that for some 6 > 0 one of the following conditions holds in Qa(Q) = { ( x ,Y + it), ( x , Y ) E Q, It1 < 61:

are analytic in Q. Here 1 is a non-negative integer, and b,, b,, c are complexvalued analytic functons, a,,, a,, are real analytic functions, and a,, - (a,,)' > OinQ. Example 4.8. The equation

has a solution that is non-analytic in y for some A E IR. This value of I is determined from the condition that the equation g" - x2kg + Ixk-'g = 0 has a non-trivial solution g belonging to the class S(IR).

114

I. Microlocal Analysis

Yu.V. Egorov

115

4.7. Generalized Analytic Hypoellipticity. In some cases it proves convenient to examine generalized analytic functions, that is, functions f ( z ) = u(x, y ) + iu(x, y ) that satisfy, instead of the Cauchy-Rimann equations, the equations

Example 4.9. Let I be a subset of the set ( 1 , ..., n}. We consider the operator

where Q is a second-order elliptic operator and x = ( x l , .. . , x,,,). A is analytically hypoelliptic if and only if I = (1, . . .,n}. 4.6. The Gevrey Classes. The properties of the second-order operators of the form r

9 =1 xjz+x,+c, j=1

where Xj, j = 0, 1, . . ., r, are first-order differential operators with real coefficients, have been investigated in the Gevrey classes. Let G,

and I

i<

= U ; u E Cm(52),SUP

= ( i l , . .. , ik),0

K

,

ij 6 r, ij E Z. We put I

and 1

=

1 -, k

1

where pij =

~ ( 1 ) j = 1 Pij

p K = inf ~ ( 1 ~ ) .

+

au + b(t)U = -, av

1, ifij= 1,2,..., r, if ij = 0.

a(t)at

?,

Assume that the fields (XII,. . .,XI")constitute a basis for vector fields in K . Set

Pu = - a(t)- + b(t)u + Y X U , at a ( au at where a(t) 2 a, > 0, and a, b are sufficiently smooth functions. Let Pu(t, x ) = 0 in the ball {(t,x): t 2 + 1x12 < R}. Then the function u can be extended with respect to t to some domain of the complex plane t is as a generalized analytic function: there exist functions U(t,s, x ) and V(t,s, x ) such that

)

~D"U(X <)M!$+'(E!)~, ~ KC 52

XI = [XiI, [ ~ ~ ~ ~ X i k ~ l ~ X i k l ~ ~ ~ l l

~

where all the coefficients are bounded, and al(x, y ) 2 a,, a,(x, y ) 2 0, a, = const > 0. Such a situation arises, for example, in the study of operators that are obtained from the Laplace operator by means of a smooth but non-analytic transformation of independent variables. Theorem 4.10 (Landis and Olejnik [1974]). Let 9 be a second-order elliptic differential operator in IR".Let

:

u E 9'(52),

Pu E G,(K)

u E G,(K),

au

av

- = --

as at V(t,0, x ) = 0. U(t,0, x ) = u(t, x), A similar theorem holds for the solutions of the parabolic equation

+9XU.

ut = (4xo)%o+ b(xo)u)xo

1
Theorem 4.9 (Derridj and Zuilly [1973]). If the Lie algebra generated by the fields X,, X , , . . ., Xr is of rank n, then

as

In this case, u is extended to the complex plane as a generalized functon of x,. By means of such extension we can obtain uniqueness theorems, theorems

on continuous dependence, Hadamard's theorems, etc. (Landis and Olejnik [1974]).

provided that y > 2/p, and the coefficients of P lie in G,(K). If the Lie algebra generated by X I , . . .,X, is of rank n, then the statement is valid for y > 1/pK. If the vector space generated by the fields Xi, [Xi, X,], where i, j = 1, . . ., r, is of dimension n at each point in 52, then the same statement is true for all y 2 2. Suppose that the operator 9 is of the form

Chapter 7 Transformation of Boundary-Value Problems § 1. The Transmission Property

where P and Q are second-order elliptic operators, and ki and 1 are positive integers. If the coefficients of P and Q belong to G,(52) and if 9 u E Gy(52)and u E 9'(52then ), u E Gy(52)for each y 1 (Derridj and Zuilly [1973]).

a

1.1. Operators in a Half-Space. Let 52 = (x E W",x , > 0}, the closure of 52, and let r be the boundary of 52. We denote by D the line x 1 = x 2 = ... = x.-~ = 0 and by D, the intersection D n O . As usual, Hs(52) denotes the space of

Yu.V. Egorov

116

I. Microlocal Analysis

Definition 1.1. An operator A(D)has the transmission property if the function +0.

restrictions, to 52, of distributions belonging to H,(IR") and fis(52) denotes the subspace of distributions in H,(52) whose supports lie in It can be shown that

a.

a(<', x,) is infinitely differentiable for x, =

Proposition 1.3. The operator A(D):Hs(52) + H,-,(SZ) is continuous for

H,(Q) = Y2(KH,(D+)) + Y 2 ( D + ,HAT))

Let us cite explicit conditions on A ( < )under which A(D)has the transmission property.

< 0. Let <' = (<,, . . .,

Proposition 1.4. Let n = 1. For the operator A ( D ) to have the transmission property it is necessary and sufficient that A( - <) = e-""A(<)for 1<1 > ltol.

be the variable dual to x', and let

s

Example 1.2. Let n = 1. The convolution operators with In x or lxla for non-integral ci or with x - ~ where , k is a positive integer, do not have the transmission property. This property is possessed by differential operators, by operators with kernels lxlk or x k , k is a positive integer, or by operators with the kernel O(-x)lxla, where tl is not an integer, and O(x)= 1 when x > 0 and O(x) = 0 when x c 0. In the general case, the function A ( 5 )can be expressed in the form

Fx,f(<',x,) = f(x)e-ix'" dx'. Proposition 1.1. Let a(<', x,)

E

crn(Q). Let

ID;ia(<', xn)I

for any j

for

E

< Cj,N,K(1+ IxnI)FN

(1)

Z,,N E Z and 5' E K , where K is a compact set in IR"-'.Assume that

A 2 1 and 15'1 2

a(l<',x,)

= Ira(<',Ax,)

X")lIffs(D+)

< C(1 + 15'1)

(2)

1. Then lla(<',

A f ( x ) = (274-" Clearly, Af E C" for x , > 0.

s

+ R:(t', t,)

where R$(<', 5,) is of the order l<,lr-p-l as + f o o and the coefficients a:(c) are homogeneous polynomials of degree k. These coefficients arise in the Taylor expansion of A ( < )in a neighbourhood of the half-lines 5' = 0, 5, > 0 or 5. < 0. 1

Proposition 1.5. The operator A ( D )has the transmission property if and only if

i

<'E IR".

a,(t') = e'(r-k)Xa:(<'),

a(<', x,)f(t')eix'5'd<'.

This condition is always fulfilled if, for example, A ( < )is a rational function

Proposition 1.2. The operator A : H,-(,/,)(T) --t H,-,(52) is continuous.

Acp = cp

for x , = 0,

a

Example 1.3. The inverse of the operator - = - + iaz ax, ax, in the complex plane z = x1 + ix, has the symbol A(<)= - i ( t l + it2)-,. The corresponding F function a(<,, x,) is of the form

a

1

Example 1.1. The Poisson operator, which gives a solution of the Dirichlet problem for x , > 0,

* * *

<,

,

The quantity r is called the degree of a. Consider the operator A : S'(T)+ 9'(52) for which

=0

+ a3c)l<"lr-1 + + a:(<')l
A ( < )= a$I<,l' s+r-1/2

where the constant C depends on a, s and r.

A(&)

-3.

HAG) = Y2(KH,P+))n Y 2 ( D + ,HAT))

s>

if s B 0 and if s

117

' '

a

in a half-space, has the symbol a(t', x,) = e-Xn16'1

!

!

of degree 0.

' F

1.2. The Transmission Property (Boutet de Monvel [1966]). Let A and A@<)= t'A(<)for t 2 1. Let c

a(<', x,) =

s

A(t)e-"nxn d<,.

Then the function a satisfies (1) and is of degree r

+ 1.

E

C"(IR")

! I'

Thus A ( D ) has the transmission property. For operators with variable symbols a(x, <) the following definition proves useful. Definition 1.2. By a Poisson operator on = 8:, we mean a continuous linear operator K : C;(T) + C"(52) for which there exists a 1. continuous extension K : b'(T) + Cm(Q), 2. sequence { s j } E IR,sJ + -m such that ~ ( fg, ,2)

= e-iMx')K (

-

fei& )(x',xn/n) C k j ( L g)nsJ

Yu.V. Egorov

I. Microlocal Analysis

for each f E C g ( f ) , g E C m ( f )which are such that g assumes real values and dg # 0 on supp f. And for each p E Z and each a, the function

1.3. Application to the Study of Lacunae. Let SZ be a domain in IR". A lacuna of a distribution f E 9 ' ( 5 2 ) is an open subset of 52 where f = 0.

118

J+-sN~,PQ

[K(/,

991) -

Definition 1.5. Let U be an open subset of 52 and xo E aU. A distribution f in 9'(52) is said to be sharp at the point xo on the side U if there exists a function g E Cm(52)such that f - g = 0 in U n V for some neighbourhood V of x o .

1kj(f, g)ASJ]

-

is bounded for 1 2 1, and uniformly bounded for bounded lx'l and x,/I when g runs through a compact set in C m ( ).f It can be shown that this definition is invariant under diffeomorphisms of the half-space 52 = IR:. This enables us to introduce analogous notions for an arbitrary smooth manifold with boundary. Let V be a smooth manifold. Let be a smooth manifold, with boundary f , having the same dimension as that of V and immersed regularly in V. Definition 1.3. A pseudodifferential operator P, defined on V , acts on if P(uo)laE Cm@)when u E Cg(a); uo denotes a continuation of u to V\Q by zero. We denote by P" the operator u --* P(uo)lQ.

Proposition 1.6. Let n = 1. Let a ( 5 ) ~ ~ = o a s - k be ( 5a) classical symbol of degree s E C. The distribution a( - x ) is sharp at the point x = 0 from the right (left)

if for 5 > O (for 5 < O), k = 0, 1, . . . a,.+(() = eins(- l)kas-k(- 5 ) Under this condition, the function a,-,(() has a holomorphic extension to the upper half-plane Im 5 > 0 so that the support of G(-x) lies on the negative semi-axis. Proposition 1.6 enables us to investigate the case of an aribtrary n too. Let P H P C (- l)"-kPm-k(x,-D)

a

-

a

Definition 1.4. A Poisson kernel on is a linear operator K : C g ( f )-,Cm(S2) such that 1. there exists a continuous extension K : 8 ' ( f )-, Cm(52), 2. for each open submanifold 52'(n'c that is isomorphic to pm and has an open boundary f ' c f the operator KO,:Cg(T')--f C:(a), defined by the R:. formula KQ.(u)= K(u)ln,is an isomorphic image of the Poisson kernel on I Let us again take = IR;, f = IR"-' and V = IR". Let P be a pseudodifferential operator on V with the symbol p(x, 5 ) c p j ( x , t). We denote by dj(x, t', t ) the Fourier transform of the function pj(x, 5 ) with respect to 5,. Since p j is a homogeneous function in 5 of degree sj, we have

be an involution of the algebra of pseudodifferential operators (for integral m). A conical Lagrange manifold A c T*O\O is symmetrical if A = - A , that is, (x, 5 ) E A o ( x , - 5 ) E A . Let A be symmetrical. A transmission is an involution T of the space I> of Fourier integrals of degree t, associated with A , under which z(Pf)= P(rf) (mod Cm) for each f E I:. Locally, integrals in I> are of the form

a)

a

(

3

d j x, At', -

-

= A S ~ + l p j (t', x, t)

for

5'

# 0,

f ( x ) = s e i s ( x * e ) a (0) x , d6,

where S is the generating function for A and a is a symbol of degree s = t n,/4 - ne/2. Since A consists, locally, of two connected parts A + and A - = - A + , it follows that every distribution f from I> can be written in the form

+

I > 0.

Theorem 1.1. The following statements are equivalent: I. The operator P acts on 11. For each integer p the operator K , : C g ( f )--f 9'(52defined ), by the formula K,(u) = P(u 0 G ( P ) ( ~ n ) ) l nis, a Poisson kernel. 111. For each j the function d j ( x , t', t ) E C" for 5' # 0, t 2 aIx,,I (a = const. > 0). IV. For each j the function pj(x, t', 5,) - e-insJpj(x, t', - t n )is flat along the manifold x , = 0, 5' = 0, t,, > 0 (that is, it vanishes together with all its derivatives).

f ( x )=

a.

When these conditions are fulfilled, we say that P has the transmission property along f . In this case, the operator P": Cg(a) -, Cm(SZ)is extended to 2 ) for . instance, P is an elliptic a continuous operator H",,,@) + ~ ~ , ~ " ( i If, operator having the transmission property, then there exists a parametrix for P and it also possesses the same property. This enables us to develop a theory of boundary-value problems for elliptic pseudodifferential operators having the transmission property (see Boutet de Monvel [1966], Rempel and Schulze [1982]).

119

s

[eis(x.e)a+(x, 6)

+ e- iscx*e)a-(x, 6 ) ] d6.

Proposition 1.7. The transmission T is locally of the form

"

5 where 1 E C\O. *

J L

The number 1 is unique but depends on the choice of S . If we move over to another representation, 1 is replaced by i-rI, where p is the signature of the matrix of second-order derivatives that defines the Maslov cocycle. For the existence of a global transmission, it is necessary that p = 0 (mod 2). Let us consider a differential operator P(t, x , D,, D,.),of degree m, which is strongly hyperbolic in t. Let E denote a fundamental solution of the Cauchy problem: PE = 0,

i f k < m - 1, 6(x - xo) ifk = m - 1.

120

Yu.V. Egorov

I. Microlocal Analysis

Theorem 2.1. k ( M ) = i ( M , N*(dM)).Here i ( M , N*(aM))is the subspace of distributions u on M that are smooth functions in M and are such that u has the representation

The distribution E can be represented in the form

s

E(x) = a(x, O)eis(x,e) do, where S(x, - 6 ) = -S(x, 0), and am-k(x,-0) am-k(x,0). Moreover, m = 1 - deg P,

= (-

121

'S +

4x7 y ) = 5 eiX5a(y,t;) dt;,

l)m-kam-k(x,0 ) if a

( x < 4,

with a E S;l,,(IR",IR), i f dim M = n 1 and ( x ,y ) are local coordinates on M and y E IR", x 3 0. It is a distribution space associated with the Lagrange manifold N*aM (see 9 3, Chap. 3).

ne = n,.

For instance, if P = P(D,, D,),then

As a corollary to Proposition 5.1, Chap. 3, we obtain the following result. If u E k ( M ) , then WF(uo)c N*(aM),where uo is the continuation of u beyond M by zero. Thus

where ll/j are the roots of the characteristic equation Pm($j(t;),t;) = 0 and aj is the symbol with degree 1 - m. A point on the cone A is called regular if at this point the projection ( x , 5) -+ x has the maximum rank n, - 1. In a neighbourhood of such a point A is a hypersurface, and E, as it comes closer to A along the normal, has always a discontinuity of the first kind. In particular, if n, is odd, then the limit of E will be finite or infinite simultaneously from both sides of A. If n, is even, one of the limits will be finite while the other infinite. Thus diffusion takes place (Girding [1971]).

5 2.

u E k(M),

W F ( u )= /a 0Cm(M).

2.2. Contracted Cotangent Bundle (Melrose [19811) Proposition 2.1. There exists a natural vector bundle FM on M and a bundle map 1: F M -+ T M such that the induced map ;*: Cm(M,F M ) -+ Cm(M,T M )is an embedding whose image coincides with K i

Distributions on a Manifold with Boundary

-

This bundle is obtained from V by means of the following equivalence relation. If m E M ,then u w 0uf(rn) = wf(m)for every f in CF(M);if m E aM, an additional condition has to be imposed, namely, d(ug)(m)= d(wg)(m)for every g in C"(M) that vanishes on aM. Thus the image of the map ;coincides with T M u T a M . The dual map ;*: T * M -+ F*M

2.1. The Distribution Spaces (Melrose [1981]). Let M be a closed smooth manifold with boundary aM. Consider the function space

C"(M) = (f

E

c ~ ( M )oaf ; = o on aM, va}.

a

It is a space in which f is a continuous function on for each differential operator P.Let Q be the linear bundle of densities on M . We set b ' ( M ) = (Cm(M,Q))'.

~

is a smooth map between the vector bundles and is, like ,; of rank n on ofn-1onaM. Proposition 2.2. The image of the map T*aM u T * M .

j =1

V ( M )= ( u E Cm(M,T M ) , v's are tangential to aM}.

Definition 2.1. The space k ( M ) c &'(M) consists of all those distributions for which u:l..

. uf;"uE H s ( M )

for all kj and for all finite collection of fields uj E V with a fixed s E IR.

aj(x, Y P Y J+ ao(x, y)xDx,

Every point p in ?*M is of the form U

can be identified canonically with

Assume that (x, y ) are, as before, local coordinates on M so that x > 0 on M and x has a zero of exactly order 1 on a M ; y E IR". Then the fields in V are of the form

t

Consider the space of vector fields

;*

and

aj E C".

I. Microlocal Analysis

Yu.V. Egorov

122

These estimates coincide with the standard ones when x > 0. The kernel of a pseudodifferential operator of this form is given by the function

The map i* induces the embedding C"(F*M) -+ C"(T*M).

And the function cp in C"(T*M) lies in Cm(F*M)if and only if a,kcplX,, is a polynomial in 5 of degree < k. Proposition 2.3. The embedding C"(M) c k ( M ) is continuous and its image dense in k ( M ) . The same is true for the embedding C"(M) c k ( M ) . We denote by & M ) the space of those distributions U E & ( M )for which UIG E k'(M). Since A ( M ) consists of distributions that are smooth in all the directions except along the normal to a M , it is natural to expect that the distributions belonging to k ' ( M ) must be smooth in the direction of the normal. Proposition 2.4. Zf u E & M ) or k ' ( M ) ,then the restriction PuldME 9 ' ( a M ) is well defined for each differential operator P. Furthermore, i ( M ) = k ( M )@ g ( M , aM),

where

9 ' ( M , a M ) = { u E &M); supp u

123

c

dM}.

s

k(z, z ' ) = (271)-"-1

ei(x-x')t+i(y-~').q

4x9 Y , x5, rl) d r dq.

5

Here z = ( x , y ) and z' = (x', y'). Replacing x and respectively, we have

k

(:

-2

Y? x Y ' 9

9

= ;k,(S, x , Y , Y'),

's

k,(s, x, y, y ' ) = 271

by s = x'/x and I = x t ,

ei(l-s)~++'(~-~')q

4x9 Y , A, rl)

dq.

Proposition 3.1. k E 9'(lR: x IR" x W:."). The distribution k , has a singularity only when s = 1 and y = y', and it decays rapidly as s -+ co. 3.2. The Transmission Property. We note that in the definition of k,, x occurs ter. It is clear from the definition of k that this kernel is infinitely

Most important for applications is the following corollary. Proposition 2.5. Let P be a differential operator on M , and let the map P : k ( M )-+ A ( M )

be surjective mod C"(M). Then u E 4'(M),

Pu E i ( M )* u E &M).

(x', y', y ) E KC IR' x IR" x IR",

0 < x < x'/2,

and it vanishes at x = 0 together with all its derivatives. In order to study the behaviour of k for x > 0 as x' -+ 0, we must examine the behaviour of k , as s +O. We assume that the following transmission conditions are fufilled: -+

s

Thus boundary values of any order are well defined for u. For instance, non-characteristic operators satisfy these conditions.

ei'Ika(x, y, A, q) dA = 0,

V k E IN.

(3)

hen we can assert that k E C" for

5 3.

Completely Characteristic Operators

3.1. PseudodifferentialOperators and their Kernels. The space of vector fields V ( M ) can be viewed as the space of first-order differential operators. Let us define Diff,,(M) as the space of differential operators generated by the elements of C"(M) and V ( M ) .Near the boundary, these operators in local coordinates are of the form p = C aa,k(X,Y)D,"(xD,)~, aa,kE c". lal+k
In microlocalization, it is natural to consider pseudodifferential operators with symbols of the form C(x, y, 5, q ) = a(x, y, x r , q), where a(x, y, I , q ) E S;l,,(Z x IR;:) and Z = E ' x IR". These symbols satisfy the estimates lDy"D,8D:D!$i(x,y, 5, ?)I

< Clxl'(1 + 151)k(l + Iql + I X ~ I ) ~ - ~ - * - I " .

( x , y, y ' ) E KC IR'

x

IR"x IR",

0 < x'

< x/2

es at x' = 0 together with all its derivatives. The first impression is that conditions (3) are very stringent. However, we Proposition 3.2. For every a' E Sy,,(Z x IR""), there exists a symbol b E S , $ ( Z x IR"") such that the symbol a = a' - b satisfies (3).

. Zf

the conditions (3) hold, then the operator A: S'(2)-+ &(2)

A * ( > ( Z ) ) c &(Z). A(CZ(2))c Crn(Z), If an operator A possesses these properties, then its symbol is defined uniquely and

I. Microlocal Analysis

Yu.V. Egorov

124

125

Using this formula, we can construct a parametrix at elliptic points. In particular, the formula implies that

3.3. Completely Characteristic Operators

Definition 3.1. An operator A is said to be completely characteristic if its kernel kA E Cm(zx z).

WFb(u)= 00u E k ( M ) ;

u E &'(M), u

E A'(M)

WF(AuIdM)c WF,(u) n T*aM.

We denote by Y;(Z) the set of such operators of order m. These operators are smoothing, A : $ ( Z ) -+ Cm(Z).The singularities of the kernels of such operators may lie beyond the diagonals also; thus, strictly speaking, these operators are not pseudodifferential operators. However, if A E Y?(Z),then

Example 3.1. Let n = 1, z = I R: x IR,,and P = XD, + iD,,. The operator P has an elliptic symbol ol(P) = 2 iq, and therefore

WF(k,) c N*A u N*(dZ x dZ),

The substitution x = e-' transforms P into -0, + iDy. Suppose that the function f(s iy) is holomorphic in the semistrip s 2 1/~,lyl < E, and If(s + iy)l < CeCls.Then the function u(x, y ) = f(s + iy) satisfies the equation Pu = 0 for 0 < x < 6, lyl < E, and has a finite order at x = 0. That is, it can be continued to an element u' that belongs to 9'((0,6) x ( - E , E ) ) . We may assume that Pu' = 0. Hence

where A is the diagonal in Z x Z and dZ x dZ = { z , z', x = 0, x' = 0). In this way we find that the singularities of kA are quite harmless. The above definitions are invariant under a smooth transformation of variables that preserves the boundary dZ. Thus we can define the space LZF(M) for every smooth manifold M with boundary. It can be shown that every operator A E Y T ( M )maps each of the spaces 9 ' ( M ) , C"(M), Cm(M),&(M, dM), &(M) and LZ2(M)into itself. These operators constitute a ring that is closed under the operation of transition to the adjoint operator. Moreover,

-

a,+,.(A.B)

om(A*)= o,(A),

and

o m , ( A d ) = om(A)IT*aM

for all u in k ' ( M ) or in B'(M).The space LZ;"(M) = A E Y T ( M )that map &'(M) into k ( M ) .

nY ; ( M ) consists of those

3.4. The Boundary Wave Front. Let u E $(M). Let sing supp,(u) = M\{m E M ; m E B, where B is an open subset in M and there is a u E A ( M ) such that u = u in B } . We define the characteristic set &(A) = { p E F*M\O; a,(A) is not elliptic in any of the cones containing p } . Definition 3.2. By the boundary waue front for u E &(M) we mean the set WF,(~)=

n {z,(A);

E A*(M)).

A

The composition of symbols in local coordinates obeys the usual rules. If A and B are pseudodifferential operators with symbols a and b and if at least one of them is proper, then the operator C = A o B (mod 9Cm(Z))has the symbol c - ca

c a ! p ! (1k

p
- p)!

dndka(x,y , A, q)APxk-PD;D:-PD$b(x,y , A, q).

'

u

E $(Z),

Pu E A ( Z )* u E k ( Z ) .

+

i"

where a E S;t,,(IR x W)for some m.

0 4. Canonical Boundary Transformation

= o,(A).am,.(B).

For every operator A in L r ( M )an operator A a E Lm(dM)can be defined such that AuldM = Aa(~ldM)

+

t

4.1. The Generating Function. Let us examine a canonical transformation of the contracted cotangent bundle T * M . Clearly, at interior points it is the usual canonical transformation. In local canonical coordinates, the symplectic form has the representation a = 5 dx

n

+ C qj dyj. j=1

Since i*(x, y , t, q) = ( x , y , 1, q), where 1 = x<, it follows that the canonical form on T * M is represented by the formula

Definition 4.1. By a canonical boundary transformation from M to M' we an infinitely smooth map x: r-+ F*M'\O, where r is an open cone in T*M\O and f * Z = d.

%can

Proposition 4.1. If 2 is a canonical boundary transformation of M into M and i f ( x , y , A, q) and (x', y', A', q') are the canonical coordinates in ' f * M an F * M , then 2*(x') = P O ( X , y , 2, q)x,

f*(W = A + P

I k Y , 1 9 rib,

where p0 > 0. In particular, the projection 2 onto T * M defined a smooth canonical transformation T*-I(r) -+ T*M'\o.

126

Yu.V. Egorov

It is clear that the surface x and hence the map

= A = 0 is

a j : T*aM n

I. Microlocal Analysis

invariant under the transformation j ,

127

When q = 0, there exists an uncountable set of non-equivalent symbols even if { P , { P , { P , X I > > f 0.

r -, T * a M

is defined.

5 5. Fourier Integral Operators

4.2. The Operator of Principal Type. To illustrate the application of canonical boundary transformations, let us examine a problem involving an operator of real principal type. Suppose that p E C" (?*M\O) assumes real values and that it is a homogeneous function in A, q of degree 1. Let p ( p ) = 0, where p E a?*M\O. Suppose that on a?*M the differentials dp and d,l are linearly independent at p. If p E T * d M , we impose an additional condition that dp and aaM are linearly independent at p. Proposition 4.2. Under the above hypotheses, there exists a canonical boundary transformation 2: ?*M 3 r -, ?*Z, defined in some canonical neighbourhood of p, such that

r),

= (O,O,X,

ii

= (iil,

0,

* ' *

, 0)

and P = i * ( V l ) .

4.3. The Differential Operator of Second Order. Let P be a second-order linear differential operator of real principal type. Suppose that the boundary aM does not have any characteristic direction for P. This implies that the forms dp and 5 dx are linearly independent at interior points while linearly independent are their restrictions to aT*M\O at those boundary points where p = 0. If aM is given in the local coordinates by the equation x = 0, then P

= g(x,

Y ) C t 2 + 54x9 Y , rl) + b(x, Y , V)l,

where g # 0 for x = 0. Proposition 4.3 (Melrose [1981]). If the boundary aM does not contain characteristic points for a real principal type operator P , then there exist local coordinates ( x , y ) such that p = *(t2 + 4x9 Y , q)). In a neighbourhood of every point (0, y , t, q ) at which r > 0 ( I < 0),p can be reduced to the form t2 + q i (respectively, t2 - q i ) by means of a canonical boundary transformation. If r = 0 at some point (0, y , t, q), we assume that 4 = { P , { P , X I > # 0.

When q > 0, the corresponding point is known as a point of diffraction, and by a homogeneous canonical transformation p can be reduced to the form

t2 - xr,2 + v],?". When q < 0, the point is known as a gliding point, and the canonical form for p is the function + xr,2 + V l ' I n .

r2

5.1. The Generating Function of the Canonical Boundary Transformation. Let M and N be smooth manifolds with boundary. Let j : ?*M 3 r-+ ?*N be a boundary canonical transformation. By introducing local coordinates ( x , y), we may assume that M=N=Z=IR+xIR"

and

r, c E; x

R; x R;.x (R;;;\O)

is an open cone and (p,0 ) E IR x IRN. A function cp E Cm(Zx IR" x IRN+'\O) is known as a phase function in I-, if it is real valued, homogeneous in (p,0 ) of degree 1 and (cpp Pcpp Pel f 0. Because the canonical boundary transformations are of a specical form, we are forced to impose rather stringent non-degeneracy conditions as follows. acp # 0 and If-acp = 0, then ae aP

det

[

qcpyy' ey*

ww

: :]

# 0.

pvpe

(WJ~

If cp is such a phase function, then under these conditions the set

i

1

c, = ( x , Y , Y', P, e) E r,,ae = o is a conical submanifold in r, of codimension N. -

The map p,:

c, 3 ( x , Y , Y', p, 0)-

(

+

-

x , y , - p-, -; x - , y', p, -? C, acp ax acp ap acp ay acp ap aY acp)

is a local diffeomorphism onto a conical submanifold in ?*Z x ?*Z. Proposition 5.1. If cp is a non-degenerate phase function, then, near the boundary x = 0, the manifold C, is the graph of a canonical boundary transformation. Conversely, the graph of a canonical boundary transformation admits locally such a parametrization.

Yu.V. Egorov

I. Microlocal Analysis

5.2. The Fourier Integral Operator. Let cp be a non-degenerate phase function, and let a E Sy,o. Consider the integral

Chapter 8 Hyperfunctions

128

r

k(z, z ' ) =

J

e i v ( z . ~ ' , x 5 . @ ) - i x4 't 2 ,

Y', x 5 , e ) d e d5,

that defines the Schwartz kernel of the Fourier integral operator associated with j . Assume that the support of a lies in the interior of the cone r, where cp is The integral k makes non-degenerate. We denote the set of these a's by Sm(rl). sense when x > 0. In order to define it for x = 0 also, we require that the formal series vanish. We set

,

ak 2k(cp)a = eiv(o,Y,Y',a,e)U ( O , Y , Y', p, e ) X k d p , axk

and formulate Proposition 5.2. The map 9:S-.O(Z x IR" x lRN+') +

nr=o S-" is surjective.

Let Sm(rl, cp) denote the subspace of symbols a from transmission conditions are satisfied. Then

Sm(r) for which the

Syr,,cp)/S-"(rl, cp) Syrl)/S-yrl). Proposition 5.3. If a E Sm(r1, cp), then k(z, z ' ) is a Schwartz kernel for the operator F : 8 ( Z )+ &'(Z). Moreover,

F(C?(Z)) c Cyz),

F(Cg(Z))c C y Z )

and FUI,=~= Fo(ulx=o)

Let A E 9"M) be an operator of real principal type. To this operator there corresponds the Hamiltonian vector field

aa x-ax an

-

aa x-aa, + j=l (:ayj ax

-

aa

).

in ?*M\O. This field is tangential to the manifold zb(A)

=

{ p E T*M\O,

U(p) = 0}

and is tangential even to its invariantly defined components ,&,(A) n a?*M and &(A) n T*aM, because x = 0 on the first of these and on the second x = 2 = 0. Theorem 5.1. If A &'(M) the set

E

6 1. Analytic Functionals 1.1. Definition and the Basic Properties. We can introduce analytic functionals in analogy with the distribution spaces (see Martineau [1961], Sato [197Oa], Schapira [1971], Hormander [1983]).

Definition 1.1. Let KC (c" and A the space of entire analytic functions in (c". By the space of analytic functionals A'(K) we mean the space of those linear forms on A for which the inequality lu(cp)I d

c w SUP 144, w

cp

E

A,

holds no matter what the neighbourhood w of the compact set K . We define A'(IR") = U x e R n A ' ( K ) . Example 1.1. The functional u(cp) = ~a,D"cp(O)/a!belongs to A'(0) if and only if la,/ d Ceclalfor all E > 0. This functional is a distribution only if the sum is finite. Every smooth curve joining the points 0 and 1 in C is a support of the functional u(q) = 1; cp(z)dz, cp E A(C).Therefore, in contrast to distributions, u E A ' ( K , ) n A ' ( K , ) does not necessarily imply that u E A ' ( K , n K 2 ) . This is, however, true when K , and K , c IR". Theorem 1.1. I f u E A'(IR"), then there exists a smallest compact set K c IR" such that u E A'(K). This set is called the support of u.

for u E Cg(z),

where F , is the Fourier integral in IR" associated with 82.

Ha =

129

S?r(M) is a real principal type operator, then for u E

That the space A'(K) is complete follows from the following Theorem 1.2. Let K Oand K be compact sets in IR",K O c K , and let uj E A'((c"). Assume that the following conditions hold: 1. For each compact neighbourhood V of K in C",the elements uJ E A'(V) for large j . 2. For each compact neighbourhood Vo of K O in (c" the difference u, - uk E A'(Vo)for large j and k. Then there exists an element u E A'(K) such that u - uj E A'(Vo)for large j

in every compact neighbourhood Vo of K O . The last condition defines u uniquely with respect to mod A'(Ko).

The next result replaces the existence of the decomposition of unity. WFb(u)\WFb(Au)

Cb(A)

is the union of bicharacteristics of the field Ha in &,(A).

,

Theorem 1.3. If K . . .,K , are compact subsets in IR"and if u E A'(K u '. . u K,), then there are uj E A ' ( K j )such that u = u1 + ... + u,.

Yu.V. Egorov

I. Microlocal Analysis

1.2. Operations on Analytic Functionals. The operations that are admissible in 9'can be extended to the space A'(lR"). 1. If u E A'(IR"), then aju E A'(IR"), and this functional is defined by the formula aju(cp)= - u(ajcp). 2. If u E A'(K), K c IR",and if the function f is analytic in a neighbourhood o of K , then the product f u is defined and given by the formula fu(cp) = u(fcp) provided that the function cp is analytic in o. 3. If u E A'(IR") and u E A'(IR"'), then u 0 v E A'(IR"+"'), and (u 0 v)(cp) = u,(vy((P(x7Y ) ) ) = vy(ux((P(x, Y ) ) ) , where cp(x, Y ) E Awn+"'). 4. Every functional K c A'(," x IR"') defines an operator X on those functions u that are analytic in a neighbourhood of the projection onto IR"' of the support of K . Moreover, X v E A'(IR"), and (Xv)(cp)= K(cp 0 v ) for cp E A(C"). 5. If u E A'(K),where K is a compact subset in IR", and iff is a real analytic bijective map of the open set o c IR" onto a neighbourhood of K , with f = h, then the pullback f *u E A'( f -'K) is defined, and

It is clear that the embedding 9 ' ( X ) -+ B ( X )is injective. The elements of B ( X ) with compact supports can be identified with those elements of A'(IR") that have supports in X . The operations on analytic functionals, described in Q 1.2, can be extended to B ( X ) . What is more, the space B ( X ) can be defined for every real analytic manifold X .

130

-'

(f*u)(cp)= u((cp

O

2.2. The Analytic Wave Front of a Hyperfunction. Let I ( < ) = JlOl=, e-(03'0d t . I ( < ) = 2ch5 if n = 1. If n > 1, this function can be explicitly written in terms of the Bessel function, and

~ ( t=)Io(l
I,(p)

+

= (27c)("-"~2e~p"-")~2[1 0(1/p)].

Let us set

s

K ( z ) = (27c)-" ei('ge)d
= u(e-i(.*c))

The function K ( z ) is defined and is analytic for z

E

C",IIm zI < 1.

Definition 2.2. For u E A'(IR"), WF,(u) is the set of those points (x,5 ) E IR" x (lR"\O) for which the function K*u = u,(K(z - t ) ) is non-analytic at the point x - i
I.

The projection of WF,(u) onto IR" coincides with the set sing supp, u. Since WF,(u) is defined at a point x by the local properties of u, Definition 2.2 can be extended to B ( X ) for any subdomain X c IR". It can be shown (Sjostrand [1982])that WF,(u) = WF,(u) for u E 9 ( X )(see Definition 4.3 of Chap. 6).

is an entire analytic function such that

I ii([)l < CeeHK(Im c)+'Icl,

Theorem 2.2. Let X be a real analytic manifold and Y an open subset of X . Then every hyperjiunction u E B ( Y ) is a restriction to Y of the hyperjiunction v belonging to B ( X ) and having support in

h)ldet h'l),

provided that cp is analytic in o. 6. For u, u E A'(IR"), the convolution u * u is defined as the value of the pullback of u 0 u under the map (x, y ) -+ ( y , x - y ) onto the function cp = 1. 7. If K is a compact set in IR"and u E A'(K),then the Fourier transform

fi([)

131

[ E C",

for each E > 0. Here HK(<) = supxEK ( x , 5 ) . Conversely, every entire function satisfying this inequality determines a unique element in A'(K).

2.3. Boundary Values of a Hyperfunction. Let X be an open set in IR"and Z

0 2. The Space of Hyperfunctions

r

an open connected cone in IR"\{O). Let Z be an open set in C" such that 3

{ z E C",Re z

EX,,

Im z E r,,0 < IIm zI < y}

for some y > 0 and every open subset X,C X and every closed convex cone

r, c r u (01.

2.1. Definition and the Basic Properties. The hyperfunctions in IR" are defined as local equivalent to analytic functionals having compact supports in IR". Definition 2.1. If X is a bounded open set in IR", then the space B ( X ) = A ' ( x ) / A ' ( a X )is called the space of hyperjiunctions in X . Just as in the case of distributions, the support of a hyperfunction u is the smallest closed set outside which u = 0.

u

Theorem 2.1. Let X j be bounded open sets in IR" and X = X j . If uj E B ( X j ) and ui = uj in X i n X j for all i and j , then there exists a unique element u E B ( X ) such that u = uj in X j for each j .

Theorem 2.3. Let f be an analytic function in Z. Then for each open subset

x:

xi4 (1) There exists an element

fxl E A'@,), independent of

r,, such

that the

analytic functional fx,(cp)- Jxl f ( x + iY)cp(X + iY) dx,

cp E A ,

has a support in every small neighbourhood of the set ax provided that y E r, and IYI are sufficiently small. This element is determined to within elements of A'(aX,), and, hence, it defines uniquely a hyperfunction in B ( X , ) .

132

Yu.V. Egorov

I. Microlocal Analysis

(2) Zf X , c X , , X , is an open set in IR",then fx, - f x , E A'(X,\X,); thus there exists a unique element fx E B ( X ) such that the restrictions of fx and fx, to X , coincide for every X , . (3) Zf If ( x iy)l 6 C l ~ l -x~E, X , , y E r,,ly( < y, then the restriction of fx to X , equals the limit in the sense of the distribution theory. (4) Iff admits analytic continuation to a neighbourhood of the set ax,, then the functionals Jx,f ( x + iy)cp(x)dx, cp E A, converge, as y + 0, y E r,,in A'@,) to the element fx, defined in (1). ( 5 ) Zf fx = 0 for some non-empty open set X , c X and i f Z is connected, then f = fx = 0. (6) WF,( fx) c X x (ro\O), where the cone rois dual to r.

Theorem 3.1. Zf P(x, D ) is a differential operator of order m, with real analytic coefficients in X , and i f u E B'(X),then WF,(u) c Char P u WF,(Pu). Moreover, i f a point (x,, 5,) is non-characteristic for P , then for each hyperfunction f E B ( X ) a hyperfunction u E B ( X )can be found such that (x,, 5,) 4 WF,(P(x, D)u - f ( x ) ) .

+

For every solution u E B ( X ) of the equation P(x, D)u = 0, we can determine the Cauchy data on a hypersurface X , provided that it is non-characteristic. To simplify the discussion, assume that u E B(X+),where X + = { x E X , x , 3 0 } and X , = { x E X , x , = O}. Let u, = u in X + and u, = 0 in X - . Then the hyperfunction P(x, D)u, has a support in X,.

Theorem 3.2. Let P ( x , D ) have real analytic coefficients.Let X , be non-characteristic. Then every hyperfunction f E B ( X ) , with support in X , , has a unique representation of the form

Definition 2.3. The hyperfunction fx E B ( X )is called the boundary value off belonging to r,and is denoted by bj-f.

f

Theorem 2.4. Zf X is an open set in IR",u E B ( X ) , and i f WF,(u) c X x

(ro\O),

+ c vj 0 D;l's(x,),

where v E B ( X ) , supp v c X,, vj E B ( X b )for 0 6 j < m, and X , = X b x (0).

As a corollary to this theorem, we have

Theorem 3.3. If P(x, D ) has real analytic coeffients in X and i f X , is noncharacteristic for P, then every hyperfunction u E B ( X + )satisfying the equation P ( x , D)u = 0 has a unique extension u, to X that vanishes on X - and is such that

As a corollary to Theorem 2.4, we can obtain the classical ''edge of the wedge" theorem.

P(x, D)u, =

Theorem 2.5. Let the functions f be analytic in

c

vj 0 D;1'6(x,)

j<m

+ iy; x E X , + y E r,lyl < y } ,

for some v,

where y > 0 and r is an open convex cone. Zf fo = brf = b-rf -, then f is an analytic function that is the analytic continuation o f f and f -.

E B(X,)

for 0 6 j < m.

3.2. The Analytic Wave Front. We now present an equivalent definition for the analytic wave front that is due to Bros and Iagolnitzer. For u E A'(IR"),let us define an entire function T,u, depending on a positive parameter I , by the formula

+

+

The following generalization of this theorem is due to Martineau.

u

Theorem 2.6. Let r,,. . .,r j be closed cones in IRn\O and 4 = IR"\O. Let X be a bounded open set in IR" and u E B(X). Then there exist uj E B ( X ) such that u = uj and WFA(uj)c WF,(u) n ( X x 4).Zf u = X u ; is another decomposition, then ui = uj + c u j , , where ujk E B ( X ) , u k j = -ujk and WFA(ujk)c X x

T,U(Z)

c

(4

= P(x,D)v

j<m

where r is an open convex cone in IR"\O, then there exists a function f, analytic in r,such that u = brf.

Z* = { x

133

= uy(e-i(z-y)2/2),

z

E

c".

T, is closely related to the Fourier transform of u, because ea"x. <)-d
rk).

: is the Fourier transform of ue-a(x-y)2/2 at the point -25. The multiplier e-'(x-y)2/2

9 3.

replaces the cutoff function used in defining the wave front of a distribution. It is natural to expect that (xo, to)E WF,(u) if and only if uy(ei'(x,T)-'(x-y)2/2 ) is O(e-"), as I + co,for some c > 0 in a neighbourhood of the point (x,, -5,). In fact, we have

Solutions of Differential Equations

3.1. The Cauchy Problem. Let P(x, D) be a differential operator, with real analytic coefficients, in an open set X c IR". Clearly, WF,(P(x, DIM)c WF,(u) for u E B ( X ) . The reverse inclusion also holds. Namely, we have

:

Theorem 3.4 (Bony [1976-19771). Let u E A'(IR") and (x,, 5,) E T*(IR")\O. Then ( x o ,5,) $ WFA(u)if and only i f there exists a neighbourhood V of the point x0 - i t o and positive constants C and c such that

I T ~ U ( Z )<~ Cerl(l~012/2-c), z E V,

I > 0.

I. Microlocal Analysis

Yu.V. Egorov

134

Let M be a subset in IR".If xo E M , then Txo(M)is defined as the set of all the x j E M . This set is known as the limits of the sequences tj(xj- x,) as tj + +a, tangential cone to M . It is a closed cone and is independent of the choice of local coordinates. Thus it can be defined for M c X , where X is a C'-smooth manifold. If F is a closed subset of a C2-smooth manifold, then Ne(F) c T*X\O is defined as the set of all the points ( x o ,5,) such that xo E F and such that there is a real-valued function f E C 2 ( X )for which df(xo)= 5, # 0 and f ( x ) < f ( x o ) for x E F. We set N , ( F ) = { ( x , 5); ( x , - 5) E Ne(F)} and N ( F ) = N,(F) u N,(F). Theorem 3.5. If u E B ( X ) and xo E X , then

WK,)c aw, x

Theorem 3.7 (Schapira [1971]). Let P be a first-order differential operator in

IR" with analytic coefficients in a neighbourhood of the origin. Suppose that the principal part P, of P does not vanish at the origin. Suppose further that there exists a function w, analytic in a neighbourhood of 0, such that P, w = 0 and Im w(x) = O o x = 0. Then for any neighbourhood Q, no matter how small, of the origin there exists an > 0 such that PB(Q) 3 He@), where H, is the space of functions that are holomorphic for IIm z1 < E . E

Tx, (SUPP 4,

8 4. Sheaf of Microfunctions

where Wo = {t E Tz(X);(x,,5) E WF,(u)} is regarded as a subset of Tz(X)\O. The proof of this theorem, due to Kashiwara, is based on Holmgren's theorem. Let us cite some of the corollaries of this theorem. Corollary 3.1. Let W, c W, be open convex sets in T'(X)\O. Assume that (1) w,n wF,(u)xo= 121; (2) Every hyperplane, with normal lying in Txo(supp u) n(- Tx,(supp u)), that intersects W, also intersects W,. Then W, n WF,(U),~ = 121. Corollary 3.2. If (x,,, E WF,(u) for all t E

5,)

EN,

(supp u), then (x,,

5 ) E WF,(u)

IR.

-

(x,,

5 + t5,)

Let, once again, X , = { x E X , x , 0 } and X , = { x E X , x, = O}. Let P(x, D) be a differential operator, with analytic coefficients, of order m in X , and X , a non-characteristic plane. Theorem 3.6. If u E B ( X + ) is a solution of the equation P(x, D)u = 0 and if (xb, 5;) $ W F , ( D ~ U ~ , ~= = 0, ~ )1,, ~. . .,m - 1, then there exists an E > 0 such that (x, 5) $ WF,(u) provided that

0 < x , < E,

lxr - xbl

+ 15' -

< E.

Remark 3.1. Analogous statement fails to hold for WF(u) even if P(D) has constant coefficients. For example, if P(D) = D , D , + 0: + D;, we can construct a solution u, E C 2 of the equation P(D)u, = 0 for which sing supp u, = {(t,0, . . . ,0),t 2 O}. Then the function u = ajuo(xl,x,, x 3 , x4 - l/j) E C2(IR4) and D4u E Cm(IR3)for x4 = 0 but sing supp u = { ( x ~0,,0, x4),where x4 = 0 or x4 = l/k, k = 1,2, 3, . . .} if aj + 0 rapidly enough. The theorems on the absence of solutions to equations for which the value of the function c,(x, 5) is negative at a characteristic point (x,, 5,) (see p. 86) can be extended to the space of hyperfunctions. For instance, the equation D , u + ix,D,u = f may not have a solution in the class of hyperfunctions even iff E C". What is more, the following result holds.

c

135

4.1. Traces of Holomorphic Functions. Let us consider in IR" the sheaves of analytic functions, infinitely smooth functions, distributions and hyperfunctions A c C , c 5V c B. The embedding of these sheaves can be described as follows. Let r be an open convex cone in IR" and o an open set in IR". Let the function f be holomorphic on the intersection of the set o x iT with a complex neighbourhood of o in (C". Then we can define the boundary value b(f )E B(o). Iflf(x+iy)l d C~y~-NforsomeintegerN,thenb(f)=lim,,,,,Erf(x+ i y ) E grand the limit exists in the space g r ( w ) . Iff and all the derivatives Oaf are bounded (or if IDaf(z)l d CaI Y I - ~ for all a and a certain N that is independent of a), then the limit exists in C". Conversely, let u E W ( w )and let {ra} be a finite family of open convex cones in IR" such that the interiors of the dual cones T : cover S"-'. Then there , C b ( f a ) .Actually, it is are f a E A ( o iTa) such that If a ( x iy)l d C l ~ l - u~ = enough that o x U (int I")covers WF,(u). int r," covers WF,(u), then there exist fa E Similarly, if u E Cm(o) and A ( o + iTa) such that fa and their derivatives of any order are bounded, and u = b(f,). If u E 9 ' ( w ) and int =I WF(u), then there exist a function cp E Cm(o)and functions f , E A(w iTa),with If a ( x iy)l d C l ~ l - such ~ , that u = CP + C b ( f a ) .

+

+

u u+

+

4.2. The Definition of a Sheaf of Microfunctions Definition 4.1. A sheaf of microfunctions C(W"x S"-') is a sheaf constructed by means of the presheaf U + C ( U )= B(w)/{uE B ( o ) ,WF,(u) n U

=

a}

for U c o x S"-'. To every hyperfunction u E B ( o )there corresponds canonically an element in C(o x S"-'). This element is zero if and only if u E A ( o ) .

Yu.V. Egorov

I. Microlocal Analysis

The corresponding definitions can also be given for sheaves 9'and C" on IR" x S"-'. Namely, to them there correspond presheaves

The values of b(f )are canonically identified on o x S"-' with a microfunction having support in o x GO. The microfunction Pb(f ) is defined in U , has a support in o x GO, and can be extended by zero beyond U to a microfunction defined in o x S"-'. Then there exists a function g E A ( o iC) such that Pb(f ) = b(g). The function g is defined in the following manner. Let C be a hyperplane in C" on which Im z1 = E. We define an operator Pz by the formula

136

U

-+

a}, Cm(IR"),WF,(u) n U = a}.

9 ' ( R " ) / { uE 9'(R"),WF,(u) n U

U -+ C"(IR")/{u E

+

=

A microfunction u E C ( U )lies in 9 ' ( u ) (in Cm(U))if for each compact subset K c U a distribution u (a function in C") can be found such that u = w in a neighbourhood of K , where w denotes the element in Cm(o x Sn-') that corresponds canonically to u. For u E C ( U ) , we denote by WF,(u) the support of u and by W F ( u ) the complement to the largest open set V for which uly E Cm(V).

c

P,f(z) =

K

~ ( zz ,- qf(3dZ.

Psf E A ( o + iG) if E > 0 is sufficiently small. We set g

= P,J the microfunction b(g) does not depend on the choice of E. The pseudodifferential operators of finite order constitute a ring sheaf 9 on IR" x S"-'. It can be shown that such operators map 9 ' ( U ) or Cm(U)into itself. If P(x, D) = pj(x,D ) and Q(x, D) = qj(x,D ) are pseudodifferential operathen their composition R = P - Q is tors in a neighbourhood of a point ( x o ,lo), a pseudodifferential operator and R(x, D) = rj(x,0,). Moreover,

c

1

Definition 4.2. A sequence { pj(x,l ) } j Eof functions, holomorphic in 52, is said to be a sequence of symbols (of a pseudodifferential operator) if it satisfies the following three conditions: 1. pj(x,5) is homogeneous in of degree j. 2. For each E > 0 and for each compact subset K c 52 there exists a constant C = CE,Ksuch that

01 6 CEj/j!,

J

E

4.3. Pseudodifferential Operators. Let 52 be an open conical subset in T*IR".

sup I P j ( X ,

137

1

rj(x,

r]) =

c

k+r=lal+j

1 @.!'

-aapk(X,

?)Dzqi(x,q).

The operator Q(x,D,),formally adjoint to P(x, D,),is a pseudodifferential operator with symbol Q(x, q) = qj(x,q), where

c

j 2 0.

3. For each compact subset K c s2 there exists a constant R = R K such that SUP K

Ipj(x, <)I

The operator P(x, 0,) =

< R-j/(-j)!,

j

< 0.

cjsz p j ( x ,0,) is characterized by the condition

P(x, D.x)a((x,V ] ) - 1)=

1 Pj(x, V ) ~ ( " ( ( X , II)- 1) Z

jc

and is known as a pseudodifferential operator. If P(x, D) = e j g m p j ( xD,), , then P is referred to as a pseudodifferential operator of finite order. P is determined by its kernel K ( x , x - y ) by the formula P(X, D,)u =

s

K ( x , x - Y ) U ( Y ) dY.

The kernel K ( x , y ) = P(x, DX)6(x- y ) can be defined by the formula K ( x , y ) = Bj(x,y ) + h(x, y), where bj(x,y ) is the inverse Fourier transform of pj(x,l )with respect to 4. If P is an operator of finite order, it can be shown that K ( x , x - y ) is a distribution and its analytic wave front is contained in the normal fibration of the diagonal. Hence pseudodifferential operator of finite order acts on distributions without enlarging their wave fronts. The action of a pseudodifferential operator P(x, 0,) can also be defined in another manner. Let o be a domain in IR" and let U contain the set o x {t.lt'l < kc,}. Let f~ A ( w + iG), where G is the circular cone { x ;x 1 > klx'l}.

c

, ,

If P(x, 0,) is a pseudodifferential operator of order m and if pm(x,q) # 0 on a set 52 c T*IR", then there exists a unique psueodifferential operator E(x, D) of finite order such that P E = E P = I.

i

'

1

11 .1

Theorem 4.1. Suppose that P(x, D ) and Q(x, D ) are pseudodifferential operators of order m and that their principal symbols coincide. Suppose further that the forms dp,(x, q) and q dx are linearly independent at those points where p,,,(x, q) = 0. Then we can find, locally, two invertible pseudodifferential operators R(x, 0,) and S(x, 0,) of finite order such that P(x,D)R(x,D)= S(x, D)Q(x,D).

P h

'

This theorem, unlike the next one, is valid in the usual theory of pseudodifferential operators too.

Theorem 4.2. Let P(x, D ) be a pseudodifferential operator of order m in a neighbourhood of the point ( x o ,qo), where xo = 0 and qo = (0, . .., 0, 1). Let the principal symbol p,,,(x, q) of this operator be equal to qy. Then the pseudodifferential equation P(x, D)u = 0 is microlocally equivalent to the equation Dyu = 0. In this way, one equation is transformed into the other by means of pseudodifferential operators of infinite order.

138

Yu.V. Egorov

I. Microlocal Analysis

For example, in the theory of microfunctions the equations

References*

are equivalent. Corollary 4.1. Zf P ( x , D ) is an elliptic operator, then every solution u, a hyperfunction, of the equation Pu = 0 is an analytic function. Theorem 4.3. Let P ( x , D ) be a pseudodifferential operator such that the function pm(x,t;) is real valued and is of principal type. Let u E B(Q) and Pu E A@). A closed set F c w x S"-' can be the support of u if and only if F is a union of bicharacteristics. 4.4. Fourier Integral Operators. Let U and V be two open sets in IR" x S"-'. Let T be an analytic diffeomorphism from U to V that preserves the contact structure (or, in other words, a homogeneous canonical transformation of degree one of one open conical domain in IR" x (IR"\O) onto another). Such a T is known as a contact transformation. Then there exists an isomorphism (which is not unique) 7 from 91"onto 91"(or TI from CI, onto Cl"), compatible with the structure of the ring sheaf on 9 (or with the structure of P, the module on C), that preserves the subsheaves 9'(or Cm) and is also compatible with the action of T on the principal symbol. In terms of formulae, to each contact transformation of a neighbourhood of the point ( x o , to)in IR"x S"-' onto the neighbourhood of ( y o , qo) there corresponds a function F ( x , y ) defined in the neighbourhood of ( x o ,yo) for which dF(xo, y o ) = (-to,qo). Let C be the surface { ( x , y); F(x, y ) = O}. Then T = 4 o p-l, where p : SyR2" + S*IR";

p(x, y , t;, q) = (x, - t);

4: S*,IR2" S*IR";

4(x, Y , 5 , q ) = (Y, q).

+

If u is a microfunction in a neighbourhood of ( x o , to),then we can define the operator P

where K ( x , y ) = h(F(x, y ) ) .The kernel K is actually a distribution, and is known as the Fourier distribution associated with the phase function F(x, y ) . B and 7 as a microlocal Fourier integral operator. It can be shown that WF(u) c WF,(u) = SzIR". The operator 7 can be examined in the space of distributions or microfunctions. f preserves microdistributions and W F ( f u ) = T o WF(u). The operator f generates an isomorphism of the sheaf 9 'under which P H Q, where Pf = FQ.

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Sato, M. [1970al

Hyperfunctions and partial differential equations. Proc. Int. Conf. Funct. Anal. Related Topics, Tokyo 1969,91-94,Zbl.208,358 Regularity of hyperfunction solutions of partial differential equations. Actes Congr. Int. [1970b] Math. 1970,2,785-794,Zb1.232.35004 Sato, M., Kawai, T., Kashiwara, M. Microfunctions and pseudodifferential equations. Lect. Notes Math. 287, 263-529, [1973] Zb1.277.46039 Schapira, P. Hyperfonctions et problemes aux limites elliptiques. Bull. SOC.Math. Fr. 99, 113-141, [1971] Zb1.209,129 Schwartz, L. Thtorie des distributions. Paris: Hermann, Zb1.37,73 [1950] Seeley, R.T. Complex powers of an elliptic operator. Proc. Symp. Pure Math. 10, 288-307, 119673 Zbl. 159,155 The resolvent of an elliptic boundary problem. Am. J. Math. 91,889-920,Zb1.191,118 [1969] Shananin, N.A. On local non-solvability and non-hypoellipticity of (pseudo-)differential equations with [1982] weighted symbols. Mat. Sb., Nov. Ser. 119, No. 4,548-563. English transl.: Math. USSR, Sb. 47,541-556 (1984),Zb1.514.35089 Shubin, M.A. Pseudodifferential Operators and Spectral Theory. Moscow: Nauka. English transl.: [1978] Berlin-Heidelberg-New York: Springer-Verlag (1987), Zbl.451.47064 Sjostrand, J. Operators of principal type with interior boundary conditions. Acta Math. 130, 1-51 [1972] (1973),Zb1.253.35076 Propagation of singularities for operator with multiple involutive characteristics. Ann. [1976] Inst. Fourier 26, No. 1, 141-155, Zbl.313.58021 Fourier integral operators with complex phase functions. Inst. Adv. Stud., Princeton [1979] 1977178, Ann. Math. Stud. 91,51-64,Zb1.446.47046 Singularitts analytiques microlocales. Asttrisque 95, 1-166,Zb1.524.35007 [1982] Sternin, B.Yu. Differential equations of subprincipal type. Mat. Sb., Nov. Ser. 125, No. 1, 38-69. 119841 English transl.: Math. USSR, Sb. 53, 37-67 (1986),Zb1.587.58046 Tartakoff, D.S. Gevrey hypoellipticity for subelliptic boundary-value problems. Commun. Pure Appl. [1973] Math. 26,251-312,Zb1.265.35023 Local analytic hypoellipticity for Obon non-degenerate Cauchy-Riemann manifolds. [19781 Proc. Natl. Acad. Sci USA, 75, 3027-3028,Zbl.384.35020 Taylor, M. Fourier integral operators and harmonic analysis on compact manifolds. Proc. Symp. [19791 Pure Math. 35, 115-136,ZbI.429.35055 Pseudodifferential Operators. Princeton, NJ: Univ. Press, Zb1.453.47026 [1981] Noncommutative microlocal analysis. I. Mem. Am. Math. SOC.313,Zbl.554.35025 [1984] Trives, F. [1961] An invariant criterion of hypoellipticity. Am. J. Math. 83,645-668,Zbl.l14,52 Introduction to Pseudodifferential and Fourier Integral Operators. New York-London: [1982] Plenum Press, Zbl.453.47027 Tulovskij, V.N. The propagation of singularities for operators with characteristics of constant multiplic[1979] ity. Tr. Mosk. Mat. 0.-Va 39, 113-134. English transl.: Trans. Mosc. Math. SOC.1981, No. 1, 121-144 (1981), Zb1.474.47025

'

F i

147

Uhlmann, G. [1977] Pseudodifferential operators with involutive double characteristics. Commun. Partial Differ. Equations 2, 713-779,Zbl.389.35047 Unterberger, A., Bokobza, J. Les operateurs de Calderon-Zygmund prtcises. C. R. Acad. Sci., Paris 259, 1612-1614, [1964] Zb1.143,369 Vasil'ev, D.G. [I9841 Two-term asymptotics of the spectrum of a boundary-value problem under interior reflection of general form. Funkts. Anal. Prilozh. 18, No. 4, 1-13. English transl.: Funct. Anal. Appl. 18,267-277 (1984), Zb1.574.35032 [1986] The asymptotics of the spectrum of a boundary-value problem. Tr. Mosk. Mat. 0.-Va 49, 167-237. English transl.: Trans. Mosc. Math. SOC.1987, 173-245 (1987), Zb1.623.58024 Vishik, M.I., Grushin, V.V. [I9691 On a class of degenerate elliptic equations of higher orders. Mat. Sb., Nov. Ser. 79, No. 1, 3-36. English transl.: Math.USSR, Sb. 8, 1-32 (1969),Zb1.177,143 Wasow, W. [1965] Asymptotic Expansions for Ordinary Differential Equations. Pure Appl. Sc. 14, New York-London-Sydney: Interscience Publishers, Zbl. 133,353 Wenston, P. [1975] On local solvability of linear partial differential equations not of principal type. Bull. Am. Math. SOC.81, No. 1, 215-218,Zb1.315.35028 [1977] A necessary condition for the local solvability of P;(x, D) P2,,-l(x,D). J. Differ. Equations 25, No. 1,90-95,Zb1.356.35011

+

.

I1 Linear Hyperbolic Equations V.Ya. Ivrii Translated from the Russian by P.C. Sinha

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

151

................................ 0 1. C"-well-posedness of the Cauchy Problem ..................... 1.1. Basic Definitions and Notation ......................... 1.2. Necessity of Hyperbolicity. &.we1 1.posedness of the Cauchy Problem ............................................. 1.3. Operators with Constant Coefficients. Algebraic Properties of Hyperbolic Polynomials .............................

155

Chapter 1. The Cauchy Problem

Regularly and Completely Regularly Hyperbolic Operators . Operators with Characteristics of Constant Multiplicity ..... Irregularly Hyperbolic Operators of General Form . . . . . . . . . Necessary Conditions for the Cauchy Problem to be Well Posed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8. Degenerate Hyperbolic Equations ....................... 1.9. Second-Order Equations in Two Variables . . . . . . . . . . . . . . . . 1.10. Systems with Characteristic Roots of Constant Multiplicity . . 1.11. Necessary Conditions for Regular Hyperbolicity of First-Order Systems in Two Variables .................... 52. Well-posedness of the Cauchy Problem in Gevrey Classes ........ 2.1. The Main Definitions ................................... 2.2. Necessity and Sufficiency of Hyperbolicity . . . . . . . . . . . . . . . . . 2.3. Operators with Constant Coefficients ..................... 2.4. Operators with Characteristics of Constant Multiplicity . . . . . . 2.5. Necessary Conditions for Well-posedness in the Gevrey Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. 1.5. 1.6. 1.7.

155 155 156 158 161 165 168 169 169 170 171 173 173 173 174 175 176 177 178

150

6 3.

V.Ya. Ivrii

Propagation of Singularities ................................. 3.1. Propagation of C"-Singularities ......................... 3.2. The Geometry of the Propagation of Singularities . . . . . . . . . . 3.3. The Construction of a Parametrix ........................ 3.4. Propagation of Analytic Singularities and Gevrey Singularities . . . . . . . . . . . . . . . . . . . . . . . . . .................

180 180 184 188

Chapter 2. Mixed Problems for Hyperbolic Operators ..............

190

6 1. Well-posedness of Mixed Problems ..........................

190 190 194

Introduction The present survey is devoted to the linear hyperbolic equations and systems. The concept of a hyperbolic equation first appeared in the case of a second-order equation

189

n

1.1. Preliminary Remarks .................................. 1.2. Operators with Constant Coefficients ..................... 1.3. Strong L,-well-posedness of the Mixed Problem (for Equations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Strong L,-well-posedness of the Mixed Problem for Systems of First Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Necessary Conditions for C"-well-posedness of the Mixed Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. Weak L,-well-posedness of the Mixed Problem . . . . . . . . . . . . 1.7. C"-well Posed Mixed Problem for Strictly Hyperbolic Equations of Second Order ............................. 1.8. Mixed Problem in the Classes of Analytic Functions . . . . . . . . 0 2. Propagation of C"-Singularities ............................. 2.1. The Wave Fronts ...................................... 2.2. Propagation of Singularities of Solutions to Dissipative Boundary-Value Problems for Symmetric Hyperbolic Systems 2.3. The Geometry of the Propagation of Singularities . . . . . . . . . . 2.4. The Propagation of Singularities of Solutions to Strictly Dissipative Boundary-Value Problems for Symmetric Hyperbolic Systems .................................... 2.5. The Geometry of the Propagation of Singularities (the Concluding Part) .................................. 2.6. The Propagation of Singularities of Solutions to Non-classical Problems ................................. 6 3. The Propagation of Analytic Singularities ..................... 3.1. The General Theory ................................... 3.2. The Wave Equation ...................................

pU =

196

199 200 203 206 207 208 208 21 1 '

214 216 217 218 219

Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

223 226

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

227

cy,j=,

with constant coefficients. It implied that the quadratic form a ( ( ) = uijtitj is hyperbolic, that is, its positive subspaces are of dimension 1 and the negative subspaces of dimension n - 1 (or vice-versa). But now such equations are referred to as strictly hyperbolic; a hyperbolic equation may have its quadratic form a ( .) degenerate (parabolically degenerate, in old terminology). In this case, the equation (0.1) has solutions of the type of moving (plane) waves u = u ( ( x , t)), where (x, <)= xot0 x,,<,,, a(<) = 0 and u is an arbitrary function of a single variable. If n = 1, the general solution of (0.1)is a linear combination of two moving waves, and on the basis of this fact we can easily solve the Cauchy problem with data prescribed on any non-characteristic curve (that is, on a curve y such that a ( N ) # 0, where N is the normal to y), and even suitable mixed problems. Moreover, the Cauchy problem is uniquely solvable and there exists a triangle of dependence; the same is true for non-homogeneous equations. When lower terms are present, the Cauchy problem and the mixed problem can be solved, in principle at least, by the method of successive approximation, just as it is done in the case of the Cauchy problem for ordinary differential equations. Similar construction is available for the case of variable coefficients also, and again the non-characteristic Cauchy problem is uniquely solvable and there is a characteristic triangle of dependence (now curvilinear). These statements do not hold for equations of other types. Therefore, apart from the algebraic definition of hyperbolicity, it is possible to give a meaningful and close analytical definition: a given equation (system) is hyperbolic if some non-characteristic Cauchy problem for it is uniquely solvable for any smooth right-hand sides and initial data and if there exists a cone of dependence. This fact was already recognised in the last century, and from that time the algebraic definition of hyperbolicity was extended to systems of first order, to equations and systems of higher orders and with a large number of independent variables in such a way that it remained meaningfully close to the analytical definition. By the above method of characteristics, a comparitively simple analysis can be given for the first-order systems i=O

A Survey of Recent Results .....................................

(0.1)

+ +

198

212

1 aijaiaju = 0, I,J=o

Aiaiu + Bu = f

(0.2)

in two independent variables, where n = 1, A , and B are d x d matrices, and u and f are column vectors of size d . The method of characteristics works (possibly in conjuction with the method of successive approximations) if A , is a non-

V.Ya. Ivrii

11. Linear Hyperbolic Equations

degenerate matrix, all eigenvalues of the matrix C = &'A, are real and if there exists a complete set of eigenvectors; when C is a variable matrix, these eigenvectors are required to depend smoothly on x. All these conditions are fulfilled when (0.2) is strictly hyperbolic, that is, when A , is a non-degenerate matrix and all the eigenvalues of C are real and distinct. We note that if C is a constant matrix and all it eigenvalues are real but there is no complete set of eigenvectors, the modified method of characteristics still works provided some additional conditions are imposed on the lower terms, that is, on B. The next important class is that of equations of second order where the number of variables is greater than 2, and, in particular, the wave equation

Petrovskij was the first to introduce the concept of a hyperbolic system of arbitrary order m. Namely, the system

152

fl

nu c (8,"- A)u =f,

A =

C 8:. i=l

(0.3)

The method of characteristics is no longer applicable to these equations, although under certain assumptions the solution can be found in the form of the integral of moving waves; by the Kirchoff's formula in the particular case of (0.3). But the method of energy integral works: (0.3)implies the energy identity rt r E(u, t ) = E(u, t o ) J,, J 2 Re 8 , ~ - fdx' dxo, (0.4)

+

R"

where E(u, t ) = J x' = (xl, . . .,x"), dx'

= dx,

2 \8,2u(t,x')12dx',

R" j = O

. . . dx,. From (0.4) there follows the energy inequality

E(u, t ) d C[E(u, t o )

+

1

R"

1

lfI2 dx' dxo (t > to).

(0.5)

The unique solvability of the Cauchy problem can be established by means of (0.5) and analytic-functional arguments. For the general equation (0.1) the identity and inequality analogous to (0.4) and ( O S ) , respectively, hold under the assumption that (0.1) is hyperbolic and the hyperplanes S, = {xo = t } are spatially similar, that is, a(x, N ) > 0, where N = (1,0, . ..,0) is the normal to $. We have assumed here that the positive subspaces of the quadratic form a(x, .) are of dimension 1 and that for Ix'I > R the coefficients of the equation are constant or else some other conditions are fulfilled at infinity. Since any surface S can be reduced to the form {xo = 0} by a change of coordinates, the Cauchy problem with data prescribed on a spatially similar surface is uniquely solvable. At the same time, it can be demonstrated that the Cauchy problem with data prescribed on a non-characteristic and non-spatially similar surface is not uniquely solvable. Thus for n 2 2 not every non-characteristic Cauchy problem for a hyperbolic equation is uniquely solvable. Therefore the term "hyperbolicity" must be replaced by a more precise term ~~~~-hyperbolicity" or "hyperbolicity in the direction N".

Pu

=

153

1 U , ( X ) D U U = f, lal<m

where a, are d x d matrices, is called xo-hyperbolic if the matrix is invertible and the characteristic equation d x , 5 ) = det P(X,

5)=0

(

P(X, 5 ) =

U(~,~,...,~)(X)

1 U,(X)P)

1a1=m

has only real roots tofor all x, 5' E IR". If, moreover, these roots are simple when 5' # 0, the system is called strictly xo-hyperbolic or, what has now become customary, Petrovskij hyperbolic. In a similar fashion, the Leray-Volevich hyperbolic and strictly hyperbolic systems can be defined (Leray [19541). Petrovskij established that the Cauchy problem for a strictly hyperbolic system is always well posed, that is, has always a unique solution. On the other hand, Lax and Mizohata proved that a non-characteristic and non-hyperbolic Cauchy problem is always ill posed. Thus the question whether the Cauchy problem for non-strictly hyperbolic equations and systems is well posed remains open. Although there is a vast number of works devoted to this question, it has not been conclusively resolved. In many cases, for the Cauchy problem to be well posed, conditions are required to be imposed on the lower terms. However, there are non-strictly hyperbolic equations and systems for which the Cauchy problem is well posed, no matter what the lower terms are; such equations and systems are referred to as regularly hyperbolic. At present, necessary and sufficient conditions are known for regular hyperbolicity of equations (but not systems) in an open domain. By the well-posedness we meant hitherto the C"-well-posedness, that is, we meant the unique solvability for any right-hand sides and for any initial data belonging to C". If C" is replaced by a narrower space, the Gevrey class G{"), 1 < x < co, it results in a weaker G{")-well-posedness.However, even for the G{")-well-posednessof the non-characteristic Cauchy problem, the system has to be necessarily hyperbolic. We remark that the condition imposed on the lower terms can be relaxed as x becomes smaller; for x < r/(r - l), where r is the maximal multiplicity of the roots of the characteristic equation, the hyperbolic Cauchy problem is always G{")-well-posed. Since hyperbolicity is closely linked with well-posedness of the non-characteristic Cauchy problem, we shall devote a large part of our survey to the well-posedness question of the Cauchy problem. Another topic of our discussion is the well-posedness of the mixed problem. This question is much more complicated because there are various types of lateral boundary (non-characteristic,(non) uniformly characteristic, characteristic only at separate points) and various types of boundary conditions, and that is why this question is not as thoroughly investigated. Thus no conclusive

154

V.Ya. Ivrii

answer is available so far to the question as to under what conditions on boundary operators is the non-characteristic mixed problem for strictly hyperbolic equation of second order C"-well posed. The third question which we deal with is closely connected with the preceding two. This is the question of the propagation of singularities (of wave fronts) of solutions of hyperbolic equations and systems as well as of the Cauchy problem and mixed problems for them. Although this problem is of comparatively recent origin, numerous results have already been achieved. Because of the limitation of space as well as of the resources of the present author, many questions, closely related with the above three, remain untouched. Namely, characteristic Cauchy problem for hyperbolic equations and uniformization, hyperbolic equations where the coefficients of Dt degenerate at certain points (at these points the equation may not have hyperbolicity directions), the construction of oscillating solutions of hyperbolic equations and systems as well as of the Cauchy problem and mixed problems for them, the construction of parametrices of the Cauchy problem and mixed problems for hyperbolic equations and systems, the Maslov canonical operator and Fourier integral operators, the theory of scattering and related questions, the application of hyperbolic problems to obtaining precise spectral asymptotics, equations and systems obtained from hyperbolic equations and systems on replacing Do by a real or a complex parameter, non-linear equations and systems and the propagation of strong and weak breaks of their solutions, etc. Many of these questions deserve a separate survey of the same length as the present one, and the author hopes that they will be dealt with in other volumes published under the present series. To conclude, we remark that the standard definition of hyperbolicity adopted in this survey is not the only possible one. First, non-hyperbolic equations such as (a," + d ) u = 0 may turn out to be abstractly-hyperbolic if they are examined in suitable function spaces of variables x' E o that are non-dense in L,(o). Second, non-standard hyperbolicity and non-standard characteristics arise when h - (pseudo) differential operators are studied. Finally, some equations, such as the Schrodinger equation (ido + x: - a:)u = 0, that are global with respect to the space variables can be taken to be hyperbolic if we examine only functions belonging to Y'(lR") with respect to x' and give equal status to the variables x' and D' in the phase space, and if by the wave front we mean a cone in lR2"\0 that shows in which direction the function g fails to be smooth and in which direction it does not decay fast enough at infinity. The author sincerely thanks M.S. Agranovich for his well-wishing and constructive criticism.

Chapter 1 The Cauchy Problem

9 1. C"-well-posedness of the Cauchy Problem 1.1. Basic Definitions and Notation. We shall use the following notation: x = (xo, x ' ) = (xo, X I ,

5 D

= (50,

5') = ( 5 0 , 51, . . ., t"),

= (Do, D')= (Do,D1,

For M c IR"+l, we denote by

.. .,X"),

. . .,D"),

Dj = - i -.a axj

a its closure and by & its interior, and set

M,' = { X M, = {X

EM, E

+ ( x , - t ) 3 0},

M , x0 = t } .

Let 52 be an open domain in IR"+l, and let X = 52 n {T- < x o < T+},where T- < T+. We put S , = 52,. The necessity of considering such domains arises due to operators that are non-hyperbolic above the final hyperplane and below the initial hyperplane (the Tricomi operator is one such operator). For a d x d matrix operator

P=

c c

aa(x)Da,

a,€ Crn(52),

IalSm

PdX, 5 ) =

lal=l

a,(W.

We call p(x, 5 ) = P,(x, 5 ) the principal symbol of P and g = det p the characterisWe examine the Cauchy problem

Pu

=f

in X,+ (TE[T-, T+)),

(1.1)

( j = 0, ..., m - 1). DiuIxT = gJ (1.2) Definition 1. The Cauchy problem (1.1)-(1.2) is said to be C"-well posed if the following conditions are fulfilled: a) For any f E Cm(XT+) and any gj E Cm(XT)( j = 0, . . .,m - 1) there exists a function u E C" (X;) satisfying (1.1)-(1.2). b) From the fact that u E C" ( X ; ) satisfies (1.1)-(1.2) and that go = = = 0, f = 0 in Xt- for all t > T, it follows that u = 0 in X ; .

Remark. In a number of works, for instance in Ivrii and Petkov [1974], a formally weaker definition of C"-well-posedness is given. This makes sense in establishing necessary conditions for well-posedness. For example, all our results

156

11. Linear Hyperbolic Equations

V.Ya. Ivrii

remain valid if we restrict ourselves in condition a) above to the zero initial data and to right-hand sides f E C z (SZ;) and require that u E Cm(X:). Definition 2. Let r = r ( 2 ) be an open convex cone in R"", with vertex 2 E X,'\X,, such that r n X : C X : . We shall say that the uniqueness condition (U,) holds if the fact that u E Cm(X,')is a solution of (1.1)-(1.2), with gjlX,,, = 0 ( j = 0, .. .,m - 1) and f = 0 in r,implies that u = 0 in r. The condition (V,) expresses the fact that the perturbation is propagated with a finite speed, and this condition enables us not to stipulate beforehand that the Cauchy problem is non-characteristic. Definition 3. The operator P is said to be hyperbolic at a point x in the direction N if a) g(x, N ) z 0, and all the roots z of the characteritic equation b) g(x, ( T N )= 0 are real for all 5 E IR"". Moreover, if for any 5, non-collinear with N , these roots are simple, then P is said to be strictly hyperbolic at x in the direction N . In particular, P is known as x,-hyperbolic (strictly x,-hyperbolic) if it is hyperbolic (strictly hyperbolic) in the direction N = dx, = (1,0, ...,0), that is, if the conditions in Definition 3 are satisfied and all the roots 5, = A(x, 5') of the characteristic equation

Fig. 1

+

g(x, 5 0 9 5 ' )

are real (real and simple) for any 5'

E

=0

I

I

For the wave equation, typical spatial type lens is shown in Fig. 1. Here the x,-axis is directed upwards, r(X)= { x , x , - X, 2 Ix' - X'l} is the light cone, directed upwards, with vertex X, and the boundary of the domain SZ where it does not intersect the boundary of X is depicted by dotted lines. The spatial type lenses for the general hyperbolic operators look similar.

Theorem 3 (Petrovskij [1938], Leray [1954], Girding [1957]). Let P be a strictly x,-hyperbolic operator in X C IR"", where X is a spatial type lens. Then the Cauchy problem (1.1)-(12) is C"-well posed for any T E[T-, T+).Moreover, for all u E C"(x), t > T and k 2-1 2 0, the following Petrovskij energy estimates hold:

(1.3)

lR"\O.

1.2. Necessity of Hyperbolicity. L,-well-posedness of the Cauchy Problem Theorem 1 (Lax [1957], Mizohata [1961]; see also Ivrii and Petkov [1974]). If the hyperplanes { x , = t } are non-characteristic and the Cauchy problem (1.1)(1.2) is C"-well posed, then the operator P is x,-hyperbolic in X ; . Theorem 2 (Mizohata [1961]; see also Ivrii and Petkov [1974]). Let and g(2, 0. If the Cauchy problem (1.1)-(1.2) satisfies the conditions (1.4) and (U,), where r = r ( 2 ) ,then P is hyperbolic at 2 in any direction N that is such that r n { ( x - 2, N ) = 0} = (2). The proofs of Theorems 1 and 2, like those of other necessary conditions for well-posedness, rest on constructing an asymptotic solution of the problem in question. This solution, constructed by assuming the invalidity of the desired condition, violates the a priori estimates that follow from the fact that the problem is well posed. On the other hand, if P is strictly x,-hyperbolic, the Cauchy problem (1.1)(1.2) is C"-well posed in suitable domains.

2 E X,'\X,

a )

+

Definition 4. Let IR"" 3 X . Let the coefficients of P be infinitely smooth in 8 0 E C" and, finally, let P be xo-hyperbolic in If a neighbourhood of g(x, v ( x ) + z N ) # Oforanyx E d2n {T- < xo < T,} andz 2 0, whereN = dxo and v ( x ) is the outward normal to 852 at x, then X is called a lens of spatial type.

x,

x.

157

tllk+m-l,l+m-l

I

6 ckI( Itu;

Tllk+m-l,I+m-l

+

j:

t'llk,l

dt'),

(1.4)

where

1

The estimates (1.4) are established, for example, by applying the method of the alternating operator (Girding [1957], Leray [1954]). After this Theorem 3 is & proved by means of analytic-functional arguments. An operator Q is said to be k a divisor operator if its order is 1 less than that of P and the characteristic roots of Q alternate with the characteristic roots of P'. The proofs, based on the i construction of a symmetrizer, were suggested by Calderon, Mizohata and Yamaguti, although the main idea in an implicit form can be traced back to the ' classical work of Petrovskij.

1 1

i'

Definition 1.5. The Cauchy problem (1.1)-(1.2) is said to be L,-well posed if it is C"-well posed and the estimates (1.4) hold. We mention the necessary conditions for the problem to be &-well posed. 'Here we are discussing scalar operators.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

Theorem 4 (Ivrii and Petkov [1974]). (i) If the estimate (1.4) holds for some k and I, then P is x,-hyperbolic and the inequality

Proposition 1. If P ( D )is Gdrding hyperbolic in the direction N , it is hyperbolic in the same direction.

lIP-l(x, 0 11 < C(-Im 50)-11<11-" v x E x, E ( a l \ l R ) x IR", (1.5) where (c + = { z E (c, fIm z 2 0}, holds. (ii) In particular, if P is a scalar operator, it is strictly x,-hyperbolic. (iii) If m = 1, P = ID, - A ( x , D')and p(x, 5 ) = I t , - a(x, <'), then the matrix a(x, 5') is uniformly diagonalizable. That is, there exists a matrix q(x, g'), uniformly bounded on X x (lR"\O) together with its inverse, such that q-'aq is a diagonal matrix. Thus in the case of scalar operators, strict hyperbolicity is a necessary and sufficient condition for the L,-well-posedness. For matrix operators, the situation is more complicated, and this case has not been thoroughly investigated. If p = p ( < ) (that is, if the principal part of P has constant coefficients),the inequality (1.5) is equivalent to L2-well-posedness. If, however, m = 1 and the matrix a(x, <') is smoothly diagonalizable, that is, if both the diagonalizing matrix q(x, <') and its inverse are infinitely smooth in a neighbourhood of x (R"\O), then the Cauchy problem will be L2-well posed. What is more, when m = 1, for the L,-well posedness it is sufficient that the matrix a is symmetrizable, that is, there exists a Hermitian matrix S(x, 5 ' ) that is infinitely smooth and positive definite in the neighbourhood of x (lR"\O) and is such that Sa is Hermitian (Friedrichs [1958]). However, as an example due to Strang [1967] shows, uniform diagonalization of a does not, in general, imply L,-well-posedness. There arise two problems that have attracted a large number of works. Problem A. Obtain conditions on the principal symbol of the operator under which the Cauchy problem is C"-well posed for all lower terms (such operators are known as regularly or strongly hyperbolic).

The converse of this statement is not true. If the homogeneous operator p(D) is hyperbolic in the direction N and Q(D)is an operator of lower order, then the operator p ( D ) Q(D) may not be Girding hyperbolic in the direction N. For instance, if P = 0: and Q = D,,then the operator P + aQ will be Girding x,-hyperbolic only when a = 0. For admissible operators Q(D) in the case of scalar operators, see Hormander [1963]. Moreover, we have the following

158

'

c

x

x

Problem B. If P is not a regularly hyperbolic operator, find conditions on its lower terms under which the Cauchy problem is C"-well posed.

+

Theorem 6 (Svensson [1970], Hormander [1963]). In order that the operator p(D) Q(D)be Gdrding x,-hyperbolic for any operator Q(D)of a lower order, it is necessary and sufficient that p satisfies (1.5).

+

In this case the Cauchy problem for any operator with principal symbol p will be L2-well posed in any spatial type lens. Until the end of this section, G will denote a scalar polynomial of degree M and g its principal part. We introduce the following sets: h(M) is the set of homogeneous polynomials of degree M ; hyp(N, M) and hypo(N, M) denote the set of homogeneous polynomials of degree M that are, respectively, hyperbolic ' and strictly hyperbolic in the direction N ; Hyp(N, M) is the set of polynomials of degree M that are Girding hyperbolic in the direction N. In view of Proposition i 1 and Theorem 6, we have 1

G E HYP(N, M ) * 9 E hYP(N, M ) ,

i

g E hypo(N, M ) = G E Hyp(N, M).

I Evidently,

5 i

g

E hyp(N,

Theorem 5 (Girding [1951]; see also Atiyah, Bott and Girding [1970,19731, and Hormander [19631). In order that the non-characteristic Cauchy problem (1.1)-(1.2) for the operator P(D), with constant coeffcients, be C"-well posed it is necessary (and sufficient i f X is a spatial type lens) that P ( D ) is Gdrding x,-hyperbolic.

(1.6)

M ) * g-'(N)g( .) is a polynomial with real coefficients,

1.3. Operators with Constant Coeficients. Algebraic Properties of Hyperbolic Polynomials. Thanks to the Fourier transformation, a complete study can be made of the Cauchy problem for operators with constant coefficients.

Definition 6. An operator P ( D ) with constant coefficientsis said to be Gdrding hyperbolic in a direction N if g ( N ) # 0 and there exists a C such that, for any 5 E IR"+l, all the roots t of the polynomial G(<+ tN),where G(5)= det P(C),lie in the strip IIm T I < C. In particular, P(D) is said to be Gdrding x,-hyperbolic if it is hyperbolic in this sense in the direction (1,0, ...,0).

159

E ~YP(N M, ) *

ag

E hyp(N, M -

I),

(1.8)

ag 9 E hyp0(N, M ) * aN E hypo(N, M - l),

ag = . N . - . That is, the differentiation of a (strictly) hyperbolic aN 'ati homogeneous polynomial in the direction of hyperbolicity results in a (strictly) hyperbolic polynomial of order one less. Let T ( g , N) be a connected component of the set ( 5 : g ( 5 ) # 0 } containing N. Then (Atiyah, Bott and Girding [1970,1973], Hormander [1963])

ag . where -0)

1 J

g

E hyp(N,

M) =. T ( g , N ) is an open convex cone,

(1.10)

V.Ya. Ivrii

160

11. Linear Hyperbolic Equations

9 E hYP(N, MI, N' E %, N ) * 9 E hYP(",

9 E hYPO(N,MI, N' E nl,N ) * 9 E hypo(", M ) , G E Hyp(N, M ) , N

Ef

(g,N)

Theorem 10. (i) Let g(x, bourhood of x. Then a) g(2, t>= o for la1

MI, (1.11)

a~"a[

+ 1/11


M ) for any x E U , where U is a neigh-

= ri

*)I,

b) the polynomial

Theorem 7 (Atiyah, Bott and Girding [1970, 19731, Hormander [1963]). (i) If G = det P and G E Hyp(N, M ) , then a fundamental solution E E Q'(IR"+') of the operator P ( D ) exists, is unique, and is such that supp E c {x: (x, N ) 2 O}. (ii) The convex hull of the set supp E coincides with f ' ( g , N). If, however, g E hypo(N, M ) , then sing supp E 3 a f ' ( g , N). For a more precise structure of supp E and sing supp E , and, in particular, of lacunae, we refer the reader to Atiyah, Bott and Girding [1970,1973]. We now discuss the structures of hyp(N, M ) and hypo(N, M). Let h,(M) denote the set of homogeneous polynomials of degree M with real coefficients.

Theorem 8 (Nuij [1968]). (i) The sets hyp,(N, M ) = hyp(N, M ) n h,(M) and hypo(N, M ) = hypo(N, M ) n h,(M) consist of two connected components on which g(N) 2 0. (ii) The set hyp,(N, M ) is relatively closed in h,(M)\{g: g ( N ) = O}; the set hypi(N, M ) is open in h,(M) and dense in hyp,(N, M). Thus every scalar hyperbolic polynomial can be approximated by strictly hyperbolic polynomials. For matrix operators, the situation is much more complicated (John [1978]). If n 2 n(d), there does not exist any first-order strictly hyperbolic d x d matrix polynomial. We shall now discuss the important concept of localization. To do this, we expand tMG(t-'t 5 ) in ascending powers oft:

+

in the n s + 1 variables ( y , q) is hyperbolic in the direction = (0, N); g(a,g)(*) is said to be the localization of g at the point (2, t). (ii) Let g(x, E hyp(N, M ) for any x E U n {xi 2 O}. Then c) a:atg(2, g) = o for 2a1 la'l 1/11 < r = rg(g(2, .)I, a' = (az,.. ., as), a )

~

c.

Theorem 9 (Atiyah, Bott and Girding [1970, 19733). Let G E Hyp(N, M ) . Then a) r,(G) = r,(g) and g , is the principal part of G,, b) 9, E hYP(N9 r,), G, E HYP(N, rc), c) N ) = &, N ) .

m,,

Suppose that g(x, c) is now a polynomial that depends smoothly on the parameter x E U,where U is a domain in IR'. Then the localization concept must take into account the dependence on x too.

+

Theorem 11 (Bronshtejn [1979]). Let g(x, E hyp(N, M ) for all x E U, N = (1,0, ...,0). If to= Aj (x, 5 ' ) are the characteristic roots of the polynomial g, then the estimates a )

, '

hold on any compact set K C U.(Note that there exists a nowhere dense closed set Z c U x IR" such that outside ,Z the roots Aj have constant multiplicity depending, possibly, on the connected components U x IR"\Z; in particular, Aj are infinitely smooth in the exterior of Z.)

+ c) = trGc(c)+ 0(tr")(t + 0),

G,(*) 0. The number r = r&G) is called the multiplicity of 5 with respect to G and the polynomial G,(c) the localization of G at c. In particular, r&G) = 0 if g ( 5 ) # 0, and G&) = g&). If, however, g ( 5 ) = 0 and d g ( 5 ) # 0, then r,(g) = 1, and g&), G&) are linear functions of

+

The polynomial g = 56 (5; - x1<:)q demonstrates that under conditions (ii) the statement a) does not, in general, hold. Finally, for polynomials depending on a parameter the following important ' results hold.

+

+

E hyp(N,

G E Hyp(N', M).

That is, the property of (strict) hyperbolicity or Girding hyperbolicity remains valid for all the elements of f ( g , N ) simultaneously. Let f ' ( g , N ) = {x: (x, 5 ) >, 0 Vg E r(g,N ) } be the dual cone to f ( g , N).

tMG(t-'<

a )

161

11 scalar In 51.4-51.9, we shall assume, without repeating it each time, that P is a xo-hyperbolic operator with a real principal symbol. We start by disE,

x

cussing Problem A which has been solved till the present time for the case of an open domain X.

t: f

1.4. Regularly and Completely Regularly Hyperbolic Operators. We note first

. that if the Cauchy problem is C"-well posed then a linear map 9: Cm(X;) x (Cm(XT))m 3 (f, go, . . . ,gm-J

-+

u E C"(XT+)

is defined that associates with the right-hand side and the initial data a solution, and for any 1 E Z+and any compact set K c X there is a q E Z+such that 9 can be extended to a continuous map acting from C8(K;) x (C8(KT))"' into C'(Kg).In view of this, we have

V.Ya. Ivrii

11. Linear Hyperbolic Equations

Definition 7. A pair of numbers ( I , q ) E Z+ x Z+ is known as the wellposedness index of the Cauchy problem at a point R E X ; if there exists a K neighbourhood of this point such that 9can be extended to a continuous map acting from C8(Kg) x (C$(KT))"'to C'(Kg).

Example 1. The operator P+ = -Di f xoD: is completely regularly hyperbolic in R* x IR. A point p = (2,5) E X x (W"+'\O) is said to be critical if d,,
162

163

Definition 8. An operator P is called regularly hyperbolic in X if for any point

R E X we can find a neighbourhood K and a number f,with the property that f < Ro if R E 8 u S , and f = Ro if R E $-, such that the Cauchy problem (1.1)-

+

(1.2) for the operator P + Q, where Q is any differential operator of order not exceeding m - 1 with infinitely smooth coefficients, is C"-well posed in K ; for T2f If the well-posedness index at every point R can be chosen independently of Q, then P is said to be completely regularly hyperbolic in X .

m

It is a simple matter to give a description of completely regularly hyperbolic operators. Theorem 12. I n order that P, an xo-hyperbolic operator in X , be completely regularly hyperbolic in X , it is necessary and suficient that d*,
(1.12)

for (x, 5 ) E X x (IR"+'\O). Moreover, the Cauchy problem for P is C"-well posed in any spatial type lens 2 C X , and the following estimates hold. (i) If ?._ = T-, T+ < T+ and x E [l, 2],12 s, then IIU; zrIIl+m-l,s+m-1

< C I I UXTIIl+m,s+m-l ; (1.13)

uniformly in t, TE [T-, T+],T < t. (ii) If ?r_ > T-, ?+ = T+ and x E [2, a],1 2 s, then

!

(a) Q = &o(x:

c

(

IIU; z T l l l + m - l , s + m - l +

1;

IIPU; Xt'Ill,s

+ 5:), o > 0;

Q = C L X , ~ ~CL, > 0; (4 Q = 5:; (4 Q = -5:; (4 Q = 0; (f) Q = -5: + 25152 + d. (b)

dt')

(1.14)

uniformly in t, T E [f-,?+I, T < t ; i f x = a, the left-hand side of (1.14) has to be replaced by IIU; z t I I l + m - l . s + m - 1 ( T +

+

where prr, pxe; = 'psx and p,, are (n 1) x (n 1) matrices of second-order derivatives, and I(.) denotes the transpose matrix. The fundamental matrix has the following properties (Ivrii and Petkov [1974], Hormander [1977]): 4 H i p , 4 ) = -Fp(P)Hq(P)9 where Hq = (aq/ag)(a/ax)- (aq/ax)(a/at)is the Hamiltonian field generated by q and { p, q } = Hpq are the Poisson brackets. b) The fundamental matrix is transformed by a canonical diffeomorphism into a similar matrix; thus its eigenvalues are symplectic invariants. c) The matrices Fp and - Fp are similar. d) If the symbol p is hyperbolic in X and p = (X, 4) is its critical point, then all the eigenvalues of the matrix FJp) are either pure imaginary or there are two non-zero simple real eigenvalues p(p) and - p ( p ) and the remaining eigenvalues are pure imginary. e) Moreover, if f o is a root, with multiplicity r, of the polynomial p(E, to,f'), then r 2 2. If X E 8 and r 2 3 or if E E S+ u S- and r 2 4, then Fp(p)= 0. If E E S+ u S- and r = 3, then either Fp(p)= 0 or rank Fp(j7) = 2 and Fp(p)has two non-zero real eigenvalues. f) Let Q(z) = $o(Fp(p)z,z), the localization of the symbol p at the point p, be a quadratic form in z E V = IR"" x IR"", where o is a bilinear symplectic form, o((x, <), ( y , q)) = (x, q) - ( y , 5). If r = 2, X E X , p is xo-hyperbolic in X and ptoc0(p)< 0, then V can be decomposed into a direct sum of symplectic orthogonal subspaces that are invariant under Fp(j7)and on each of which Q has one of the following canonical forms:

- t)1/2*

The necessity of condition (1.12) has been established by Ivrii and Petkov [1974] and its sufficiency by Ivrii [1976a] by the method of alternating operator. Corollary. If P is completely regularly hyperbolic in X , then it is strictly hyperbolic in X and the multiplicity of its characteristic roots does not exceed 2 if (x, 5 ' ) E (S+ u s-) x (IR"\O).

Moreover, there is exactly one subspace of the type (b) u (d) u (f); the subspaces of the types (a) - (e) are two dimensional, while those of the type (f) four dimensional. To the subspace of the type (a) there correspond eigenvalues k i o of the fundamental matrix and to the type (b) the eigenvalues fp; the eigenvalue zero corresponds to the remaining types of subspaces.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

f') If, however, r = 3, X E S+ u S- and p is x,-hyperbolic in X , then in the decomposition of V there may be subspaces of types (b) and (e) only; moreover, the number of subspace of type (b) cannot exceed 1.

X E X and K be a sufficiently small neighbourhood of X that depends only on p. Zf

164

Definition 9. An x,-hyperbolic operator P with real principal symbol p is called effectively hyperbolic if at any critical point p its fundamental matrix has non-zero real eigenvalues fp, p > 0. At critical points p, we also introduce the subprincipal symbol of the operator by the formula

. . .." 1

P"

= pm-1

+ -2 1 pxjtj; j=o

this is also invariant. Ivrii and Petkov [19741 conjectured that an operator is regularly hyperbolic if and only if it is effectively hyperbolic, and established its validity in one direction.

Theorem 13. Zf the operator P is regularly hyperbolic, it is effectively hyperbolic. Corollary. Let the operator P be regularly hyperbolic in X . The multiplicity of its characteristic roots does not exceed 2 if (x, 5') E k x (lR"\O)and if (x, 5 ' ) E (S+u S-) x (lR"\O), the multiplicity does not exceed 3. Theorem 14. Let P be effectively hyperbolic in X . Suppose that one of the following three additional conditions is satisfied: (i) x = 8,S+ = S- = 0. (ii) p can be factorized in the neighbourhood of each of its critical points p, that is, p = p l p 2 in the neighbourhood of p, where p l , p 2 E C" and p l ( p ) = p 2 ( p ) = 0. (iii) p can be factorized in the neighbourhood of each of its critical points p = (x, 5 ) such that 5, is a characteristic root of multiplicity 3. Then P is regularly hyperbolic in X and the Cauchy problem is C"-well posed in any spatial type lens ZCX . Note that under assumption (ii) all the eigenvalues of FJp) are zero except, possibly, the two f{ pl, p 2 } (p). Under assumption (ii) (as also under certain other conditions when n = l), Theorem 14 has been proved in Ivrii [1976a], while under assumption (i) in Iwasaki [1983a, 19841 and in Melrose [1983]. Under assumption (iii), it can be proved by combining the results of Ivrii [1976a] and Iwasaki [1983a, 19841. The proofs carried out in Ivrii [1976a] and Iwasaki [1983a, 19841 are based on the transformation of the operator P to an operator with "nice" lower terms by means of integro-pseudodifferential operators and on the energy estimates for the resulting operator. The proof of Melrose [1983] is based on the study of operators that are strictly hyperbolic for x, # 0 but degenerate at xo = 0; the Cauchy problem for the general operators is reduced to the one for degenerate operators by means of arguments based on strictly hyperbolic approximation and the uniqueness theorem. As a by product of the proofs, we have

165

Theorem 15. Suppose that the hypotheses of Theorem 14 are fulfilled. Let the inequality Ilm PW< ( L - n - 2)lAp)L where f p w e non-zero real eigenvalues of F,(p), holds at any critical point then any pair (q m,q L ) will be a well-posedness index in K .

+

p E K x (IR"+'\O),

+

The following assertion well supplements Theorems 13 and 14.

Theorem 16 (Ivrii and Petkov [1974]). Let (X, 4) E X x (lR"+'\O) be a critical point of p. Zf FJX, 4) has non-zero real eigenvalues + p , then for the pair (1, q) to be a well-posedness index, it is necessary that IIm P"X,

< Cn(q + m - Olpl,

?)I

where C is an absolute constant.

Example 2. The operator P = -D$+x~D;+

f:oj(D,?+xj'D;)+

j=l

n-1

Dj' ( O < k < n - l , o j > O ) j=k+l

is regularly hyperbolic in IR"+l.

Example 3. The operator P+ = (-0,"f xOD:)D, is regularly hyperbolic in R* x IR.

Example 4. The operators PI

=

Pz

=

-0:

+ c wj(D,? + xj'D,") + 1

-0,"

+ DOD1 + x ~ D ;+ 1wj(Dj' + xj'D;) + 1

k

n-1

j=l

j=k+l

Dj' (0 < k

k

r-1

j=2

j=k+l

(1 < k P3 = -0,"

< n - 1, oj > 0), D,?

< n - 1, oj > 0),

k

n-1

j=2

j=k+l

+ xlDoDn + Df + 1 oj(D,? + xj'D,') + 1

Dj"

(1 < k < n - l , o j > O ) are x,-hyperbolic in IRn+' but are not regularly hyperbolic in X if X n (x, = - . * = xk = 0 } # 0,where s = 1 for Pl, P3 ands = 2for P2. We now turn to Problem B.

V.Ya. Ivrii

166

11. Linear Hyperbolic Equations

1.5. Operators with Characteristicsof Constant Multiplicity Definition 10. An operator P is said to be an operator with characteristics of constant multiplicity (locally) in X if in the neighbourhood of every point p E { p = 0 } c X x (lR"+'\O) its principal symbol can be expressed in the form p = eq'. Here e ( p ) # 0, q is a real symbol, q ( p ) = 0, and Hq(p) # 0 is noncollinear with

a

5-, and r E Z'\O and depends, in general, on the connected a5

component of the characteristic set { p = O}.

Definition 11. An operator P , x,-hyperbolic in X,is said to be an operator with characteristic roots of constant multiplicity if all its characteristic roots 5, = Aj(x, 5 ' ) are of constant multiplicity rj E Z+\O, rl + *.. + r, = m (and then they depend smoothly on (x, 5 ' ) E X x (lR"\O)). Proposition2. (i) An x,-hyperbolic operator is an operator with characteristic roots of constant multiplicity in X i f and only i f its principal symbol can be represented in the form (1.15) p = ao(x)py ... p p , where ao(x)does not vanish, rj E Z+\O, and where pi are strictly hyperbolic polynomials in X no two of which vanish simultaneously. (ii) The operator P , x,-hyperbolic in X , is an operator with characteristics of constant multiplicity in X i f and only i f its principal symbol can be represented in the form (1.15), where p j are completely regularly hyperbolic polynomials in X no two of which vanish simultaneously.

-=

'

B

Remarks. a) If condition (L) is satisfied in 8 x (lR"+'\O), it is also satisfied in

b) If qc0(p)# 0, we may assume, without loss of generality, that q = 5, A(x, 5') and Q = Do - A(x, D'),where A(x, D') is a pseudodifferential operator

q(x, dcp(x))= 0

(1.17)

in a neighbourhood of X and are such that dcp(X) = 4, and for any f E C" P(e'"0f) = O(pm-') as p + +a

(1.18)

in a certain neighbourhood of X. The real functions cp satisfying (1.17) are known as phase functions.

Theorem 17. Let P be an x,-hyperbolic operator in X having characteristics of constant multiplicity. For the Cauchy problem for P to be C"-well posed, it is necessary (and sufficient i f X is a spatial type lens) that condition (L) holds.

(1.16)

x x (lR"+'\O).

with respect to x' with real principal symbol (which is a characteristic root of P). Then we may assume, without loss of generality, that in condition (L) A , are differential operators with respect to x, and pseudodifferential operators with respect to x' of order m - r. c) If P is xo-hyperbolic in X and qt0(p)= 0, then X E S , u S-, qeoto(p)# 0 and Hq(p) # 0 is collinear with Hxo.In view of the Weierstrass basic theroem, we may then assume, without loss of generality, that q is a polynomial in 5, of degree 2, where the coefficient of 5: is 1, q is positive homogeneous in 5 of degree 2, Q and A,'s are, respectively, differential operators of order 2 with respect to x, and pseudodifferential operators of order m - r k with respect to x'. d) In view of Proposition 2, the x,-hyperbolic operators that have characteristics of constant multiplicity and satisfy condition (L) can be characterized in the following inductive manner. Suppose that such operators have already been characterized in (1.15) for maxjrj r, and let p' = n j i Z , = , p j ,p = p'p''. Then P satisfies condition (L) if and only if it can be expressed in the form P = P'P" + A , where P' and P are differential operators with principal symbols p' and p", respectively, P" satisfies condition (L) and A is a differential operator of order m - r. e) Let dtq(p) # 0. Then condition (L) is satisfied in a neighbourhood of the point p if and only if for all (or for some, it is immaterial) real functions cp E C", that satisfy the eikonal equation

+

Thus for hyperbolic operators, Definitions 10 and 11 coincide if S , = S- = 0 but Definition 10 is somewhat more general when S+ u S- # Qr. Let us formulate the Levi-Strang-Flaschka condition that turns out to be a necessary and sufficient condition for well-posedness. Without loss of generality, we may assume that in Definitions 10 and 11 q is a homogeneous positive symbol in 5 of degree 1. Definition 12. An operator P with characteristics of constant multiplicity r in a neighbourhood of the point p = (X, 4) E X x (lR"+'\O) is said to satisfy the Levi-Strung-Flaschka condition (L) in the neighbourhood of this point if

where Q is a pseudodifferential operator of order 1 with principal symbol q and A , are pseudodifferential operators of order m - r, to within a pseudodifferential operator with symbol equal to zero in the neighbourhood of p (Taylor [1981], Tr6ves [1980]).

167

;

The necessity of condition (L) has been established by Flaschka and Strang [1971] and its sufficiency by Chazarain [1974] by constructing a parametrix for the Fourier integral operator. It was done under the assumption that P has characteristic roots of constant multiplicity. However, the necessity of condiion (L) in the general case of the operator having characteristics of constant multiplicity follows from this result in view of Remark a), and the sufficiency can be easily established, in view of Remark d), by means of the estimates (1.4), (1.11) and (1.12). For operators with characteristic roots of constant multiplicity, we need only the estimate (1.4). We note that when r 2 4 the proof of Flaschka and Strang [1971] does not go through unless condition (U,)is assumed to hold (see the note to the Russian translation of this work). But we can do away with this condition if we follow the method of Ivrii [1976b].

V.Ya. Ivrii

11. Linear Hyperbolic Equations

1.6. Irregularly Hyperbolic Operators of General Form. Let us formulate, first of all, the most general necessary conditions on the lower terms. The following assertion well supplements Theorem 13. Theorem 18 (Ivrii and Petkov [1974], Hormander [1977]). Let P be an x,-hyperbolic operator in X with a real principal symbol p. Let p E X x (W""\O) be a critical point of p. Suppose that all the eigenvalues of F,(p) are pure imaginary. In order that the Cauchy problem for P be C"-well posed, it is necessary that the following two conditions are satisfied: a) Im p S ( d = 0, b) IPS(P)lG 4Tr+Fp(p), where Tr+F,(p) = lpj1 and & ipj are all the non-zero eigenvalues of F,(p) (with regard to their multiplicity). The Ivrii-Petkov-Hormander conditions a) and b) turn out to be very precise. Theorem 19 (Ivrii [1977], Hormander [1977]). Let P be an x,-hyperbolic operator in X , with real principal symbol p. Suppose that the following conditions are fulfilled: C) Z = { p = d p = 0} is a C"-manifold; d) p E Z =. TpZ = Ker F,(p), Ker F i ( p )n Ran F,'(p) = 0, Spec F,(p) c ilR (then there exist E = f 1 and a zo(p) E Ker F J p ) such that o(zo, z) = 0, z 4 Ker F,(p) =.~o(F,(p)z, z) > 0). e) It is possible to choose a zo E C"(Z). I f the condition a) and the condition b') Ips(p)l < 4Tr F,(p) (in contrast to b), the inequality here is strict) are fulfilled, then the Cauchy problem for P is C"-well posed in any spatial type lens z CX . Theorem 20 (Hormander [1977]). Let P be xo-hyperbolic in X . Assume that the following conditions, apart from condition c), are also satisfied: f ) p E Z * TpZ = Ker F,(p) c Ran(p) (that is, Z is an involutive manifold). €9 P E L - = > P Y P ) = 0. Then the Cauchy problem for P is C"-well posed in any spatial type lens C X . Remarks. Theorem 19 remains valid under more general conditions (Ivrii [1977], Iwasaki [1984]). Under conditions of Theorems 19 and 20 very strong a priori estimates hold (Ivrii [1977], Hormander [1977]) that are established by the method of the divisor operator; in comparison to strictly hyperbolic operators, the loss in smoothness equals and 1, respectively. For model operators, we can prove that the Cauchy problem is C"-well posed also under the conditions a) and b) (but not b')), but the loss in smoothness as compared to strictly hyperbolic operators will be 1. Under the hypotheses of Theorem 20, the condition g) turns out, in view of Theorem 18, to be necessary and sufficient for the Cauchy problem to be C"-well posed. Example 4'. Let PI be one of the operators of Example 4, and let X ; n { x s = ..* = xk = O} # In order that the Cauchy problem for the Operator P, cD,,(c E (c) be C"-well posed, it is necessary (and sufficient if X is a spatial type lens) that c E (c and IcI < G =

1.7. Necessary Conditions for the Cauchy Problem to be Well Posed. We do not have results, apart from Theorem 18, that are both general and precise enough. All the same, the following result is fairly general and in a number of cases proves to be precise enough.

168

Theorem 21. Let X E X ; , f = (0, ..., 0, 1) and q 8$,8p(X,

+

a.

cjoj.

. . .,q,,)E [O, 1)". Let

4) = 0

for

(1.19)

and

qoP(% B # 0.

(1.20)

If the Cauchy problem for P is C"-well posed in X ; , then

a;ap,(X,

<) = 0

(1.21)

for lal

+ ( B - a, 4 ) < r(1 - 40) - (m - 4.

By results a) and b) of Theorem 10, we have

+

4

= (q,,

169

Corollary. I f 4, is a characteristic root of multiplicity r, then in order that the Cauchy problem be C"-well posed, it is necessary to impose conditions on all the members of P which are of order greater than m - yr, where y = i f E E 8 and y = 5 i f X E s+v s-.

4

: b

In a particular case, Theorem 21 was proved by Ivrii and Petkov [1974]; for r > 4, it was assumed that condition (V,) holds. However, the method of Ivrii [1976c] enables us to prove it in its present form. By the same method, Mandai i recently established a more general theorem. In his formulation, Mandai We . note that both ! used Newton's polyhedra instead of half-spaces in Z+('"+') Theorem 21 and Mandai's theorem are non-invariant under the change of [ i coordinates.

I 1

1 \ \ L

Example 5. In order that the Cauchy problem for the operators

+ a,(x)D, + a,(x)D, + az(x) (0 E X : c a'), Pz = D; T xik+'x;'D: + bo(x)Do+ b,(x)D, + bz(x)

P, = D i

- xtkx:'D:

(OEX; c R * x lR) ( k , l E Z + )

' I

be C"-well posed, it is necessary (and sufficient if X ; is a spatial type lens) that a , = x:-'x;ii, and b, = xtx:6, with a",, 6, E C". 1.8. Degenerate Hyperbolic Equations. Among non-strictly hyperbolic equations we identify degenerate equations that are either strictly hyperbolic or have characteristic roots of constant multiplicity in the exterior of initial or final or

V.Ya. Ivrii

11. Linear Hyperbolic Equations

intermediate hyperplanes. For a number of years only these equations were investigated, and there is a vast literature devoted to them. We mention one of the most specific results.

is the Weierstrass polynomial in xo and E(0) > 0. Let us expand the function t p ( x , )in Puiseux series for + x l > 0. Then

170

Re t,(xl)

=

C:~(+X~~'~(~), j>O

Theorem 22 (Olejnik [1970]). Let P = 0; -

171

5 DiaijDj+

i,j - 1

bjDj j=O

where c:j E IR and p ( p ) E Z+\O. For an arbitrary function f (x),which is analytic in a neighbourhood of 0, we introduce the functions f,(x) = f ( x o Re t,(x,), x l ) and for them define the Newton polyhedra at the points (0, +O). To do this, we expand f, in series

+ c,

+

where aij = aji.Let the quadratic form n

a(x, 4') =

C i,

j=1

aijtitj

be positive definite for xo # To E [T-, T+]. If the lower symbol b(x, 5') = satisfies the inequality Ib(x, <')I2

a

< co(xo - TO)-'-a(x, 8x0

and denote by N + (f,) the convex hull of the set

bitj

5 ' ) + W x , 5'),

Iff

then the Cauchy problem for P is Cw-well posed in any spatial type lens.

+ b(x)D1 + c(x),

P = D; - exp( -2x;')Df

= IR+ x

IR,

+ Bx;'+'+~(- In x,)'

We now examine first-order systems. In 0 1.10 and 9 1.11, P denotes a d x d matrix operator, A ( x , D') =

= ID,

- A(x, D')

2 aj(x)Dj+ a'(x), j=l

'

exp( - x;')D,,

0.

N,(X~E,,)c i~&i,) for all p = I, . . .,2m.

where the function 1 has a zero of infinite order at x , = 0. By Theorem 21, in order that the Cauchy problem for P be C"-well posed, it is necessary that b has a zero of infinite order at xo = 0. But neither this necessary condition nor the sufficient condition following from Theorem 22 are precise.

Example 6 (Yagdzhyan [1980]). Let X

=

Theorem 23. In order that the Cauchy problem for P be C"-well posed in some neighbourhood of 0, it is necessary and sufficient that B(x) = x:E(x), where is analytic and

More recent works on this topic are devoted to equations of order greater than 2 and to equations with infinite order of gluing of characteristic roots. A typical example of the latter is the operator P = 0; - A2(xo)D:

= 0, we set N,(f,)

and p(x, 5 ) = I t o - a(x, 5') and a(x, 5 ' ) are principal symbols of P and A, respectively, and g = det p.

1.10. Systems with Characteristic Roots of Constant Multiplicity. Let 5, = , A(x, 5 ' ) be a characteristic root with the following properties: ' a) In a neighbourhood of the point (X, 4') E X x (lR"\O) the root A(x, 5') is of constant multiplicity r. b) The matrix a"(x,5 ' ) = a(x, 5 ' ) - IA(x, 5 ' ) is of a constant rank d - r + x in the neighbourhood of (X, 4'). Since for x = 0, the corresponding part of the Jordan's normal form of the matrix a is diagonal and it does not spoil the well-posedness, we should consider only the case x 2 1. Let q ( x )be a phase function, that is, a real function satisfying the eikonal equation j

and 0 < s, I, k E IR,B E C. A necessary and sufficient condition for the Cauchy problem to be Cw-wellposed is { I > 0 or 1 = 0, k < 0 } if B 4 IR and { I > -s or 1 = -s,k
1.9. Second-Order Equations in Two Variables. These equations are the simplest. Assuming the coefficients to be analytic and S+ u S- = 0,Nishitani [1984a] carried out a complete analysis of Cw-well-posednessof the Cauchy problem for these equations. Namely, let the operator P = D; - A(x)Df

+ B(x)Dl + c(x)Do+ R ( x )

be xo-hyperbolic in a neighbourhood of 0 (that is, A ( x ) 2 0), and let A(x),B(x) be analytic functions, with A(0) = 0 and A f 0. Then, in view of the Weierstrass ( x the ) neighbourhood of 0, where basic theorem, A ( x ) = x ~ ' ~ ( x ) E in

!

I

cp,,(x) = 4x5 dx,q(x)),

d X 4 3 =4

(1.22)

in a neighbourhood of X. Let us first examine the necessary conditions for C"-well-posedness that were established by Petkov for r = 2, 3. Let R,(x, 5 ' ) and L,(x, 5') (k = 1, . . .,r - x ) be smooth vector functions that constitute a basis in Ker ii and Ker a"*, respec-

V.Ya. Ivrii

11. Linear Hyperbolic Equations

tively; when x = r - 1, we shall omit the index k = 1. Since the identity ( L , R ) = 0 is valid for x = r - 1, r 2 2, we can find a smooth vector function h(x, 5 ' ) such that iih = R. If, moreover,

The case where only r = 2 is constant while x = 0, 1 may be variables has been discussed in Demay [1977]. Let ""p denote the matrix of cofactors of p, 1 = ps. ""p - 1 { p, ""p},1' = { ""p.p s - 1 { ""p,p } , where p s is the subprincipal

172

( L , ( x , d,,cp), P(R,(x, d , d ) = 0,

(1.23)

173

symbol of P, and let

a smooth vector function H ( x ) can be found such that d,,cp)H(x) If r = 3 and x

=

= P(R(x,d

x4).

denote the matrix Poisson brackets.

1, we may assume, without less of generality, that ( L j , R k ) =

8j28k2*

Theorem 24 (Petkov [1975]). Zf conditions a), b) and (1.22) are satiaied, r = 2,3 and x 2 1, then in order that the Cauchy problem for P be C"-well posed, it is necessary that (1.23) is satisfied in a neighbourhood of X. Furthermore, for r = 3, x = 2, it is necessary that (L(x,d , , ~ ) ,P(h(x,dx,cp)) + H ( x ) ) = 0,

(1.24)

( L ( x , d x 4 , P ( H ( x ) ) )= 0

(1.25)

in a neighbourhood of Jz, while for r = 3, x = 1 it is necessary that 91.2.92,l

=0

(1.26)

Theorem 26. Let all the characteristic roots Aj(x, 5') of P be of constant multiplicity rj = 1, 2 in X x (IR"\O). Assume that there are smooth matrices sj(x, 5 ' ) and $(x, 5 ' ) such that 50

= Aj(X,

5'h

rj

=2

* l(x, 5 ) = P ( X ,

S)Sj(x, 5'1,

u x , 5 ) = g x , 5')P(X, 5).

(1.27)

Then the Cauchy problem for P is C"-well posed in any spatial type lens. 1.11. Necessary Conditions for Regular Hyperbolicity of First-Order Systems in Two Variables. We conclude the present section with the only necessary condition, known to us, for the Cauchy problem to be C"-well posed for systems with variable coefficients and with characteristics of variable multiplicity. Theorem 27 (Petkov and Kutev [1976]). Let n = 1. Let the operator

in a neighbourhood of E, where q

, k

=

( L j ( x ,d,cp), P(Rk(x,d,.cp))).

We can put these necessary conditions for well-posedness in an almost equivalent form as we did with the Levi-Strang-Flaschka conditions for scalar operators. Let x = r - 1 or x = 1; in the latter case suppose that R , E Ran ii. We introduce Condition (L). For any real function cp satisfying (1.22) and any f E Cg there are scalar functions & ( X ) and vector functions vk(x)such that gk@) = 0 and

P = ZD, + a(x)D, + b(x) be x,-hyperbolic in a neighbourhood of X E IR'. Let 3 be an eigenvalue of a@) having multiplicity r 2 2 with rank(a(Jz) - %) = d - 1. Set p = (X, -1,1). For regular hyperbolicity of P in the domain X 3 Jz, it is necessary that the matrix Fg(p) has a pair of non-zero real eigenvalues. Otherwise, all its eigenvalues are zero, and for the Cauchy problem for P to be C"-well posed in X ; 3 X, it is necessary that ( L , , a'R,) - ( L o ,a'R,) - 2i(L0, b(Jz)Ro) = 0, where R,, R,, Lo,and L , are vectors such that

= O(p-")

iiR, = 0,

as p + +a

in a neighbourhood of X. It can be demonstrated that condition (L) is equivalent to (1.23) for r = 2, x = 1, to (1.23), (1.22), (1.25) for r = 3, x = 2, and tn (1.23) and (1.26) for r = 3, x = 1; 91,2 = 0 or Y2,,= 0 in the neighbourhood of x. Theorem 25 (Petkov [1978]). Let all the characteristic roots Aj(x, <') of P be of constant multiplicity rj in X x (lR"\O). Let condition b) hold for any j , with xi E (0, 1, rj - 11, and at each point (x, 5') E X x (IR"\O) let condition (L) hold for any j , with xj 2 1. Then the Cauchy problem for P is C"-well posed in any spatial type lens.

0 2.

E*L, = 0,

iiRl

= Ro,

a*L1 = Lo,

Well-posedness of the Cauchy Problem in Gevrey Classes

2.1. The Main Definitions. As we have seen in the preceding section, the Cauchy problem is C"-well posed for non-strictly hyperbolic equations and systems only when, in general, some conditions on the lower terms are fulfilled.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

If we desire the Cauchy problem to be well posed for any hyperbolic operator with analytic coefficients, we must give up C"-well-posedness and replace C" by some narrower class of functions but strive for this new class to be as wide as possible.

are non-characteristic. Zf x > 1 and the Cauchy problem (1.1)-(1.2) is G{")-well posed, then P is xo-hyperbolic in X;.' Z however, the Cauchy problem (1.1)-(1.2) is locally G{"}-wellposed, then P is xo-hyperbolic in X;, where X is some neighbourhood of X,.

Definition 13. Let x E [l, co) and 1 E Z'U,. By an inductive Gevrey class G'(")(X)we mean the set of functions u E C!&(X) such that for any compact set KC X there are quantities R = R,, C = C, for which the following inequalities hold: ID"u(x)l < CRl"l(laI!)", Vx E K , a E Z+("+'): a, < 1. (1.28)

What is more, if { x } = (1) and the Cauchy problem (1.1)-(1.2) is G(')-well posed, then P is x,-hyperbolic in X;. At the same time, by the Cauchy-Kovalevskaya theorem, for { x } = (1) any non-characteristic Cauchy problem is locally G(')-well posed. On the other hand, the hyperbolicity is a sufficient condition for the Cauchy problem to be well posed in a suitable Gevrey class.

174

By a projective Gevrey class G'(")(X)we mean the set of functions u E C;:JX) such that for any compact set KC X and any R > 0 there is a C = C,,R for which the inequalities (1.28) hold. Evidently, GI(") c G'(")c G'("1) for x < x,, and G'i"} c G'l{"}for 1 > I,, and all the embeddings are proper (throughout, {. } will mean either or We denote by Grn(')and respectively, the space of analytic functions and that of entire analytic functions. (a)

Theorem 29 (Bronshtejn [1980]). Let P be x,-hyperbolic in X , S+ = S- = 0, and let X ; be a spatial type lens. Let x* = r/(r - l), where r is the maximum multiplicity of characteristic roots of P. If the coefficients of P are in G"(")(X)(G"(")(X))and x E [l, x*)(x E (1, x*]), then the Cauchy problem (1.1)(1.2) is G(")-wellposed (G(")-wellposed). Zf, however, x = x*, then this problem is locally G(")-wellposed.

.))a(

Definition 14. Let the coefficients of the operator P belong to G'{"}(X).The Cauchy problem (1.1)-(1.2) is said to be G'{"}-wellposed if the following conditions are satisfied Existence condition a). For any f E G'{")(X;),g j E G(")(X,) ( j = 0,. . .,m - 1) satisfying (1.1)-(1.2). there exists a u E G'+'"{"}(X,+) Uniqueness condition b) of Definition 1. The Cauchy problem (1.1)-(1.2) is said to be locally G'{"}-well posed if the following conditions are satisfied Existence condition a'). For any f E G'{"}(X;), gj E G{")(X,), ( j = 0, . . ., m - 1) there exists a neighbourhood X of X , and a u E G'+"{"}(X;-+) satisfying (l.l)-(l.2) in &+. Uniqueness condition b) of Definition 1 in each of the domains X' mentioned in the preceding condition. We shall examine the G(")-well-posedness,G(")-well-posednessand local G(")well-posedness. Since GI("),in contrast to GI("),are countably-normed spaces and since in conditions a) and a') above the requirement u E G'+"{"}(X;) is, apparently, meaningfully equivalent to the requirement that u E C"'(X:), the local G(")-well-posedness has no new effects in comparison to the G(")-wellposedness. Because till now we have not come across even a single example where the (local) G'{")-well-posednessdepends substantially on I, we take some liberty and omit the letter 1 in the well-posedness notation. 2.2. Necessity and Sufficiency of Hyperbolicity. As we already mentioned, for the non-characteristic Cauchy problem to be well posed in Gevrey classes the hyperbolicity is a necessary condition. Theorem 28 (Ivrii [1976b], Komatsu [1977], Nishitani [1978]). Let the coefi ficients of the operator P belong to Grn{"}. Assume that the hyperplanes {x, = t }

175

As confirmed by examples (see 9 2.3 and 9 2.4), in none of these assertions can the conditions imposed on x be, in general, relaxed. When S+ u S- # fa, we have a less precise result. '

Theorem 30 (Ivrii [1975]). Let the operator P be x,-hyperbolic in X and X ; be a spatial type lens. Let the characteristic symbol g of the operator be analytic in X x (IR"+'\O), and, finally, let the coefficients of P belong to G"{"}(X). Then the assertions of Theorem 29 remain valid for x* = (2r - 2)/(2r - 3). Remark. For r = 2, both the formulae for x* coincide. We now examine the following problem. Obtain conditions for the Cauchy problem to be well posed in a given Gevrey class. We shall assume below, without mentioning it each time, that P is x,-hyperbolic in X and that X,+ is a spatial type lens. 2.3. Operators with Constant Coefficients. For such operators the wellposedness conditions can be easily obtained by means of the Fourier transformation. Theorem 31 (Hormander [1963]). Let P ( D ) be a (matrix) operator with constant coefficients and g ( N ) # 0, N = (1,0, ... ,0).Then for the Cauchy problem for P to be G(")-wellposed, it is necessary and sufficient that for any a > 0 there exists a C = C, such that

vC'

E

IR"G(C,, 5')

= 0 => IIm

Col

< al<'ll/" + C,

(1.29)

where G(C) = det P(C). For the problem to be G(")-well posed or locally G(")-well posed, it is necessary and sufficient that (1.29) holds for suitable a and C. This result and the Seidenberg-Tarski theorem together give the following

176

11. Linear Hyperbolic Equations

V.Ya. Ivrii

in the neighbourhood of X for any real function larly, condition (L'J signifies precisely that

Corollary. Let g E hyp(N, M ) but G $ Hyp(N, M). Then the Cauchy problem for P is G(")-wellposed for x E [l, x*] (and for these values only) and is G<")-well posed as well as locally G("'-well posed for x E [l, x*] (and for these values only), where x* = x*(C) is a rational number, x* 2 r/(r - 1). When G E Hyp(N, M ) , we can take x* = 00.

C" and any h E C". Simi-

(1.31)

+

in the neighbourhood of X for any real function E C" and any h E C". Finally, the implications ( L ) (L,) * (L'J (L,,) turn out to be valid for x > x', and it can easily be shown that of all these conditions substantially distinct are only the conditions (L),(Liij)with 1 d j < i d r, i, j E Z+.

Theorem 32 (Ivrii [1976b], Komatsu [1977]). Let the operator P be xohyperbolic in X and have characteristics of constant multiplicity. Let the coefficients of P belong to G"(")(X)(Gm(X)(X)), and let X: be a spatial type lens. Then for the Cauchy problem (1.1)-(1.2)to be G(")-wellposed (G(")-well posed), it is necessary and sufficient that the condition (L,) (condition (LL))is satisfied in X;. For the problem to be locally G(")-wellposed, it is necessary and sufficient that the condition (L',) is satisfied in X;', where X' is some neighbourhood of X,.

(1.29) Therefore for any N # 0, we can introduce the classes of polynomials HYPW, M , x) and =

E

P(exp i(qp + Ic/p"")* h) = O(p"'-r(l-l/x) a s p - +co

We note that the well-posedness conditions (1.29)may be replaced by the following conditions:

H ~ ~ (M N, ,(x))

+

177

n H ~ P ( NM, ,

X'CX

These classes decrease as x increases, and we can establish for them Properties (1.7)and (1.1 1).Theorem 7 can also be proved, but now the fundamental solution lies in gi,},the space of generalized functions over Gm{")n C,. But Theorem 31 will become valid only after the localization concept is modified. In what follows, we shall examine only scalar operators.

2.4. Operators with Characteristics of Constant Multiplicity. We can prove for such operators also necessary and sufficient conditions for the problem to be well posed in Gevrey classes. Suppose that in a neighbourhood of the point p = (X, 4) E X x (IR"" \O) the principal symbol of P is of the form p = aq', where

a

a@) # 0, q(p) = 0, and where Hq(p)# 0 and is non-collinear with t-, r E Z'\O.

at

We shall say that the generalized Levi condition (L',) is satisfied in this neighbourhood if the representation (1.16)holds to within a pseudodifferential operator with symbol zero in the neighbourhood of p; here Q is a first-order pseudodifferential operator with principal symbol q and A, are pseudodifferential operators of order not exceeding m - r k/x. If the orders of A, are strictly less than m - r + k/x for k 2 1, we shall say that the generalized Levi condition (L,) is satisfied. What is more, just as in 0 1.5, all the operators can be taken to be differential operators with respect to xo and pseudodifferential with respect to x'; Remark a) of 0 1.5 remains valid. If dcq(p)# 0 and rp is a real function that satisfies the eikonal equation in a neighbourhood of X and is such that drp(X) = f , then the condition (L,) means precisely that

+

P(exp i ( q p + +p'/")-h) = p"-'('-l/,) exp i ( W X

(1

M!

'

+

+ +P"")

drp)(d+)"h o(1)) as p + +co

(1.30)

2.5. Necessary Conditions for Well-posedness in the Gevrey Classes. The following result is analogous to Theorem 21. i

1

, I

Theorem 33 (Ivrii [1976c]). Let the operator P have analytic coefficients. Let X E X ; , 4 = (0,. ., 0,1) and q = (qo, . . .,4") E [0,l)n+'. Assume that conditions (1.19)and (1.20)are satisfied. (i) If the Cauchy proboem (1.1)-(1.2)is G'")-wellposed, then

(1.32)

+

for la1 ( B - a, q ) < r(1 - qo) - (m - s)x/(x- 11, s c m. (ii) If qo = 0, X E X , and the Cauchy problem (1.1)-(1.2)is locally G(")-well ' posed, then i

d.pp,(Z,

4) = 0

(1.33)

+

for la1 ( p - a, q ) (r(1 - qo) - (m - s)x/(x - I), s < m. (iii) If qo > 0, X E X , and the Cauchy problem (1.1)-(1.2)is locally G(")-well j piosed, then (1.32)holds. ' (iv) I f the Cauchy problem (1.1)-(1.2)is G(")-wellposed, then (1.33)holds. 1

i i

It follows from this theorem and from a), b) of Theorem 10 that for the Cauchy problem to be G(")-wellposed or to be locally G(")-well posed for X E X , (G(")-well posed), it is necessary to impose suitable conditions on all the members of P of order not lower (greater) than m - yr(x - l)/x,where y = $ if X E 8 and y = 3 if X E S , u S-.

Definition 15. Let the operator P have analytic coefficients. The Cauchy problem is said to be G(xl-regular (locally G{"}-regular)if it is G{")-wellposed (locally G{"}-wellposed) for the operaor P + Q, where Q is any operator, with coefficients in G"{"I, of order not greater than m - 1.

178

11. Linear Hyperbolic Equations

V.Ya. Ivrii

Example 8. Let P = Pl + cD, be the operator as in Examples 4 and 4,and 51. Let X, n {xl = = xk = O} # We recall that the condition that c E [- G, a] is necessary and sufficient for the Cauchy problem for P to be C"-well posed. It can easily be shown that a necessary and sufficient condition for the G(")-well-posednessis x < 2, while for the G(")-well-posedness or for the local G(")-well-posednessit is x < 2.

Theorem 34 (Ivrii - [1976c]). Let the operator P have analytic coefficients, and let X E X:, 5 = (0, ..., 0, 1) and q = (qo,. . ., 4.) E [0, l)"+'. Assume that d t a , 8 p ( ~t) , = Ofor la( + (fi - a, q ) < 0. (i) If the Cauchy problem (1.1)-(1.2) is G(")-iegular,then x < x* = a/(a - 1). (ii) If qo = 0, X E X, and the Cauchy problem (1.1)-(1.2) is locally G(")-regular, then x < x*. (iii) If qo > 0, X E X, and the Cauchy problem (1.1)-(1.2) is locally G(")-regular, then x < x*. (iv) If the Cauchy problem (1.1)-(1.2) is G(")-iegular, then x < x*.

a.

c E C\[-G,

Example 9 (Ivrii [19781). Let (i) 0 E X, and P

The following result has no analogue in the C"-well-posedness theory.

z

P = 0:

t)= 0

where Z =

< 0,

(a, P ) #

(a, B),

= (o,O,. . . ,O), and that

a:oa;oP(%

= r/(r - 1).

where p, rj E Z'\O, Lj # I, for j # k and 0 E X l . We set r = maxrj, x1 = r/(r - l), x2 = m/(m - 1 - 1/p), where m = rl + ... + r,, and, finally, x* = min(x,, x 2 ) . If x1 < x 2 , a necessary and sufficient condition for the G(")-regularityof the Cauchy problem is x < x*, while for the G(")-regularity or local G(")-regularity for 0 EX, it is x < x*. If, however, x1 > x 2 , then a necessary and sufficient condition for the G(")-regularity or for the local G(")-regularity for 0 E X, is x < x*; a necessary condition for the G(")-regularity is x < x* and a sufficient condition is x < x*. If x = x*, the Cauchy problem is G(")-well posed for any operator whose principal symbol is p and coefficients are analytic and depend only on xo. Finally, if x1 = x 2 , a necessary and sufficient condition for the G(")-regularity or for the local G(")-regularity for O E X, or for the G(")regularity is x < x*.

By Theorem 34 and the results a) and b) of Theorem 10, for the Cauchy problem to be G(")-regularor locally G(")-regularfor X E X&G(")-regular), it is necessary that x < x*(x < x*), where x* = r/(r - 2) if X E X and x* = r/(r - 3) ifXES, US-. Remark. If the corresponding inequality x < x* or x < x* does not hold in any one of the assertions (i)-(iv) of Theorem 34 or of Theorem 35, then for the Cauchy problem to be suitably well posed it is necessary that Pm-l(X,= 0.

z)

2.6. Examples. Since fairly general results, apart from Theorems 29 and 30, concerning well-posedness in Gevrey classes of the Cauchy problems for operators with characteristics of variable multiplicity are not available, we confine ourselves to examining a few examples that are, in our view, quite instructive.

Example 7 (Ivrii [1978]). Let 0 E X, c IR2, and let P = 0: - x;"Df

+ a(xo)xGDl,

- (-x,)~"D~

Example 10 (Ivrii [1978]). Let n = 1, and let

z) # 0.

If the Cauchy problem (1.1)-(1.2) is G(")-regular, then x < x*

+ a(xo)xGDl,

where v, 2p E Z ' , v < p - 1, and a is analytic on [T-, T,], with a(0) # 0. Then a necessary and sufficient condition for the G(")-well-posedness,as well as for the local G(")-well-posednessin case (i), is the inequality x < x* = (2p - v ) / ( p - v - l), while for the G(")-well-posednessit is x < x*.

for la1 + ( B - a, 4 )

= D,Z - x;"Df

or let (ii) 0 E S , and

Theorem 35 (Irvii [1976c]). Let the operator P have analytic coefficients, and let X E X:, = (0, . .. , 0, 1) and 4 = (qo,... ,qn)E [0, 1)"". Assume that a;a,8p(R,

179

+ a(x)x;D1,

where p, v E Z ' , v < p, and a is analytic in X, with a(0) # 0. For the Cauchy problem to be locally G(")-well posed or to be G(")-well posed, a necessary and sufficient condition is x < x* = (2p - v ) / ( p - v). If p > 1, a necessary and sufficient condition for the G(")-well-posednessis x < x*. If, however, p = 1, then a sufficient condition for the G(")-well-posednessis x < x* = 2 and a necessary condition is x < 2. Thus the question whether the Cauchy problem for P is G(')-well posed for p = 1 and v = 0 remains open.

1

Example 11 (Ivrii [1978]). Let (i) 0 E X, and

or let (ii) 0 E S , and

n (5; - q(-X0)2"15'lZ)rd, S

p(x, 5 ) = 52

j=2

where 2p, rj E Z'\O, Aj # 1,for j # k and Ij> O V j . Then all the conclusions of the preceding example remain valid.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

Remark. If v 2 p in Example 7 or v 2 p - 1 in Example 9, then the Cauchy problem is C"-well posed. If in Examples 10 and 11 r = 1 and rn = 2, p = i, 1 or m = 3, p = *, then the Cauchy problem is C"-regular. In all the Examples 7 to 11, the fact that the Cauchy problem is C"-well posed (regular) implies that it is G{")-wellposed (regular) provided that the coefficients of the operator belong to the corresponding Gevrey class.

where Char P = { g = det p = 0} c T*Q\O is the characteristic set of the operator P and m is the order of P. We start by examining the propagation of singularities of symmetric hyperbolic systems for which the Cauchy problem is undoubtedly well posed. Let

180

Example 12 (Yagdzhyan [1978]). Let 0 E X ,

P = D; - exp( - 2x;')D:

= S-

P = ID, - A(x, Dr),

181

(1.36)

where A is a differential or a classical pseudodifferential operator (Taylor [1976, 1979, 19811)with principal symbol a(x, tr); a(x, 5 ' ) is a Hermitian matrix.

and let

+ Bxb exp( - fix;')&,

Theorem 36 (Ivrii [1979c]). Let Q = (T-, T+)x o and T E (T-, T+). Let C(T-, T+),9(o)), P u E Cm(Q)and #In, = u. Then

where B E C\O, 1 E lR,fi < 1. Then for fi c 1 and x c (2 - fi)/(1 - fi), and for fi = 1 and any x,the Cauchy problem is G(")-and G(")-well posed; at the same time, if fi = 1, it is C"-well posed not for all 1 (see Example 6).

uE

WF(u)ln, c z-'WF(v) u (N*Q,\O),

(1.37)

where I : Char PIn, + T*QT is the natural map, and N*Q, is the normal bundle of 0,. Moreover,

Example 13. Assume that fi = 0, 1 E Z+ in the preceding example. Then a necessary and sufficient condition for the G(")-well-posedness,the local G(")-wellposedness or the G(")-well-posedness is that x < 2.

WF(u)l

c I-'

WF(v),

(1.37')

if P is a differential operator.

tj 3.

;

Propagation of Singularities

; We mean by singularities various types of wave fronts (C", G{")or analytic). It should be kept in mind that the hyperbolicity of an operator is a concept that is local in x but global 5, while at the same time the wave front is a microlocal concept, that is, is local in both x and 5. Therefore for describing the propagation of singularities the hyperbolicity of the operator is not as important a property as are its other microlocal properties, such as microhyperbolicity. However, hyperbolicity is a strong constraint too (for instance, the localization of a hyperbolic operator is itself hyperbolic), and hyperbolic operators cannot belong to several microlocal types. We confine our attention to only those microlocal types that are admissible for hyperbolic operators.

3.1. Propagation of C"-Singularities. We first recall the basic concepts of the microlocal analysis (Hormander [19711, Duistermaat and Hormander C19721, Taylor [1981], Trbves [1980]). The wave front of u is a closed conical set Definition 16. Let u E 9'(Q). WF(u) c T*Q\O: (X, 4) 4 WF(u) is there is a function 'p E Cr(Q), with p(X) = 1, and a conical neighbourhood r of such that

i

1'

(1.38)

t

B

11 ~

1i

for all s E IR, here F = Fx+<denotes the Fourier transform. For each s E IR,the same condition defines WFS(u),the wave front of order s. We recall that

'

WF(Pu) c WF(u) c WF(Pu) u Char P,

(1.35)

WF"(Pu) c WF"+'"(u)c WF"(Pu) u Char P,

(1.35')

Theorem 36 enables us to reduce the study of the propagation of singularities of solutions to the Cauchy problem to that of the propagation in the whole space. Suppose now that P = P(x, D)is a classical pseudodifferential operator of first order. Assume that the following is satisfied: Condition a). The matrix p ( x , t),the principal symbol of P, is Hermitian. Theorem 37 (Ivrii [1979c]). Suppose thut condition a) is satisfied. Let T*Q\O 3 V be an open conical set. Let 'p E C"(V) be a real positive definite homogeneous function (in 5 ) of degree 0 such that

(1.34)

(1 + ISI)SF('pu)E

The same assertions hold for WFs too.

then

WF(Pu) n V n {'p < 0 } n Char P = 0,

(1.39)

WF(u) n 8 V n ('p < 0 } n Char P = 0,

(1.40)

WF(u) n V n {'p < 0 } n Char P = 0.

(1.41)

The same conclusion holds for WF" too.

This theorem bears a strong resemblance to the uniqueness theorem for the Cauchy problem in a spatial type lens. But instead of the spatial type lens, a subset of lR:+', we now have V n {'p < 0 } , which is a subset of the phase space, and instead of the base of the lens we have the set a V n {'p < 0}, while the statements of the type u = 0 in X are now replaced by the statements of the type u 0 in W (that is, WF(u) n W = 0).

-

182

11. Linear Hyperbolic Equations

V.Ya. Ivrii

In other words, the relations p' E K * ( p , V ) between the points p, p' E V are transitive and symmetric. We can now reformulate Theorem 37 in the following equivalent form.

Assume that the following is satisfied: Condition b). The condition (1.38) holds for cp = x,. Then for any real function the condition (1.38) can be rewritten in the equivalent form (HEH:okg)(PI .(H:og)-l ( P )

V k = 1, ...,r(p) = dim Ker p ( p ) ,

'0

p E Char P n V,

Theorem 38 (Wakabayashi [1984]). Suppose that conditions a) and b) abooe hold. Let T*D\O 3 V be an open conical set and ji E Char P n K If

(1.42)

where

z

(H:og)(P) 09

E

(1.47)

WF(u) n K * ( p , V )n LJV= 0,

(1.48)

Like Theorem 37, this theorem also resembles the uniqueness theorem for the Cauchy problem, only now in a characteristic conoid the role of which is played by K * ( p , V). Theorems 36, 37 and 38 enable us to estimate from above the wave front of the fundamental solution of the Cauchy problem. Namely, let E = E ( x , y ) E C m ( X y9'(Dx)) , satisfy the conditions

P ( x , D,)E

TpT*Q: VC E f(p)a(z, C) 2 0 }

=I

~ ( x- y),

SUPP E c {xo 2 Y o } .

be the dual cone, where a is a symplectic form.

TWF(E)c diag(T*X\O) u { ( p , p ' ) E (Char P n T*X\O)', p

E

K * ( p , V ) c K'(p, V ) ,

(1.44)

that is, for any W-neighbourhood of K'(p, V )we can find a W,-neighbourhood of p such that p

E

W, n Char P => K'(p, V ) c W,

p " K*(p', ~ V ) - ~ " EK * ( p , V ) , Vp, p'

E

V p'

E K'(p,

V ) o p E K'(p', V ) .

(1.45) (1.46)

(1.51)

K+(p', T*X\O)),

where T M = {(x, 5, y , q): (x, 5, y , -q) E M } , M 2 = M x M and diag M 2 = { ( p , p), p E M } c M 2 is the diagonal of M 2 .

It turns out that Theorems 36,37, 38 and 39 remain valid for other types of operators, described in 91, too. Namely, suppose that one of the following , conditions holds. P is strictly hyperbolic or else has constant coefficients and is I Girding hyperbolic; the scalar operator P is completely regularly hyperbolic or else has characteristic roots or at least characteristics of constant multiplicity and satisfies the Levi-Strang-Flaschka condition or the hypotheses of one of the Theorems 4, 19 or 20; the first-order matrix operator P satisfies the hypotheses of one of the Theorems 25 or 26. Then the conclusions of Theorems 36, 37, 38 or 39 remain valid if we carry out the following modifications. (i) In Theorem 36, we should now set %

P'P

(1.50)

Theorem 39. Let IR"'l3X be a spatial type lens and P a first-order differential operator satisfyng (1.47) and (1.48). Then

We shall analyse below the notion of the generalized bicharacteristic in various particular cases. We can clearly take t = x, for a parameter on the generalized bicharacteristic. We now assume that T*Q\O 3 V is an open conical set and that PE Char P n K Let K * @ , V ) be the union of all generalized bicharacteristics issuing from p in the direction of increasing (decreasing)x, and lying in K It can easily be shown that (1.43)

(1.49)

If P is a symmetric x,-hyperbolic differential operator of first order and R"+l3 X is a spatial type lens, then such a function exists, is unique and satisfies the dual system in y .

Definition 17. A generalized bicharacteristic of the operator P is a connected dP Lipschitz curve y = { p = p(t) E Char P } such that - E P ( p ) for almost all t . dt

K * ( p , V )are closed sets,

WF(Pu) n K * ( p , V ) = 0,

then WF(u) n K'(p, V ) = 0. The same conclusions hold for WF" too.

(a,"a!g)(P)= 0

< r = r ( p ) when conditions a) and b) are satisfied. for la1 + Theorem 37 can be stated in an equivalent form. Let p E Char P and let g p ( .), the localization of symbol g at p (see b) of Theorem lo), be a polynomial on TpT*Q. It can easily be shown that this polynomial is hyperbolic in the direction Hx0(p) if conditions a) and b) hold. Assume that T,T*Q =I f(p) = f ( g p ( * ) ,and Hx0(p)), containing H,,(p), is a connected component of the set {C, g,([) # O } . Under these assumptions (1.42) is equivalent to the inclusion H,,,(p) c f(p). Finally, let f"(p) = {Z

183

u = D&u(,,;

j = 0, . . ., WI - 1.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

(ii) If P is hyperbolic only in X , then we should now regard u as an element of W ( 2 )and replace WF(u) by WF(u) n (T*X\O); this concerns Theorems 36 and 39. (iii) In Theorem 37, we replace condition (1.38) by (1.42). (iv) In the statements concerning wave fronts of finite order, we should write

Theorem 40. Let T*Q\O 3 V and V, be open conical sets. Suppose that g = egs in V, where e # 0, s E Z+\Oand g 1 is a real symbol of the principal type, that is,

184

Wp(pu),

5'Wp+m-i-l

(uj) and WFs+m-L-l(u). Here

185

a

g 1 = 0 =E- HBI# 0 and is non-collinear with l -.

at

L = 0 for strictly and

(1.53)

Suppose further that P is a matrix operator such that g , ( p ) = Oadim Kerp(p) = s

j=O

completely regularly hyperbolic operators, L = f for operators satisfying the conditions of Theorem 19, L = 1 for operators satisfying the conditions of Theorem 20, L = r - 1 for scalar operators with characteristics of constant multiplicity, L = x and L = 1 for operators satisfying, respectively, the conditions of Theorems 25 and 26, etc. It should be kept in mind that if we choose t = x, for the parameter, then the generalized bicharacteristic will be a Lipschitz curve only if x, E (T-, T+).In order that the curve be Lipschitz near x, = T,, we must take for the parameter on it t = -(T+ - x,)"~ and t = (x, - T-)'12,respectively. In conclusion, we remark that Theorems 37 and 38 remain valid also for more general operators than the ones mentioned above. In particular, the conditions imposed on the lower terms that guarantee the desirable propagation of singularities are far weak than the well-posedness conditions for the Cauchy problem (Ivrii [1979b1).

or P is a scalar operator satisfying the Levi-Strung-Flaschka condition or, finally, P is a first-order matrix operator satisfying the hypotheses of Theorem 25 or 26. a) Zf V 3 y is a segment of the bicharacteristic of the symbol g 1 and WF(Pu) n y # 0, then either WF(u) n y = Qr or WF(u) 3 y. b) Zf V? y is a segment of the bicharacteristic of g 1 and the endpoints of y do not lie in V, c V, then we can find a function u E W(Q)such that WF(Pu) n V, = Qr and WF(u) n V, = ye n V,, where M, = {(x, At), (x, 5) E M , I > 0} is the conical hull of M . Therefore we have the following precise description of the wave front of the fundamental solution of the Cauchy problem. Theorem 41. Zf the hypotheses of Theorem 40 are satisfied in V = T*X\O and ifP is x,-hyperbolic in X and X is a spatial type lens, then 'WF(E) = diag(T*X\0)2u { ( p , p ' ) E (Char P n T*X\0)2, p E y:(p')}. (1.54)

3.2. The Geometry of the Propagation of Singularities. In this section, we shall discuss the structure of the sets K*(p).We first note that if in a neighbourhood of p the characteristic symbol of the operator is of the form g = egy .. . g:, where e ( p ) # 0, gj are real and g j ( p ) = 0, sj E Z+\O, then cp satisfies (1.42) for g if and only if it satisfies the same for each gj. In particular, if C is a connected component of the set { g = 0} and if g = egs in a neighbourhood of C, e # 0 on C and p E C, then the sets K*(p), constructed for g and g,, coincide. If g ( p ) = 0 and gso(p) # 0, then (1.42) implies precisely that dcp > 0, where dxo ~

d ~

denotes the differentiation along the bicharacteristic of g, that is, the curve

dx, y = { p = p ( t ) }along which

(1.52) Therefore if the symbol g is strictly (or completely regularly) x,-hyperbolic in X , then K * ( p )coincides with y$(p), the bicharacteristic of g issuing from p in the direction of increasing (decreasing) x,. Thus we obtain a unique description of the propagation of singularities for x,-hyperbolic operators with characteristics of constant multiplicity. What is more, we can do away with the hyperbolicity condition and establish the following

A curve y containing its limit points and in a neighbourhood of which the hypotheses of Theorem 40 are satisfied and which is a bicharacteristic of the symbol g 1 will be called a regular bicharacteristic. A closed set M c V will be called a limiting bicharacteristic in V, C V if there is an open set V2, V, C V2 C V , and - a sequence of regular bicharacteristics Yk, with endpoints not in V,, such that =M n that is, if M n is the smallest closed set M such limk+m(Yk n that i&k+m (Yk n c M . From the arguments based on the Baire's category theorem it follows that assertion b) of Theorem 41 remains valid for limiting 1 bicharacteristic also. In many cases, this result enables us to show that Theorem i 39 is precise. It can easily be shown that if g is an x,-hyperbolic symbol, then there exists a nowhere dense closed subset Z in C = { g = 0} such that the multiplicity of the characteristics of g is constant in C\Z, although the same depends, in general, on the connected component of C\Z. In view of Theorems 40 and 41, the main interest lies in the propagation of singularities in a neighbourhood of Z. We first assume that g can be factorized, that is,

v,)

v,)

vl

v,,

L

g

= egi'

.. . g p ,

(1.55)

where e # 0 and g j are real symbols of the principal type. Suppose that one of the following conditions holds: k=2,

91 = 9 2 = 0 * { g 1 , g 2 ) # 0 ;

(1.56)

V.Ya. Ivrii

186

k 2 3,

Vj g j = 0

11. Linear Hyperbolic Equations

-

complete analysis of the general situation (at a point) is still possible if the multiplicity of characteristics does not exceed 2. These are the situations that turn out to be the general case for narrower classes of systems of chrystaloptics and the theory of elasticity in anisotropic media. Let Z be a stationary set of g , that is, let

gj,to > 0

Suppose further that for any set X c { 1, . . .,k}, the facts that g j ( p ) = 0 V j E X and that 1= (lj)jEis a non-zero vector with non-negative components imply CxA # 0, where (1.58) C x = ( { g i , gj})i,js;v (this condition will be fulfilled if, in particular, Vi,j

gi=gj=O,

i<j*{gi,gj)>O,

gj = 0 * gj,c0 > 0).

Then it can be shown that any generalized bicharacteristic of g is a polygonal bicharacteristic. That is, the curve y, parametrized by xo, consists of segments of bicharacteristics of g j , the endpoints (and only the endpoints) of these segments lying in C; at points of C a "transition" to bicharacteristics of other factors may take place; C n y is a locally finite set. Thus in the above-mentioned cases, and in some other cases as well (Ivrii [1979c]), K' turn out to be "bicharacteristic trees" with branches at points of C. We note that in (1.57) and (1.58) we require definite signs and not merely non-vanishing of the quantities concerned. Otherwise there may take place infinitely many transitions on a finite interval for the polygonal bicharacteristics and there may appear generalized bicharacteristics lying entirely in Z and being bicharacteristics of the linear combinations of the factors (with non-negative coefficients)but not of the factors themselves. If, however, the factors are in involution, that is, if gi=gj=O=-{gi,gj}=O Vi, <

< i,

Hgil,* . . HBit, 9

g '.I =

and

= gi, = 0 *

187

C = { d g = 0} c { g = O}.

(1.60)

To start with, we assume that at each point p E C the fundamental matrix Fg(p)has two non-zero real eigenvalues +@) (that is, g is effectively hyperbolic). Then it can be shown that for any point p E Z in some neighbourhood V of Z there exist two C"-curves yj(p) = { p = pj(t)} ( j = 1,2) such that pj(0) = p, pj(t) # Z for t # 0 and such that y j is a bicharacteristic of 9 for t # 0 (but, of -(O) course, this parametrization does not coincide with the one in (1.52)), and dpj dt is a non-zero eigenvector of Fg(p)corresponding to the eigenvalue (- l ) j p ( p ) . What is more, for a parameter on yj outside xo = T+ we may take t = xo. We denote by yf@) that part of yj(p) on which xo xo(p). Suppose, finally, that Cj = yj(Z)and Cj' = yF(C). Then we have the following results. Cj are closed conical sets, Zl n C, = Z, Zl u C, c C, and Cl u C2= C if and only if g can be factorized. If C is a C"-manifold of dimension d, then Cj are C"-manifolds of then y$(p) has no limit points in dimension d 1. If p E V n C\(CIT u CZT), C and K'(p) = y:(p); if p E C, then K'(p) = y:(p) u y:(p), and, finally, if p E ZT\C, then p E y?(p) for some point p E C and K'(p) = yj(p, p) u K*(p), where yj(p, p) is a segment of yj(p) between the points p and p. Thus under the above assumptions, generalized bicharacteristics are regular bicharacteristics not passing through points of Zand polygonal bicharacteristics having branches at points of C. [ We now assume that the hypotheses c)-d) of Theorem 19 hold with p = g . We have the following results. ! a) If p E V n C\C, V is a neighbourhood of Z, then yg(p)has no limit points , in C, and, moreover, its distance from Z remains all the time of the same order;

+

1

(1.59)

@/at are linearly independent

and if g , ( p ) = = g k ( p ) = 0, then K'(p) are bicharacteristic sheets, that is, pieces of a k-dimensional manifold Y 3 p spanned over the vector fields Hgj(j = 1, ...,k) and bound by (k - 1)-dimensional manifolds 3 p spanned over the vector fields Hgj(j = 1, . . .,k, j # i); for K'(p) we take the piece on which xo 2 x,(p). In the general case, the dimension of the bicharacteristic sheet K'(p) equals the number of factors that vanish at the point p ; the same factors also vanish on K'(p) but the other factors do not vanish on K'(p). Assume that the factors are divided into different groups in such a manner that factors belonging to the same group are in involution but those belonging to distinct groups are not. If { g i , g j } > 0 when g i belongs to the group with a smaller number than g j , then K'(p) are trees consisting of bicharacteristic sheets of different dimensions corresponding to distinct groups of factors. The geometry of the propagation of singularities becomes much more complicated when the characteristic symbol g cannot be factorized, and to analyse the behaviour of bicharacteristics becomes a far more difficult task. However, a

' 9

K*(P) = Y:(P). b) If p E C, then any generalized bicharacteristic passing through p lies in C

and satisfies the inclusion -E

dt

P ( p ) n Ker Fg(p).

(1.61)

Let us replace condition e) of Theorem 19 by a stronger condition, namely, e') uIz has a constant defect d (then d 2 1). Then the manifold C is fibered into d-dimensional bicharacteristic C"-submanifolds A = A ( p ) 3 p : p E A TpA = Ran Fg(p)n Ker Fg(p);and in this case 01, = 0. Let p E A . Then the restriction of the localization g p ( . ) to Ker F:(p) is a well defined quadratic form on the factor space KerF:(p)/Ker Fg(p). Since there exists a unique isomorphism between this factor space and T*A,

V.Ya. Ivrii

11. Linear Hyperbolic Equations

we obtain a symbol h defined on T*A which, by statement b) of Theorem 10, is strictly x,-hyperbolic of second order. Then K'(p) lie in A ( p ) and are conoids of dependence for h. In particular, in the general case d = 1, the bicharacteristic manijiolds (they are the limiting bicharacteristics as well as the generalized bicharacteristics) are the curves y = { p = p ( t ) } defined by the inclusions

E ( x , y ) is a Fourier integral distribution in the sense of Melrose and Uhlmann [1979b], that is, is a Fourier integral distribution with a Lagrange manifold having self-intersections. For scalar operators with characteristic roots of constant multiplicity that satisfy the Levi-Strang-Flaschka condition a parametrix was constructed in Chazarain [19741 by means of Fourier integral operators, while for first-order systems satisfying the hypotheses of Theorem 25 in Petkov [1978]. For various classes of effectively hyperbolic operators the parametrix has been constructed in Kucherenko and Osipov [1983], Alinhac [1978], Yoshikawa [1977,1978a, b, 19801. For operators with factorizable principal symbols for the case when the factors are in involution and also for operators satisfying the conditions of Theorem 20, parametrices have been constructed in Sjostrand and Uhlmann [1979a, b]. For diagonalizable first-order systems the parametrices have been constructed in the form of series in Kucherenko [1974]. A more detailed study led Kumano-go and his students to the construction of the theory of Fourier integral operators with multiphase functions (Kumano-go and Taniguchi [19791, R. Lascar [1981]).

188

dP

-E

dt

Ker F,(p) n Ran F,(p).

(1.62)

Since now through every point p E Z there passes exactly one generalized bicharacteristic, assertions a) and b) of Theorem 40 hold for generalized bicharacteristics in this case and the inclusion (1.51) turns into an equality. Now there arises the question as to how precise is the description of the propagation of singularities obtained above. We have in mind the inclusion (1.51) in the first place. The existing results (Ivrii [1979a, b], Uhlmann [1982]) are not exhaustive nor even somewhat general, but enable us to presume that in the general case (1.51) turns into an equality for almost all lower terms, although for some lower terms there may be abnormally poor propagation of singularities. What is more, if g can be factorized into two factors, not in involution, then for almost all lower terms all the singularities, and not only the typical ones, are propagated in a prescribed manner (Ivrii [1979a]).

3.3. The Construction of a Parametrix. The length of the present survey paper does not permit us to give an account of the construction of a parametrix for various cases, and we confine ourselves to a cursory look at the basic results. For strictly hyperbolic operators and for diagonalizable hyperbolic systems of first order, with characteristic roots of constant multiplicity, the parametrix was constructed by many authors by means of the oscillatory integrals of a certain type. The investigation of these integrals and operators led Hormander [1971] to the development of the theory of Fourier integral operators (and distributions). Similar reasonings connected, however, not with the construction of the parametrix but rather with the global construction of individual asymptotic solution of the Cauchy problem with oscillatory initial data had earlier led Maslov [1965, 1973, 1976, 19833 to the construction of a canonical operator. The further development of these two theories ran parallel to a great extent, although each one of them has its own deep individual features. It should be noted that not the fundamental solution E ( x , y ) is a Fourier integral distribution in the strict sense but a distribution U(x, y ) such that P(x, Dx)U = 0, DiUlxo=yo = iSj,m-16(x- y ) l

( j = 0, ..., m - 1).

(1.63) (1.64)

If the coefficient of Dg in P is 1, then (1.65)

I t

i

189

3.4. Propagation of Analytic Singularities and Gevrey Singularities. We first define the wave fronts WF{")(u),x E [l, a), corresponding to the Gevrey classes.

Definition 18. Let u E 9'(sZ). Then (X,4) 4 WF(")(u)if there exists a function r of 4 such that the inequality cp E G'")n Co(sZ),with cp(X) = 1, and a concial neighbourhood

IF(cpu)(t)(< C exp(-AItl"")

for

5 Er

(1.66)

holds for some C and A > 0. Similarly, Definition 19. Let u E 9'(Q). Then (X,4) 4 WF(")(u) if there exists a function G ( " )n C0(Q),with q(X) = 1, and a conical neighbourhood r of 4 such that (1.66) holds for any A > 0 and suitable C = C,. cp

E

The extension of these definitions to the case x = 1 is not automatic. For distributions there are three equivalent definitions of the analytic wave front (or of the essential spectrum or of singular spectrum); these are by Sato, by Bros and Iagolnitzer, and by Hormander. Here we present the one given by Bros and Iagolni tzer. Definition 20. Let u E 9'(sZ). Then (X,4) 4 WF,(u) = WF(')(u) if there exists a function cp E C,"(sZ),cp = 1 in a neighbourhood of X,and a conical neighbourhood I' of 4 such that the inequalities IF(cpu)(x,<,211 < CN(l

+ I
v t E r,

1 E [O, Yltll, (1.67) hold for suitable constants A > 0, y > 0 and CN,and for all N E Z ' ; here (Fu)(x,t, 4=

s

exp(-i < y , 5 )

-

Alx

-

y12)u(y)d y

191

V.Ya. Ivrii

11. Linear Hyperbolic Equations

is the Fourier-Bros-Iagolnitzer transformation. We note that WF{")(u) and WF(u) can also be defined in terms of this transformation.

tinct variants, namely, in a strong sense and in a weak sense. As we already remarked, the well-posedness question for the mixed problem is far more complicated than the one for the Cauchy problem, because it depends on three objects. Namely, on the operator P itself, on the boundary values and on the boundary surface itself. What is more, the class of admissible operators P gets somewhat enlarged because on the other side of the boundary surface or below the initial or above the final hyperplanes the operator may fail to be hyperbolic. Let o be an open domain in IR" with a C"-boundary do. Let X = [T-, T+]x 0 be a cylindrical domain with respect to x , with lateral boundary S = [T', T+] x do.Let v(x)denote the inward normal to S and S + = { T,} x 0 the base of X , -m < T- < T+ < +a. Put N = dx,. We shall represent ao in a local coordinate system by the equation x , = 0 and w by the inequality x , > 0. We examine the following mixed problem:

190

It can easily be shown that WF{")(u)are closed conical subsets of T*SZ\O, and that WF(")(u)3 WF("')(u) 3 WF("')(u) 3 WF(u) for 1 < x < xl. What is 'more, WF{")have basic properties of the usual wave fronts: n,WF{")(u) = sing SUPP{X)(4,etc. For x > 1, the propagation of singularities for hyperbolic operators was investigated by Wakabayashi [1983a]. The propagation of analytic singularities was studied by many authors, but, in our view, Wakabayashi [1983b] succeeded in putting these results for hyperbolic operators in the most convenient form.

Theorem 42. Let P be an x,-hyperbolic operator in SZ whose coefficients lie in G"{')(SZ). Assume that at least one of the following three conditions is fulfilled: (i) x* = r/(r - l), where r is the maximum multiplicity of the characteristic roots of P . (ii) P is a scalar operator with characteristics of constant multiplicity and the condition holds. (iii) The coeflicients of P are constant and (1.29) holds with x = x*. W e have the following results. (i) If { x } = (x), x E [l, x*] or if { x } = ( x ) , x E (1, x*], Pu E G"{"I(SZ) and v = (DiuInT, j = 0, . . .,m - 1) are the Cauchy data, then (1.37") WF{")(u)l,, c z-l WF{")(u); (ii) If { x } = (x), x E [l, x*) or if { x } = ( x ) , x E (1, x*], p T*S2\0 3 V is an open conical set and WF{")(Pu)n K * ( p , V ) = fa,

E

Char P n V, and

WF{"}(u)n K * ( p , V )n d V = fa,

Pu = f inX,+, (2.1) B k U = (pk ( k = 1, . . . ,p) On s;, (2.2) ( j = 0, ..., rn I), D&Ixr = gj (2.3) where P and Bk are scalar differential operators, with C"-coeficients, of order rn and mk, respectively, 0 < mk < m - 1. We also consider the mixed problem Pu = f

in X,+,

Bu = cp on S:,

(2.1') (2.2')

(2.3') where P is a d x d matrix differential operator of first order with C"-coefficients and B = B ( x )is a p x d C"-matrix. Thus right from the beginning we exclude from our study those cases where the lateral surface has corners or edges or where different types of boundary conditions are prescribed on its different parts. Before defining a C"-well posed mixed problem, we remark that for no mixed problem with p > 0 (that is, other than the Cauchy problem) does infinite smoothness of the right-hand side, the boundary and initial data imply infinite smoothness of its solution unless compatibility conditions of infinite order are satisfied. Namely, we say that Ulx, = 9,

( 1.47') ( 1.48')

then WF{")(u)n K * ( p , V ) = fa.

Corollary. Let P be a scalar operator with characteristics of constant multiplicity in SZ and with coefficients in G"{"}(Q). Let P satisfy the condition (L;*) with either { x } = (x), x E [l, x * ) or { x } = ( x ) , x E (1, x * ] . Then the assertions a) and b) of Theorem 40 remain valid for WF{"}too.

Chapter 2 Mixed Problems for Hyperbolic Operators

{ f, cpk(k = 1, .. ., p), g j ( j = 0, .. ., m - l)}E C" satisfy compatibility conditions of infinite order if there is a u E C" such that (2.1)-(2.3) are satisfied to within functions having zero of infinite order for x , = T.

5 1. Well-posedness of Mixed Problems

Definition 1. The mixed problem (2.1)-(2.3) is said to be C"-well posed if the following two conditions are satisfied: a) The problem (2.1)-(2.3) has a solution u E Cm(X,')for any

1.1. Preliminary Remarks. In the present section we consider the question of the well-posedness of mixed problems. By well-posedness we mean the Cmwell-posedness as well as the L,-well-posedness in two of its substantially dis-

{f,(Pk(k = 1, ..., p), g j ( j = 0, ..., m - l)} E C" that satisfy the compatibility conditions of infinite order.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

b) The fact that u E C" satisfies, for any t > T, (2.1)-(2.3) with go = = gm-l = 0 and with f = 0, (pk = 0 ( k = 1, . ..,p) for xo < t, implies that u = 0 for xo < t.

that is, for the solvability of the problem the number of boundary conditions must not be too big. We remark that the adjoint problem does not always exist. However, if the original problem is, for instance, normal (see below), the adjoint problem does exist. Similar arguments apply also to the problem (2.1')-(2.3')but now (2.7) and (2.9)are to be replaced, respectively, by

192

For the problem (2.1')-(2.39, the definitions of compatibility conditions of infinite order and of C"-well-posedness are analogous. In both cases, we shall assume that the hyperplanes {xo = t } are non-characteristic, that is, g(x, N ) # 0

vx E x ,

(2.4)

where g is the characteristic symbol of P. It can easily be shown that for the mixed problem (2.1)-(2.3)(or (2.1')-(2.3')) to be C"-well posed it is necessary that P is xo-hyperbolic in X;; the proof coincides word for word with the corresponding proof for the Cauchy problem. Therefore we shall assume that P is xo-hyperbolic in X. Then for x E S, 5 E IR"+', z E C \ R , the polynomial g(x, 5 zN av(x)) (in a) has no real roots a. What is more, the degree of this polynomial, by b) of Theorem 10, is independent of (5, z) E IR"+' x ((C- \R); we denote it by M(x). Thus the number of roots o of this polynomial with Im o 2 0 is also independent of (5, z) E IR"+l x ((C-\lR); we denote it by M'(x), and set M+(x)+ M - ( x ) = fi(x). Moreover, if M is independent of x E S , then M*(x) depends solely on the connected component of S. Using the method of Ivrii and Petkov [1974], we can easily prove that for the a priori estimate, which is a consequence of the C"-well-posedness of (2.1)-(2.3) or of (2.1')-(2.30, to hold it is necessary that

+

p 2 M+(x)

at every point x E S,+.

(2.5)

P*u= h inX,+, = ll/k

( k = 1, ..., p')

(2.6) On S:,

suppucXn{T<x,
(pu,u)X

- (u, p*')X

=

c

(Bk%

bku)s

-

i

y

Ii

I

I

!

(2.8)

:

c

( Y k U , cku)s

(2.9)

k=l

for any u, u E Cm(X),with DAuls- = 0 and Diul,+ = 0 ( j = 0, . . . , m - 1). Here (., .)M denotes the scalar product in L,(M), and Bk and y k are certain differential operators of order m - mk - 1 and m - m; - 1, respectively, in which mi is the order of Ck.From this a priori estimate one can deduce by the same methods the inequality p' 2 M-(x),

x E s,,

(2.5')

(2.10)

If in a neighbourhood of X E S; rank p ( x , v(x))is constant and equals s, then in this neighbourhood the adjoint problem is defined with p' = d - s - p, and (2.5),(2.5')imply

P'

k=l

- (YU, C 4 ,

rank B(x) = p, Vx E S;.

'

P

S

(2.7')

where C , B and y are, respectively, p' x d, p x d and p' x d matrices. Evidently, a necessary condition for (2.1')-(2.3')to be C"-well posed is that

(2.7)

that is, for the problem which is such that

on S,+,

(Pu, 4 x - (u, p*4, = (Bu,P 4

+

If the number of boundary conditions is too small, then an asymptotic solution violating this estimate is constructed. On the other hand, the fact that the problem (2.1)-(2.3)is solvable implies a priori estimate for the adjoint problem

CkU

Cu = ll/

193

,

M + ( Z )< p

< M+(Y)+ d - s - A?@).

(2.5")

Hence, if fi(x) = d - s (which, for instance, holds when S is non-characteristic), we obtain p

= M+,

(2.5"')

a condition that is necessary for the problem (2.1')-(2.3') to be C"-well posed. We know that the mixed boundary problem for an equation can be reduced to that for a first-order system, and, moreover, since the non-characteristic mixed boundary problem (that is a mixed problem with a non-characteristic lateral boundary) is reduced to a similar problem having the same number of boundary conditions. In this reduction the C"-well-posedness of the original problem is equivalent to the C"-well-posedness of the resulting problem. Therefore (2.5'")is a necessary condition, under the assumption that the boundary surface is non-characteristic, for both the problems (2.1)-(2.3)and (2.1')-(2.3')to be C"-well posed. And all the arguments remain valid provided that the solution is required to exist and to be unique in the same class of functions; for instance, in C'(m < 1 < 00). Unfortunately, the arguments of this section have not been carried out accurately in any of the works known to the present author. In the case of characteristic boundary, the number of boundary conditions may depend on the lower terms of P and on the required smoothness of the solution, as, for instance, for the operator P = 002 - X1D:

+ aD1,

u E (C.

(2.11)

V.Ya. Ivrii

11. Linear Hyperbolic Equations

In order to overcome this difficulty, the concept of a uniformly characteristic boundary is introduced. A surface S = {'p = 0}, with a 'p # 0, is said to be uniformly characteristic if for small a the surfaces {'p = a } are characteristic, and of the same multiplicity. This concept is very important on account of the Maxwell system.

We assume that p = M', that is, that (2.15) is a square system. By the Lopatinskij determinant of (2.1)-(2.3) we mean the determinant of (2.1 5):

1.2. Operators with Constant Coefficients. In this and the next section, we follow Sakamoto [1970a, b, 19803. In the present section P = P ( D ) and Bk = Bk(D) are operators with constant coefficients, X = IR" x K+ = {x E IR"+', x, 2 0} is a half-space and S = IR"= {x E IR"", x, = 0} a hyperplane. Let the operator P ( D ) be Girding x,-hyperbolic. This is a necessary and sufficient condition for the C"-well-posedness of the non-characteristic Cauchy problem, and, hence, a necessary condition for the C"-well-posedness of the mixed problem. We assume that the hyperplanes {x, = t } are non-characteristic. Then the polynomial G (to,5", 5") has no real roots 5. for 5'' = (t1,..., t,-,) E IR"-' and for Im to < - C; recall that G(5) = det P(5). Assume that the hyperplane S is non-characteristic, that is, g(v) # 0, where v is the inward normal to S . Then, for 5'' E IR"-' and Im to(- C, G(t,, t",5,) has M' roots t,,with Im 5, 3 0. Denoting these roots by A:(, 5") ( j = 1, ...,M') and setting

We can define a more general determinant 9 ( q TO), where q = (to, 5") E IR" and 0 belongs to the hyperbolicity cone T(g,N ) of P(D).

194

n ( tn

M'

G,(O

=

j =1

- A,?(<,,

we have G ( 5 ) = cG+(t)G-(<), c # 0. We perform (formal, for the time being) Fourier transformation with respect to X" -+ 5'' E IR"-' and Fourier-Laplace transformation with respect to xo -+ to, Im to< -C. Then (2.1)-(2.3) goes over to a boundary-value problem for an ordinary differential equation with constant coefficients: (2.12)

Bk(50, t",Dn)alxn=O = @k ( k = . - * ) (2.13) where the right-hand side F of (2.12) is determined by the right-hand side of (2.1) and by the initial data, while @ k are determined by the boundary conditions and the initial data. The general solution of (2.12), with F = 0, that decays as x, -+ +a is of the form =

M+ s C 2x1 cj

eiXnrn<',-'G;'(t)d t , ,

t")= det

(&

Bk(t)ti-1c;'(5)

dtn) k ,j = l ,

..., p

.

+

Theorem 1. For the problem (2.1)-(2.3) to be C"-well posed, it is necessary that 9 ( q + z N ) # 0 Vq E IR", Im 7 < -C,

(2.16)

with a suitable C, . In particular, if P ( D ) = p(D) and Bk(D) = bk(D)are homogeneous operators, it is necessary that

l(r]

+ 7N)# 0

vr] E IR",7 E C-\IR,

(2.17)

where 1 is the Lopatinskij determinant for { p , b,, .. .,b,,}. For non-homogeneous operators also we can define 1 by its principal parts. We assume that 3q E IR": l(q - iN) # 0.

(2.18)

Then 1 is the principal part of 9 (If I( - iN) = 0, then 1 is defined precisely as the principal part of 9.)

t")),

P(t0, t",Dn)a = F,

9(tO?

195

(2.14)

j='

where y is a closed contour in C that goes round all the roots A:(tO, ("1 ( k = 1, . . ., M + ) once in the counterclockwise direction and cj are arbitrary numbers. Substitution of (2.14) into (2.13) yields the following system of algebraic equations:

Theorem 2. For the problem (2.1)-(2.3) to be C"-well posed when (2.18) holds, it is necessary that 1(-"

# 0.

(2.19)

The proofs of Theorem 1 and 2 are based on the construction of asymptotic solutions. It can easily be shown that (2.16) and (2.19) imply (2.17). Theorem 3. If (2.16) and (2.19) hold and the boundary conditions are normal, that is, if j > k*mj > mk and

bk(v)# OVk,

(2.20)

then the boundary value problem (2.1)-(2.3) is C"-well posed.

Assume that (2.16) and (2.19) hold. Let T,be a connected component, containing N , of the set (0 E r(g, N ) , I( - i0) # O}. Then

r, is an open convex cone,

+ 70) # 0 9(r] + 70) # 0 l(r]

(2.21)

< 0,

(2.22)

-= -C,

(2.23)

V q E IR", T: lim z

31 E IR,

7: Im

z

(compare with the similar properties in 4 1.3,Chap. 1).

11. Linear Hyperbolic Equations

V.Ya. Ivrii

196

holds with the constant independent of u, to,t or y, where

Finally, let

r+ = r ( g , N ) n ( 5 + OV,r E r,,0 E W} and let r; be the dual cone and'r = r' n { x , 2 O}.

Ilu; Kl14,y=

PEj = 0 for SUPP

and K(t,,,t)= K n {to < xo < t}. The other norms Ill'; Srlllq,7and 111.; S(tO,t)JJ/q,yare defined in a similar fashion, only that the derivativesare now taken only along S. It is easy to see that if the boundary is non-characteristic, then the estimate ' . Moreover, for s = 0 analogous estimate (2.25,) implies (2.25,) for all s E Z is obtained if one employs the norms used by Sakamoto [1970a, b, 19801. Namely, Ilu; Kllb,y= Ile-yxou;Kllq, and similarly, for Illu; KIllh,Y;the index y = 1 is omitted. Let 9be the set of infinitely smooth right-hand sides and initial and boundary data having compact supports and satisfying the compatibility conditions of the closure of % in the space infinite order. Let FS,

x , > 0,

Ej c {xo 2 O},

(2.24)

BkEjlX.=, = d i j s ( X 0 ) d ( X r r ) (k = 1, ..., p), then supp Ej c r:. The proofs of Theorems 3 and 4 are based on the application of Fourier transformation in the complex plane with respect to the variables (x,, .. .,x , - ~ ) and on proving a Holmgren type uniqueness theorem. In the latter, the fact that the problem is normal plays an important role; this guarantees that the adjoint problem, satisfying all the conditions of Theorem 3, exists. Theorem 4 and Theorm 7 of Chap. 1 together describe the support of the solution to the problem (2.1)-(2.3) All these arguments apply also to the problem (2.1r)-(2.3r)with the only difference that the condition (2.20) is replaced by (2.10) and the inequalities 9 ( q z) # 0 and l(q + T O ) # 0 by the conditions that the row matrices

+

+ + +

B " O P ( ~ T e + O V ) (mod G+(q B"O~(? ze BV) (mod g+(q

+ + av)), + + OV))

are, respectively, linearly independent; here "'(. ) denotes the matrix of cofactors.

1.3. Strong L,-well-posedness of the Mixed Problem (for Equations). As in the preceding section, here also we follow Sakamoto [1970a, b, 19803. The strong L,-well-posedness (H-well-posednessin the terminology of Sakamoto) is the strongest form of well-posedness.We denote by Hsvarious Sobolev spaces. We shall assume in what follows that either ao is compact or ao = { x , = 0 } and that all the coefficients of P and Bk are constant for lxrl > R. For simplicity, we assume that T+ co.

-=

Definition 2. The mixed problem (2.1)-(2.3) is said to be strongly L,-well posed if it is C"-well posed and if, for all u E C", all y 2 0, to, t E [T, T+], to <: t and all s E Z ' , the a priori estimate X(t,,r)IIm-l+s,y+

IIU;

XrIIm-l+s,y +

IIU; S(to,t)IIm-l+s,y

1 y4-t"111e-Y"~D"u; L2(K)ll, I4cq

Theorem 4. Zf the hypotheses of Theorem 3 are satisfied and E j e Cm(E+,9(lR")) is a solution of the problem

J~IIU;

197

be the set of right-hand sides, initial and boundary data having smoothness as mentioned and satisfying the compatibility conditions of orders. It can be shown that if S is non-characteristic and the boundary-value problem is normal, then the compatibility conditions are of local nature, and when ao = {x, = 0}, the compatibility conditions of order s are the p ( s + m - 1) - E m k conditions

(

k

)

that connect the values on S , of the function DzD.""f(ao + a,, < s - l), D2gj(an< s + m - 2 - j ) , Dz@'(txo < s + m - 2 - mk) and their derivatives along S,. These conditions enable us to express @q&, in terms of the remaining functions and their derivatives along s,. It follows from the definition that if (2.1)-(2.3) is strongly L,-well posed and if t {f,g j ( j = 0, . . . m - I), pk(k = 1, . . . PI} E E , I Y then the problem (2.1)-(2.3) has a solution I I I

n

s+m-1

uE

j=O

Cj([T, T,],

Hs+"'-'-j (0)) c HS+m-'(XT+);

1 moreover, D"u E HS+m-l-l"I(Sg) for all a, la1 < s + m - 1. +

Theorem 5. Let the estimate (2.25,) hold for some s. Then (i) P is strongly x,-hyperbolic in X ; . (ii) Either at no point is S; non-characteristic or else p = m, s = 0. (iii) p 2 M'. (iv) Zf p = M + , then the uniform Lopatinskij condition holds: I(x, 5

+ TN) # 0

vx Es ,; 5 E IR"+l, I(x, 5 ) # 0 v x E ST+, where I(x, l )= I(x, 5 - OiN).

z

(2.26) E

(c-\IR,

5 E IR"+l\lR.V,

(2.27)

11. Linear Hyperbolic Equations

V.Ya. Ivrii

198

199

L,-well-posedness. The first of these is of the form

Conversely, we have Theorem 6. Let the operator P be strictly x,-hyperbolic in X , S be noncharacteristic and let p = M'. If the uniform Lopatinskij condition holds on S , then the estimate (2.25) is true. The proof of Theorem 6 uses the method of the divisor operator, which in the case of a boundary-value problem turns out to be much more complicated than for the Cauchy problem. On account of (2.27) and (2.22), N in condition (2.26) may be replaced by any vector 0 E T(g,N). What is more, if the boundary-value problem is normal and if the hypotheses of Theorem 6 are satisfied, then the same conditions are fulfilled for the adjoint problem. Therefore, by means of functional-analytic arguments, we can establish the following

or Ilu; Xtlls + Ilu; S(t0,t)llSG CClIu; Xt0lls

IIPu; &lIs

dt'

+ IIIBu; S(t0,t)lllSl (2.28;)

Theorem 8. Let the estimate (2.25,) hold for some s. Then (i) The operator P is x,-hyperbolic in X: and the matrix p(x, 0, 5') is uniformly diagonalizable in X ; (it is assumed throughout this section that p(x, N ) = I). (ii) Either at no point is S; non-characteristic or else p = d and s = 0. (iii) p 2 M'. (iv) If p = M', then the uniform Lopatinskij condition holds: a) For all X E S:, 5 E IR"+l, and r cC-\lR, the rows of the matrix BCop(x,5 r N av) are linearly independent (mod g+(x,5 rN + ov)). b) For all x E S:, k E lR"+l\lR*v,the rows of the matrix Bcop(x,5 + av) are linearly independent (mod g+(x,5 + av)), where g+(x,q ) = g+(x,q - OiN).

+

D & U I ~ ;= \ ~O; ( j = 0,. . . , m - I),

(k = 1, . . .,p)

+

+

Conversely, we have

imply that u = 0 in X'.

Theorem 9. Let the operator P be x,-hyperbolic and the matrix p(x, 0, 5') be smoothly diagonalizable in X . Suppose that S is non-characteristic, p = M + , and that on S the condition (2.10) and the uniform Lopatinskij conditions (a) and b)) hold. Then the mixed problem (2.1)-(2.3) is strongly L,-well posed and, moreover, the conclusion of Theorem 7 regarding the uniqueness holds.

In conclusion, we note that we could have required that the estimate Ilu; X t l l m - l + s

I'

for all to, t E [T, T,], to < t, y > 0, u E C" and s E Z+, with constants independent of u, to, t or y.

Theorem 7. Let the operator P be strictly x,-hyperbolic in X , S be noncharacteristic, and let p = M'. Assume that the uniform Lopatinskij condition holds on S and that the boundary-value problem is normal. Then the mixed problem (2.1)-(2.3) is strongly L,-well posed. Furthermore, let X: 2 X , a x ' = X ; v XT+v S' v S , where S c S, S' E C". I f at each point x E S' the outward normal v'(x) to S' lies in T(g,N), then the facts that u E H m ( X ) ,Pu = 0 in X' and

Bk~Is,\s;= 0

+

+ Ilu; S(to,t)llm-l+s

The proof of this theorem rests on deducing the energy estimate by the symmetrizer method (Kreiss [19701) and then applying functional-analytic arguments. Note that the symmetrizer method is much more complicated for the mixed problem than it is for the Cauchy problem. We also note that the problem for strictly hyperbolic equation, examined in the preceding section, can be reduced, after localization, to a problem for a pseudodifferential system of first order.

(2.28,)

instead of (2.25,), holds for all to, t E [T, T,], to c t, u E C" and s E Z+with a constant that is independent of to, t or u. Using the Duhamel's principle, we can easily show that the estimates (2.25,) and (2.28,) are equivalent; moreover, (2.28,) implies (2.28,) for an arbitrary s E Z+. Using (2.28,), we find that the smoothness condition on the right-hand side can be relaxed:

Dif E Ll([T, T,], H"-'(o)) ( j = 0, ..., s).

' '

1.4. Strong L,-well-posedness of the Mixed Problem for Systems of First Order. In a series of works (Friedrichs [19581, Lax and Phillips [19601)devoted to the symmetric systems with "non-negative" boundary conditions, there appeared L,-estimates in two forms leading to the notions of strong and weak

1.5. Necessary Conditions for C"-well-posedness of the Mixed Problem. Using the methods of $ 2 of Ivrii and Petkov [1974], we can easily prove the following

Theorem 10. Let X E S:, p = M+(X)and 1(X, 5 + r N ) = 0 for some 5 E lR"" and r E C-\lR. Assume that l(X, 5 + ' N ) f 0 on C - . Then the mixed problem (2.1)-(2.3) is not C"-well posed.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

Corollary 1. Let X E S:, p = M+(X) and 1(X, N ) # 0. Then for the problem (2.1)-(2.3) to be C"-well posed, it is necessary that the Lopatinskij condition (2.26) holds with x = X.

the wave equation with Neumann boundary conditions. On the other hand, the wave equation with homogeneous Neumann condition possesses the classical energy integral and for it the existence and uniqueness theorems hold. Similar is the case with many problems for the Maxwell system for which there are no strongly L,-well-posed boundary-value problems at all because the boundary is characteristic. In view of this situation, in their works Agemi and Shirota C1970, 19711introduced various other definitions for L,-well-posedness.

200

Theorem 11. Let X E S,' and p = M+(X).Assume that condition a) is saitsfied for x = X, 5 = 4 and z = 7 E C-\lR. Then for the C"-well-posedness of the mixed problem (2.1')-(2.3') it is necessary that the Lopatinskij condition holds for x = X, = and for all z E C \ l R .

Definition 4. The mixed problem (2.1)-(2.3) is said to be weakly' L,-well posed if it is C"-well posed and if for all to, t E [T, T+], to < t, y 2 0 and u E C" such that Bkuls = 0 (k = 1, ... , p), s E z+,the a priori estimate

Corollary 2. Let X E S$ and p = M+(X).Assume that condition a) is satisfied for x = X and 5 = N . Then for (2.1')-(2.3') to be C"-well posed, it is necessary that condition a) is satisfied for x = X and all ( E R"+'and z E cC-\lR.

J"J tt U ;x(to,r)It

m-I+s,y

Definition 3. Let X E S,'\S, and r be an open convex cone with vertex X; r\X c {xo < KO}. We say that the problem (2.1)-(2.3) satisfies the uniqueness condition ( U r )if the facts that P u = 0 in X,' nr and D&UIxTnj-

=0

BkUIS+Tnr=O

( k = I, ...,/A)

imply that u = 0 in X : n r. Theorem 12. Let X E S,'\S, l ( X , .)

and p

+0

= M+(X).Suppose

that

on lR"+' - iT(g, N).

(2.29)

holds with the constant independent of to, t, y or u.

As compared to (2.25,), the member Ilu; S(to,t)llm-l+s,y on the left-hand side is now absent and a condition has been imposed on u that makes the members, absent on the right-hand side, vanish. We shall assume throughout that S is non-characteristic and the boundary-value problem is normal. Then it is not difficult to prove that (2.32,) implies (2.32,) for all s and that the problem (2.1)-(2.3) has a solution

For the problem (2.1)-(2.3) to be C"-well posed and condition (Ur)to be satisfied, it is necessary that (2.30)

+ IIU; xt It m-1 +s,y (2.32)

'

( j = 0, . .., m - l),

s+m-1

E

n

j=O

C ~ C TT+], , ~ s + m - i - j (4)

for any

{ f, g j ( j = 0, .. . ,m - l), cp,(k = 1, . . .,p ) } E

and, in particular, that l ( X , N ) # 0.

(2.31)

ve E r(g,N): {(e,x - X) 2 o} nT = {x},

n{ q k E Hs+m-mk-1/2

~s+m-3/2-lal

(S,' 1

for la1 < s + m - 2. Thus, in comparison to the strong L,-well-posedness, the required smoothness of boundary data increases by 1/2, while that of DauIs decreases by 1/2. Moreover, the concept of L,-well-posedness with the loss in smoothness by q E Z+was also introduced. Apart from the C"-well-posedness, the estimate

and, in particular, for 8 = N . The present author is not aware whether the proofs of these results have been published in their general form somewhere. Corollary 1 of Theorem 10 has been proved by Wakabayashi [1980]. 1.6. Weak L,-well-posedness of the Mixed Problem. The uniform Lopatinskij condition, introduced in 9 2.3 and 0 2.4, is an extremely severe restriction on the class of problems under discussion because it does not hold, for example, for

(S,+)}.

Then, by the embedding theorem, Dau

Theorem 13. Let X E S,'\S, and p = M+(X).Suppose that condition a) holds for x = X and for some 5 = 4 E lR"" and z = 7 E C \ R . For (2.1')-(2.3') to be C"-well posed and condition ( U r ) to be satisfied, it is necessary that condition b) holds for x = X,5 = 8

20 1

(2.33) must be satisfied for all u E C" having the property that Bkuls= 0 (k = 1, . . .,p) and u = 0 for xo < T. Even for q = 0, this definition is weaker than Definition 3, at least formally. When P and Bk are homogeneous operators with constant coefficients and o = lR"-' x lR+,necessary and sufficient conditions for the Agemi and Shirota termed this problem, in contrast, as strongly L,-well posed.

V.Ya. Ivrii

202

L,-well-posedness with a loss in smoothness by q were given in terms of the Green's function of the problem (2.12)-(2.13), and it was shown that for q = 0 the cone of dependence of the mixed problem coincides with the intersection of X: and the cone of dependence of the Cauchy problem (for q 2 1 this result is, in general, not true). the weak L,-well posedness of Finally, it was shown that if w = R"-'x E+, (2.1)-(2.3) implies the weak L,-well-posedness of the problem obtained by freezing the coefficients at any point X E S:, and, in particular, implies (2.19). This enables us to obtain conditions that are necessary for the weak L,-wellposedness, which in many cases turn out to be sufficient as well. Let us examine in some detail the strictly hyperbolic operators of order m = 2. Let o = R"-'x R+ and S be non-characteristic and non-spatially similar. Then for x E S , 5'' E IR"-' and 5, E C \ R , the polynomial p(x, to,5", 5.) has two roots t,, = A*(x, to,5") one with positive and another with negative imaginary parts. Then p = 1, and l(x, 5 0 , 5") = b(x, 5 0 , 5", A'(x,

50,

<"I),

where b is the principal symbol of B. We recall that a necessary condition for the problem (2.1)-(2.3) to be C"-well posed is that the Lopatinskij condition holds. Namely, l(x, to,5") # 0 for x E S:, 5" E R"-'and 5, E C-\R (we assume that (2.19) holds, that is, that l(x, - i, 0) # 0).A necessary and sufficient condition for the problem to be strongly L,-well posed is the uniform Lopatinskij condition, namely, l(x, lo,5") # 0 for x E S:, 5" E R"-'and 5, E CC-. This condition is fulfilled, in particular, for the Dirichlet boundary condition with B = I; it is assumed that the boundary condition is normal.

Theorem 14 (Miyatake [1975]). Let

B=

2 bj(x)Dj+ b'(x),

then either c

5" E IRn-l,

*

Then the problem (2.1)-(2.3) is weakly L,-well posed.

We note that if (2.35) is violated at some point X E S:, we can impose other technical conditions that guarantee weak L,-well-posedness (see Miyatake [1975], where (2.34) is given in various other forms). It turns out that if conditions (2.34) and (2.35) or some of the above-mentioned technical conditions are satisfied, then the conoid of dependence of the mixed problem (2.1)-(2.3) will be the intersection of X: and the conoid of dependence of the Cauchy problem. We also note that (2.34) implies that the Lopatinskij determinant 1 does not vanish at hyperbolic points of T*S:\O but may vanish at tangent points and at elliptic points (the classification of points of T*S\O is given in the next section). We have already discussed the question of the loss in smoothness in the case of weak L,-well-posedness. But in some cases the actual loss is smaller. Let P = 0 be the wave operator and X = IR x o.Consider the maps: supp v c s,+-+ u E C"(X): Pu = 0,

u E C"(S),

If w is strictly concave (that is, if o is an "exterior" domain with a non-zero Gaussian curvature of boundary), then A is an Airy type pseudodifferential operator of class L&3 (Melrose [1978]). This is a classical pseudodifferential operator outside the tangent set 9 and the inverse operator A-' is an Airy type What is more, A-' is a classical pseudodifferential operator of class L&,. pseudodifferential operator of order (- 1) outside 59. Therefore for a solution of the problem

Pu=O,

where b,, = 1. (i) I f P and B have constant coefficients, then a necessary and sufficient condition for the problem (2.1)-(2.3) to be weakly L,-well posed is the inequality

E ST+,

= 0 or Im b, = . . = Im bn-' = 0 in some neighbourhood of X, where c = Re b, - PEnE, . P;in.

j=O

vx

203

11. Linear Hyperbolic Equations

5,

E

c-

(2.341

(of course, in this case 1 and A+ do not depend on x). (ii) I f P and B have variable coefficients, then (2.34) is a necessary condition for the problem to be weakly L,-well posed. (iii) Assume that the coefficients of P and B are constant for ( X I ( > R and that (2.34) as well as the following additional condition hold: rf X E s;, c(X) = 0, (2.35)

a"l an s = w

suppucx;,

the estimate

[Iu;

x;

llkfl

-t

st-

Ilk+'

cllw;

st-

llk+l/3

holds for all k E Z?. This estimate cannot be improved upon. 1.7. C"-well Posed Mixed Problem for Strictly Hyperbolic Equations of Second Order. Let us begin with the results of Ikawa and Soga for the wave equation. Let o c IR"", ao be compact and v the inward unit normal to do.Let P = 0 and B = &, bjDj + b', where bjvj = 1 and bj are real. A necessary and sufficient condition for strong (weak) L,-well-posedness is the inequality b, P < 0 (b, + fi < 0), where = bj' - 1)'I2. By Theorem 10, a necessary condition for the C"-well-posedness is that b, < 1 for n 2 3 and b, < (1 + fiZ)'" for n = 2.

+

(cy=l

204

205

11. Linear Hyperbolic Equations

V.Ya. Ivrii

A domain X is said to be bicharacteristically strictly convex (concave)if

Theorem 15 (Ikawa [198l]). Let n = 3. Let w be a strictly concave domain. For the mixed problem (2.1)-(2.3) to be C"-well posed and to have a finite propagation speed, it is necessary and sufficient that bo < 1.

X,

that is, if

Now let n = 2. Let bo = 0, bj be independent of xo, z = z(s) = blv, - bZvl, A = A(s) = blvl + bzvz and x r = xr(s),where s is a parameter on am. We drop the earlier assumption that 1E 1.

"41

= 0,

{ p , x,} = 0

H:x,

>< 0,

2 0. The behaviour of bicharacteristics in bicharacteristically

ax,

convex and concave domains is depicted in Fig. 2 and Fig. 3, respectively.

+

Theorem 16 (Ikawa [198l]). Let 1 1. Let o be a strictly concave domain. For the problem (2.1)-(2.3) to be C"-well posed, it is necessary and suficient that z and dz/dS do not both vanish simultaneously.

xnf

Theorem 17 (Soga [1980]). For the problem (2.1)-(2.3) to be C"-well posed, it is necessary that at no point of aw does 1(s) change its sign. Theorem 18 (Soga [1980]). Assume that z(s) does not vanish and I(s) does not change its sign on aw. If [1(~)]'/~ is a smooth function, then the mixed problem (2.1)-(2.3) is C"-well posed but does not have, in general, a finite propagation speed.

x

We now turn our attention to far more general results of Eskin. Let P be a strictly x,-hyperbolic operator of second order and B a differential operator of an arbitrary order M . We examine the mixed problem in a somewhat different form: Pu = 0, supp u c x;, Buls = cp. (2.36)

S Fig. 2

Theorem 19 (Eskin [1979180, 19811). If the domain X is bicharacteristically strictly concave and i f conditions (2.26), (2.38) and (2.39) hold, then the mixed problem is C"-well posed. Furthermore, if cp E Hfo;M"(S) and cp = 0 for xo < T, then u E H",,(X).

Suppose that S is non-characteristic and non-spatially similar. Then locally, in a suitable coordinate system,

X

= {xn 2 O},

P(X,

5 ) = 5.' - q(x, 5 0 , t"),

Theorem 20 (Eskin [1979/80]). Let the domain X be bicharacteristically strictly convex. Let conditions (2.26'), (2.38) and (2.39) be satisfied. Assume that in a neighbourhood of any point p E 9 n { I = 0} the condition

where q = 0 * qto # 0.

(2.37)

Let T*S\O 3 %? = {q > 0},6 = {q < 0 } and 9 = {q = 0} be, respectively, a hyperbolic, elliptic and tangent set (or set of hyperbolic, elliptic and tangent points, respectively). Let l(x, 5 0 , 5") = b(x, 50, t",A+(%

50,

where Im

l(x, to,5") # 0 for Im to< 0

VP = ( x , 5 0 9 5 ' 7 E 9 n { I t o l

(2.26) i

50,

t")= 0,

(x, to,(?)

E2

u8

l(x, 5 0 , 5") = 0, ( x , t o , 5") E 9 =>

ai -(x,

50,t") # 0,

(2.38)

a50

ab ag. (x, t o , t")# 0.

(2.39)

+ 157 = 1}

(2.40)

is fulfilled; here 5, = y(p) is a smooth root of the polynomial b(p, 5,) such that y ( p ) = 0. Then the mixed problem (2.36) is C"-well posed. Furhermore, if cp E HfOYM"(S), cp = 0 for xo < T and 0 < t - T is sufficiently small, then u E Hfoc(x;).

2 0 for

holds and that

I(x,

IRe Y(P)l In-llRe Y(P)l sign 4ro(P)G CIIm Y(P)I2

57)

be the Lopatinskij determinant and ' 1 = &-,/ Im to< 0. Assume that the Lopatinskij condition

Fig. 3

Theorem 21 (Eskin [19791803). Let the domain X be bicharacteristically strictly convex. Let conditions (2.26), (2.38) and (2.39) be satisfied. Assume that { y , q} (p) # 0 at any point p E 9 n (1 = 0 } in a neighbourhood of which (2.40) is not satisfied. Then the mixed problem (2.36) is C"-well posed. Furthermore, if cp E H/o;M+L+l(S),cp = 0 for xo < T and 0 < t - T is sufficiently small, then u E Hfo,(X;), where

V.Ya. Ivrii

206

and the maximum is taken over all the points of which the condition (2.40)fails to hold.

11. Linear Hyperbolic Equations i j E 9 n( 1

These assumptions ensure strong L,-well-posedness for the mixed problem in the quadrant ( x o > 0, x , > O } that is obtained from the original problem by removing all differentiation operators other than Do and 0,and the variables x l , . . .,x , - ~are regarded as parameters. Under these assumptions, we can construct P-characteristic surfaces Ak that contain So and are orthogonal to the vectors (&(x),0, . ..,0, 1) at points of So. When k 6 M + , these surfaces lie in X i for xo 2 0 and divide X i into M + 1 sectors T,(k = 0, 1, . . .,M + ) (see Fig. 4).

= 0 } in a neighbourhood

On the other hand, we have

Theorem 22 (Eskin [1979/80]). Let the domain X be bicharacteristically strictly convex. Assume that there exists a sequence of points p k € 9 n { I toI I ["I = 1>, p k + p, such that fqc,,(p) > 0, fRe Y ( p k ) < 0,

+

(Imr(pk))2 In

IRe

y(pk)l

(Re Y ( p k ) ) - ' *

as

+

201

+

O0

and

S

I {Y, 4 ) ( p k ) l

< CIRe

r(pk)ld,

> O.

Then the mixed problem (2.36) is not C"-well posed. We conclude the present section with the remark that for non-strictly hyperbolic equations lower terms also figure in the well-posedness conditions. Thus, if x = IR' x E+, P = D,Z - Dl - x ~ D :+ cD1,

c E (c,

(2.41) k

and B = D, + aD,, then the mixed problem is C-well posed for Re a < 0 and for Re a = 0, c E IR; in the remaining cases it is not well posed. When B = I , a necessary and sufficient condition for the well-posedness is that c E IR (Zajtseva [1983]). Similar, though more complicated, well-posedness conditions have been obtained by Zajtseva [1987] for the operator

+ cD1,

x, Fig. 4

(2.42)

i

1.8. Mixed Problem in the Classes of Analytic Functions. We formulate Duff's theorem 119593, which is an analogue of the Cauchy-Kovalevskaya theorem for the Cauchy problem. Let o = ( x , > 0 } and T = 0. Let P and Bk (k = 1, .. .,p) be scalar differential operators of order m and mk, respectively, whose coefficients are analytic in a neighbourhood of 0. Assume that

1

P

= 0: - Dl - x$D,

n

c E C.

1

1

m

P(X, t o ,

where Aj(x) are real, Aj # M + + 1. We also assume that

jlk

0, t n ) =

j=1

( t o - Aj(x)tn),

(2.43)

for j # k, Aj c 0 for j iM + , and Aj > 0 for j

>

Theorem 23. Let conditions (2.43) and (2.44) be satisfied. Let the functions f, (pk (k = 1, . . . , p) be analytic in a neighbourhood of 0. Then in the neighbourhood of 0 there exists a function u E L, satisfying (2.1) in X i in the sense of the theory of generalized functions and the boundary conditions (2.2) on S,'\So and the initial conditions (2.3) on Xo\So. Moreover, its restrictions to can be continued to functions that are analytic in a neighbourhood of Further, iff, gj and (pk satisfy the compatibility conditions of order s, then 0. u E Cs+m-2and the solution is unique. NoG that the compatibility conditions of infinite order enable us to determine the analytic functions (pk uniquely, and in this case the solution u can be obtained by the Cauchy-Kovalevskaya theorem. We also note that if the mixed problem (2.1)-(2.3) is C"-well posed (in general, the coefficients of P and Bk are no longer analytic), conditions (2.43) and (2.44) hold, and the functions f, g j and (pk belong to C", then in a neighbourhood of So the solution is infinitely smooth in each of the sectors up to its boundary and belongs to Cs+mn-2 iff, g, and (pk satisfy the compatibility conditions of order s. gj ( j = 0, . . .,m - 1) and

1 1

E

p = M', the boundary-value problem is normal, and the polynomials

bk(x,to,0, 5.) are linearly independent (mod p+(x,to,[,,)),

I

where (2.44)

0 2.

Propagation of C"-Singularities

In this section we examine the propagation of Cm-wavefronts of solutions to the boundary-value problems for hyperbolic equations and systems. However, we shall not discuss the well-posedness question for these problems.

208

V.Ya. Ivrii

11. Linear Hyperbolic Equations

2.1. The Wave Fronts. Let X = X x IR = IR" x R' 3 (xo, ..., x,-~, x,), S = X ' x 0 and 8 = X\S. Let 1 : T*XIs --f T*S be the natural map. We denote by Lm(X')the space of classical pseudodifferential operators of order m on X and set L""(X) = Cm(IR+,Lm(X')).We shall discuss only operators with compact supports. We introduce the function space

where (., .>+ denotes the scalar product in L2(lR+,ad), the Hermitian product in (EL and yu = (u, D,u, ...,On m-1 U ) I ~ , = ~ are the Cauchy data of u on S, D = md, and T ( p )is a D x D smooth Hermitian matrix, with rank T(p*)= D, that can be expressed in an obvious manner in terms of ak(p). Let D , and d , denote, respectively, the number of positive eigenvalues of the matrices T ( p * ) and ao(p*). Then D, = md/2 for even m and D, = (m - l)d/2 + d , for odd m. Let B = (Bo,. . . ,Bm-l)be the boundary operator and Bk E L&,,(s). Let bk, b = (bo,. . ., bm-l),the principal symbols of Bk and B, respectively, be D, x d and D, x D matrices. We assume that ( a ,

%"(X) = {u, D i u E L2(E+,H S - ' ( X ) ) Vj E Z '}.

Definition 5. By the boundary wave front (boundary wave front of order s) of we mean that set WFb(u)c T*S\O (WFi(u) c T*S\O): a distribution u E 9'(8) p $ WFb(u) ( p # WFi(u)) if we can find an operator a(x, D')E Lo'(X) which is elliptic at the point p and is such that au E C m ( X )(au E %'(X)). Unfortunately, the invariance of such a definition under a change of the variable x, is very doubtful. However, in many cases this invariance can be established for solutions of systems with smooth right-hand sides. Another definition of the boundary wave front can be found in Melrose [1981]. We also introduce the complete wave fronts

rank b(p*) = D+,

- Tb) + P(P)b(P)+ b*(p)P*(p) is a non-negative definite matrix in the neighbourhood of p*. We already know that for u E 9'(8)

= WF'(U)u z - ~WF~(U).

Thus "gluing" of T * X on S appears already in the definition: two points 8 and 8' E T*XIs are identified if ztl = 10'. 2.2. Propagation of Singularities of Solutions to Dissipative Boundary-Value Problems for Symmetric Hyperbolic Systems. We consider a d x d matrix operator m P= AkD,"-k, (2.45)

1

k=O

, t1 1

where A , E L$(X). Here the notation " d l x d2" stands for the size, and "d x d" will be written as "d". Let ak and

1

m

p=

1

akt,"-k

k=O

be the principal symbols of A , and P, respectively. Let T*S\O 3 p* be the point in a neighbourhood of which the discussions will be carried out. We assume that

(that is, S is non-characteristic at p*), and that

f

P* = P

(2.47)

(that is, the system is principally symmetric). Then for all u, v E Y(IR+, ad)and all p E T*S\O lying in a neighbourhood of p*, we have ( P h D,)u, v>+ - (4P ( P , D,)o>+

I

(2.46)

rank ao(p*) = d

=

i ( T ( p W , YO),,

(2.48)

( T ( p ) U , U>D < 0 VU E Ker b(p) (2.49) in some neighbourhood of p*, that is, there the boundary operator B is nondegenerate and is dissipative. Then there exists a D x D, smooth matrix P ( p ) such that

WFf(u) = WF(u) u Z-lWFb(U), WF;

209

WFs(u) c WF"-"(Pu) u Z, where C = Char P = (9 E T * X , g = det p = 0} is the characteristic set of operator P ; outside this set P is elliptic. Let E ( p ) denote the solution space of the system ( p , DJu = 0 on ,'RI the solutions decaying as x, +a. Definition 6. The problem ( P , B), satisfying conditions (2.46) and (2.47), is elliptic at a point p E T*S\O if p 4 zZ and the map b(p)y:E ( p ) + Cpis a bijection. Here p = dim E ; p = D, if (2.47) holds and p I$ zZ. If Cbdenotes the set of points where the problem (P, B) is non-elliptic, Zb is known as a characteristic set of the problem (P, B). By the standard methods of the theory of elliptic boundary-value problems, we can easily show that if (2.46) and (2.48) hold and u E C m ( R + ,9'(X)), then c WF:-'"(pu) U WF"i'2(Byu) U Cb. (2.50) WF~(U) = zk ( k = 1, . . . , N ) be all the real and distinct roots of the Finally, let characteristic equation g(p*, t,) = 0. Theorem 24 (Ivrii [1986]). Assume that conditions (2.46), (2.47), (2.48) and (2.49) hold. Let (pk E C m ( E + x (T*X'\O)), cp E Cm(T*S\O) be red, positive homogeneous functions in (' of degree 0 such that cpl = * * * = cpN = cp on T*S, cp(p*) = 0 and ({P?V k } ( p * , z k ) v ? v> (2.51) k = 1, .. . , N , v v E Ker p(p*, zk)\o,

<,

'

Re({p, q } ( p * ,

Dn)u,

O>+

-

Re i(B(p*){b,(P}(p*)Yv?y v > D

V v E E ( p * ) n Ker b(p*)y\O.

'

(2.52)

V.Ya. Ivrii

210

If

11. Linear Hyperbolic Equations

Remark 3. Assume that (2.46)-(2.48) hold. Let dim Ker p(p*, zk) = 1 for all k = 1, .. . , N . Let rk denote the multiplicity of zk, a root of the characteristic equation. Assume that Ek is the space of generalized-exponential solutions on ]R+ of the system p ( p * , D,)v = 0 that correspond to the characteristic root zk and to the following growth condition as x,, + +a:

u E Cm(R+,9 ' ( X ' ) )and

p*

4 WFbS-m+l(Pu) u WF"+"'(Byu),

(2.53)

WF"(u)n{cpk < 0 ) n wk= @ Vk = 1, ..., N ,

(2.54)

WFi(u) n (cp < 0 } n W = (ZI,

(2.55)

211

u = 0(x2-"")

where w k and W are sufficiently small neighbourhoods of the points (p*, zk) and p* in T * X and T*S, respectively, then p* 4 WFi(u). The same assertions remain valid for the wave fronts of infinite order also.

for even rk, (2.58)

u = O ( X ~ - ~ ) / ~for ) odd r,

and for

In analogy with the established terminology of 5 3.1, it is natural to call P, satisfying (2.47) and (2.51) with a given k, a microhyperbolic operator at the point ( p * , zk) in the direction Hqk(p*) and the problem (P, B), satisfying (2.46)-(2.49), (2.51) and (2.52),microhyperbolic at the point p* in the multi-direction

while u = O ( X ~ - " / ~ ) for odd rk and ak > 0,

(2.58")

where (pk satisfies (2.51). We set = El 0 E , 0 ... 0 EN 0 E(p*). If the strict dissipativity condition is satisfied, then Remark 1. If the non-degeneracy condition E(p*) n Ker b(p*)y = 0,

n Ker b(p*)y = 0.

(2.56)

Conversely, if (2.46)-(2.48) hold and

is satisfied, then (2.52) of Theorem 24 is empty and then the propagation of singularities does not depend on any concrete choice of the boundary operator B (though for some B there may be abnormally poor propagation of singularities). This theorem has been proved roughly under such an assumption by Ivrii [1979d, e, 1980al. If p* 4 zC,then the study of the propagation of singularities of solutions to boundary-value problems reduces to that of propagation of singularities for a certain operator W E L!,2(S).Its characteristic set is Z,,. The Rayleigh waves are a phenomenon of this very type in which W is a symmetric operator having characteristics of constant multiplicity 2. In the general case, there is a complicated combination of the singularity propagation. A detailed examination in the particular case of the wave equation can be found in 5 2.6.

i 1

I

Remark 2. If the condition of strict dissipativity

(2.57) holds at the point p*, then (2.56) also holds. What is more, in this case lower smoothness is required from the boundary data if (2.53) is replaced by p*

# WF~-'"+'(Pu)u WFs(Byu),

yielding, in addition, that p* $

u

k=O

(2.53')

1

1 k

!

m

WF"k(D,k~I,).

(2.59)

dim Ker p ( p * , z k ) = 1 for all k

=

1, ..., N ,

then the problem satisfying (2.59) can be reduced to a strictly dissipative problem for a symmetric system of first order.

2.3. The Geometry of the Propagation of Singularities. Assuming that the non-degeneracy condition (2.48) holds, we list a number of concrete results. We start with the assumption that P has simple characteristics. Analogous results also hold for operators with characteristics of constant multiplicity, but in this case g is not the characteristic polynomial of P but a polynomial of principal type whose zeros coincide with those of the characteristic polynomial. We shall assume throughout, without stating it each time, that the right-hand side and the boundary data are infinitely smooth in the desired domain I/ c T*X\O of the phase space, that is, it does not intersect WFf(Pu) or z-~WF(BUI,). We immediately find that if the only bicharacteristics are those transversal to S , then the wave fronts of the solutions arrive on S along the incoming bicharacteristics and move away from S along the outgoing ones. In this way they are propagated along bicharacteristic billiards having branches, in general, on S. We can easily drop the transversal bicharacteristics from the microlocal analysis, which indeed we shall be doing for the convenience in formulation, without stating it each time. Suppose that over the point p* E T*S\O there lies exactly one tangent bicharacteristic, that is, the characteristic polynomial g ( p , t,,)has for p = p* exactly one multiple real root 5, = zl. Let 8* = ( p * , zl). We recall that since the characteristics of P are simple, it follows that g,(8*) # 0. We also assume that

V.Ya. Ivrik

11. Linear Hyperbolic Equations

z1 is a root of multiplicity 2 of the characteristic polynomial, that is, g(O*) = (Hxng)(fl*) = 0, Hing(O*)# 0. To be definite, we assume that H:ng(O*) > 0. If the domain X is bicharacteristically strictly convex at 0*, that is, if (H,2xn)(O*)< 0, then in the neighbourhood of z-lp* the wave fronts are propagated along bicharacteristic billiards of g as well as along boundary bicharacteristics, namely, along bicharacteristics, defined on T*S\O, of the boundary symbol gb(p) = g ( p , y(p)), where 5, = y ( p ) is a root of the polynomial { g , x,} { p , t,} such that y ( p * ) = zl.A boundary bicharacteristic is the z-projection of a curve, lying in T*XI,, along which

of singularities in bicharacteristically concave domains near the points of tangency. This is due to the requirement that the function rp should not change when the bicharacteristic is reflected from S , that is, that this function should not depend on 5,. True, some implicit and weak dependence was assumed for N 2 2 because we had considered a collection of functions rpk. We now drop this requirement and impose a more natural condition that rp does not decrease when the bicharacteristic is reflected from S. However, we are forced to examine a considerably narrow class of systems and only strictly dissipative boundaryvalue problems for them. We assume that the following conditions are satisfied. a) In a neighbourhood of L-'p* in T*Xls, the characteristic symbol can be factorized

212

dP - = Hg(P) + XHXn(P), dt

(2.60)

N

g = ( 9 , xn} = 0;

this curve will also be referred to as a boundary bicharacteristic. Note that x = (Hixn)(H:ng)-l.A boundary bicharacteristic will be termed a C"-boundary bicharacteristic if H i x , < 0 on it. Such bicharacteristics are limits of those bicharacteristic billiards which press against the boundary more and more tightly. In particular, for the wave equation bicharacteristic billiards are the usual billiard trajectories, while boundary bicharacteristics are the geodesic boundaries. If X is bicharacteristically concave, that is, if Hfx, 2 0, then C"-singularities are, perhaps, not propagated along boundary bicharacteristics (see the next two sections). Therefore, in this case Theorem 24 does not furnish a precise description of the propagation of C"-singularities. If there are several tangent bicharacteristics over p * , each satisfying the assumptions made above, and if X is a bicharacteristically strictly convex domain relative to each of them, then there arise several boundary symbols g b , & and there may be different types of propagation depending on their tion". The simplest case is the one where all the boundary symbols g b , &

~~~~~~~~~-

( k = 1,. . ., N) are in involution and the vector fields HBb.l, ..., Hgb." ('7

U

are 85 linearly independent. Then there arises the propagation of singularities along N-dimensional bicharacteristic sheets that are constructed in accordance with the aggregate of boundary symbols in exactly the same way as was done in 0 3.2, Chap. 1, in accordance with the aggregate of factors. Ivrii [1979d, e, 1980al has also analysed the propagation of singularities for the case where one tangent bicharacteristic lies over p* and the root 5, = z1 of the characteristic polynomial is of multiplicity 3. The propagation of singularities of solutions to boundary-value problems for systems with characteristics of variable multiplicity has been studied to a lesser extent, and for concrete results the reader is referred to Ivrii [1979d, e, 1980al. 2.4. The Propagation of Singularities of Solutions to Strictly Dissipative Boundary-Value Problems for Symmetric Hyperbolic Systems. Theorem 24, though fairly general, does not enable us to study conclusively the propagation

213

g =e

n

k=l

(2.61)

h?,

where e is an elliptic symbol and hk are real polynomials in 5, of degree mk = 1, 2 with the stipulation that hj and h,, j # k, do not both vanish simultaneously and that hk = 0 * {hk,xo} > 0. b) In a neighbourhood of p* there exists a real, positive homogeneous symbol q of degree 0 such that q ( p * ) = 0, (q, xo} ( p * ) # 0 and for q ( p ) 3 0 all the roots 5, of hk(p,5,) are real, whereas for q ( p ) < 0, mk = 2, all the roots of hk(p,5,) are non-real. Then both the roots of h,(p, 5,) coincide if q ( p ) = 0 and mk = 2. Condition b) clearly rules out the presence of bicharacteristic boundary sheets of dimensuion greater than 1. Theorem 25 (Ivrii [1980b]). Suppose that conditions a) and b), conditions b) of Theorem 37, Chap. 1, and conditions (2.46), (2.48) and (2.49) hold in a neighbourhood of z - l p * . Let rp E C"(T*X\O) be a real positive homogeneous function of degree 0 that satisfies (1.38) (of Theorem 37, Chap. 1) and the conditions hk(O')= hj(O")= 0, {hk, x n } ( e ' ) 2 0 2 {hj, xn)(e), let

=

q=O

in the neighbourhood of

z-lp*.

p*

z e rp(e')2 rp(e);

(2.63)

on z - l p * n { g = o } If u E Cm(]R+,W ( X ' ) )and

# WFi(Pu) u WF"(Byu),

WF;(u)n { g = 0} n {rp < 0} n W =

0,

where W is a neighbourhood of z-lp* in T * X , then p*

(2.62)

4 WFE(u) u

u WF"-k(D,k~I,). m

k=O

(2.64)

215

V.Ya. Ivrii

11. Linear Hyperbolic Equations

We do not know whether this theorem remains valid for wave fronts of infinite order. However, applying it to concrete cases, we obtain results for wave fronts of finite as well as infinite order. For N = 1, we can easily obtain a global version of this theorem that is also true for infinite-order wave fronts. Theorem 25 is applicable to strictly hyperbolic operators of second order. Namely, to operators of the form

If E C and z-'zV c V is a neighbourhood of we denote by K'(e, V) the union of all Cm-raysthat issue from 8 in the direction of increasing (decreasing) xo and lie in It can be shown that K'(8, V ) have the properties (1.43)-(1.46). Further, if the domain X is bicharacteristically concave or its boundary S is bicharacteristically non-degenerate, that is, if the set

214

P

=

-D:

+ Q,

Q E L2'(X),

e

e,

Qm = { p E 3, ( H , k ~ , ) ( p=) 0

Vk E Z+}

(2.65)

that are such that a) q, the principal symbol of Q, is real and q 2 0 =. E { q , xo} > 0, E = & 1 fixed, if the boundary operator B = yo& y', y j E Lj(S), satisfies (2.59). This condition in this case is equivalent to the two uniform Lopatinskij-type conditions

+

(2.66)

is empty or, more generally, each point p E 9" is an interior point of some bicharacteristic of g lying in Q (in particular, if g is an analytic function), then through each point 8 E C there passes a unique maximally extended C"-ray y(@. On this ray the exceptional points (the points where the bicharacteritic is reflected from S and the points where transition takes place from the bicharacteristic to the C"-boundary bicharacteristic and vice-versa) constitute a locally finite set. In the general case, however, COD-rayshave branching on Qm, and together with them the wave fronts too have branching. Theorem 26. Let the above-mentioned assumptions be fulfilled. Let z-'zV c V be an open set, V n Z C C and E C n V. I f

IZ > 0 * b(8 + AiHJ # 0,

\

I

(WF,(Pu) u z-'WF(Bul,)) n K - ( & V) = @,

where p and b are principal symbols of P and B, respectively. As we have already noted, such problems lead to strictly dissipative boundary-value problems for first-order systems satisfying the conditions (2.46)-(2.48) and condition b) of Theorem 37, Chap. 1. When y o = 1 and y' E Lo(S),such a reduction is not possible in the neighbourhood of points of tangency. However, in this case Melrose and Sjostrand [1978,1982] have obtained similar results, as we describe below. 2.5. The Geometry of the Propagation of Singularities (the Concluding Part). We assume in this section that the operators P and B satisfy the hypotheses of Theorem 25. We start by describing the propagation of singularities when the characteristic polynomial is of the form g = -5: + q, where the symbol q = q(x, <') satisfies condition a) of Theorem 25. Under these assumptions WF,(u) c WF,(Pu) n WF(Bu(,)u % u 3,

e

(2.67)

(2.68)

where, we recall, 2,8,and Q are, respectively, hyperbolic, elliptic and tangent sets. We shall consider the restriction of complete sets. We shall examine the restriction of the complete wave fronts to the manifold C = { g = 0}, identifying the equivalent points of Zls. Definition 7. By a C"-ray we mean a Lipschitz curve y = (0 = e ( t ) } , with parameter t = xo,lying in C that in the neighbourhood of almost all its points is either a bicharacteristic or a C"-boundary bicharacteristic.

WF,(u)n

K ( 8 , V ) n d V = 0,

(2.69) (2.70)

then

WF,(u)nK-(8, V ) =

a.

(2.71)

We now assume that the hypotheses of Theorem 25 hold and that the decomposition (2.61) holds with N 2 2 in a neighbourhood of i-'p* in T * X . Assume that the domain X is bicharacteristically strictly concave with respect to h, ( k = 1, . .. , M ) and M = N - 1 or else M < N - 1 and X is bicharacteristically strictly convex with respect to h, ( k = M 1, .. .,N ) . In the first case, the singularities are propagated along billiards that consist of bicharacteristics h, ( k = 1, . .., M - 1) and Coo-raysh,. In the second case, singularities are propagated along billiards consisting of bicharacteristics h, ( k = 1, . . ., N ) and boundary bicharacteristics, that is, bicharacteristics of thC boundary symbol. For M = 0 Theorem 24 can also be applied to the present situation. Therefore in this case (2.57) can be replaced by (2.56). We observe that for bicharacteristically strictly concave domains there is, in addition to the implicit study of propagation of singularities, an explicit construction of a parametrix (Eskin [1976], Melrose [1975], Taylor [1976a, b, 19791) that is based on Ludwig's constructions and uses Airy functions as the components of oscillatory integrals. In bicharacteristically strictly convex domains there is "semi-explicit" construction (Eskin [19771). There are also several works devoted to the reflection and refraction problems that fit into the given scheme.

+

216

11. Linear Hyperbolic Equations

V.Ya. Ivrii

2.6. The Propagation of Singularities of Solutions to Non-classical Problems. In this section we shall examine the propagation of singularities of solutions to the non-classical problems. That is, to these boundary-value problems for strictly hyperbolic equations of second order that do not satisfy the conditions (2.66) and (2.67). Let P be an operator of the form (2.65), B = 0, M , M E L'(S) and p its principal symbol. Then the sets of points where (2.66) and (2.67) are violated + p = 0} c 2 u B and A+\A', where are, respectively, the sets A - = {(E&

+

A+ = { i f i + p = O} c 6 u 4 A' = { q = p = O}

=A

-nQ =2 n B

(we identify the points of T*XsIs with their I-projections). Evidently, A- U A + c A = (4 = p2}. If p E 6 and p 4 WFb(Pu),we can easily show that p 4 W F ( ( D , - a)uls),where L'(S) 3 a is an operator with principal symbol a, = i f i in a neighbourhood of p. If, in addition, p 4 WF(BuI,), then p 4 WF(au), where ZI = uIs and a = i(a + M ) E L'(S) is an operator with principal symbol + ip and with characteristic manifold A+\A'. It is also easy to show that if p E 2,p 4 WFb(Pu)and K - ( p , V )n W F ( u )= 4, where V is a sufficiently small nighbourhood of the set z-lpnZ, then p 4 W F ( ( D , - fl)uls),where L'(S) 3 fl is an operator with principal symbol fl, = - E& in the neighbourhood of p. If, in addition, p 4 WF(Bul,),then p 4 WF(bu), where b = fl M is an operator with principal symbol -E& + p and characteristic manifold A-\A'. Thus outside the tangent set 92 the study of the propagation of singularities of solutions to boundary-value problems reduces to that of solutions to pseudodifferential equations on S. On the other hand, B satisfies (2.66) and (2.67) outside A + u A - . Therefore our main interest must lie in the propagation of singularities only near the set A'. It is evident that there a and b are operators of principal type.

-6

+

Definition 8. By a generalized Cm-ray we mean a Lipschitz curve y = (6' = 8 ( t ) } c Z u A + u A - , with parameter t = x,,,that in the neighbourhood of almost all its points 8 E Z u A-\Ao is either a C"-ray or a bicharacteristic of b and in the neighbourhood of each of its points 6' E A+\Ao it is a bicharacteristic do of a, and, finally, if at each of its points 8 E A' it satisfies the equation - = dt ?fqQ r z 0. Let E V n (Z u A') and T*X\O 3 V be an open set such that z-'zV c K We now denote by K*(g, V ) the union of all generalized Cm-raysissuing from in the direction of increasing (decreasing) xo and lying in It can be shown that K'(8, V )have the properties (1.43)-(1.46) if, at least, one of the following conditions is satisfied: o # O on A' (2.72) Rep=oImp,

217

(then the operators a and b are elliptic near A'); (4, Rep} > 0,

~ { qImp} , < 0 on A'

(2.73)

(then a and b are subelliptic operators near A'); Imp-0,

{q,Rep} > O

on A'

(2.74)

(then the operator a is elliptic and b is an operator of real principal type near A', A is a C"-manifold and A - is a Cm-manifoldwith edge A'; the bicharacteristics of b, parametrized by xo, start from A'); E R e p G O if q + ( I m p ) ' = O

(2.75)

(then the operator b is elliptic). The inclusion (2.67) remains valid if one of the conditions (2.72) to (2.74) holds. In the general case the inclusion WFb(U) c

wFb(PU) U WF(BU1,)U 2 u 9 u A +

(2.76)

holds. Theorem 27. Assume that (2.65) and one of the conditions (2.72)-(2.75) hold. Let V be a sufficiently close neighbourhood of V n A' such that z-'zV c V, 6' E V. Then (2.69) and (2.70) imply (2.71). Under assumptions (2.72)-(2.74) this theorem has been established by Ivrii [1981] while under (2.75) by Melrose [1975]. Thus under assumptiions (2.72) and (2.73) the propagation of singularities remains classical, but under (2.74) there appears additional propagation of singularities along bicharacteristics of b and additional branching of singularities at points of A-. If (2.75) is assumed to hold, there appears additional propagation of singularities along bicharacteristics of a and branching at points of A', though, possibly, not at all points. We note that under (2.75) a does not always have a real principal symbol a , . Therefore by a bicharacteristic in this case we mean the curve along which a = 0, where z( p ) is a non-vanishing complex-valued function.

4 3. The Propagation of Analytic Singularities We devote a separate section to the study of propagation of analytic singularities because in the case of the mixed problem it differs significantly from the propagation of C"-singularities. At the same time, we shall use the notation of the preceding section. We basically follow the works of Sjostrand [1980a, b, c, 19811.

218

V.Ya. Ivrii

11. Linear Hyperbolic Equations

3.1. The General Theory. Let g , containing 0, be an open domain in IR"+'. Put X = 2 n (x, 2 0},S = 2 n {x, = 0}, and 8 = X\S. Let d ( X ) and d ( S ) denote, respectively, the space of functions that are real-analytic in X and on S. We introduce the analytic wave fronts

If V is a sufficiently small neighbourhood of p* in T*S\O and 0 < E is sufficiently small, and if p E V, t E (0, E ) and z is a root of the characteristic polynomial g ( p ito, z) that is close to zk, then Im(z - itCk)# 0 ( k = 1, . ..,M ) , by virtue of (2.31). Now suppose that R ' ( p , t ) is the set of roots z of this polynomial that are close to zk with some k = 1, . . ., M and with Im(z - itCk)> 0 and R"(p, t ) is the set of roots z of the same polynomial that are close to zk with some k = M + 1, . . ., N and with Im zk > 0. We denote by E ( p , t ) the space spanned over generalized exponential solutions of the system p ( p ito, 0,)v = 0 corresponding to R'(p, t ) u R"(p, t), the set of roots of the characteristic polynomial.

WF,(u) = WF,(uIj) c T*2?\O, wFb,(U)

=

u

WF,(D,kuI,) c T*S\O,

WF,-,(u) = WF,(U) u l-'wFb,(U). Let P be a d x d matrix differential operator of order m with coefficients in d ( X ) . Put B = (El, .. ., EN),where Bk are 1 x d matrix differential operators of order mk with coefficients in d ( S ) . Let p, bk and b be the principal symbol of P , Bk and B, respectively, and let g = det p be the characteristic symbol of P . We assume that S is a non-characteristic surface. Definition 9. The operator P is said to be elliptic at a point 0 E T*X\O if g(0) # 0. The problem (P, B) is elliptic at a point p E T*S\O if P is elliptic on z-'p and the map +

c'

(2.77)

is a bijection; in particular it is necessary that dim(Ker p ( p , 0,) n cp(lR+)= p.

Theorem 28. (i) If Pu E d(8), then WF,(u) c C. (ii) If Pu E d ( X )and yBu E d ( S ) , then Cb.

Definition 10. The operator P is said to be microhyperbolic at a point 0* E T*X\O in the direction u = (ux, uc) if there exists a neighbourhood V of this point in IR2"+2 and an E > 0 such that g(O + itv) # 0 V0 E V, t E (0, E). (2.78) E T*S\O. Let zk(k = 1, ...,N ) be the distinct roots of the polynomial C,), where z l , . .., z M E IR and z ~ +. .~., zN , E C\R. Let u = ( u i , 0); E JR2" and Ck E IR ( k = 1, . . .,M ) . Assume that P is microhyperbolic at the points

Let p*

g(p*,

uk

= (p*, z k )in

= (Ui,

0,U;, ( k )

the directions

(k = 1,. .. , M ) .

yb(p

+ ito, D,): E ( p , t ) + C'

(2.80)

is a bijection for sufficiently small I/ and E > 0, then the problem (P, B) is microhyperbolic at the point p* in the multi-direction (o, C1, . . .,

cM).

Theorem 29. Let p* E T*S\O. Let zk (k = 1, . . .,M ) be distinct real roots of the polynomial g(p*, z). Suppose that 0: = (p*, zk) E z-'p* and V and V, are neighbourhoods of p* and 0* in T*S and T * X , respectively. Let (pk = (pk(x,t') E d(V,)( k = 1, . . ., M ) , cp = cp(x', 5') E d ( V ) be real, positive homogeneous functions of degree 0 such that q1 = . * . = cpM = cp on T*S, cp(p*) = 0 and the problem ( P , B ) is microhyperbolic at p* in the multi-direction H,, -__ acp1

")

(

( p * ) . If Pu E d ( X ) ,yBu

Ed

ax,

,... ,

( S ) and

ax,

Let T*X\O 2 C and T*S\O 2 Cbbe the set of points where P and (P, B) are, respectively, non-elliptic. We shall call 2 and C b the characteristic set of P and ( P , B), respectively.

0:

Definition 11. If the condition (2.79) holds and if the map

-

Here, and in what follows, y denotes the restriction operator to {xn = 0).

wFba(u)

+

+

k e Z+

yb(p, Dn):Ker P(P,0,) n~ ( w ' )

219

(2.79)

wF,(u) n V, n { (pk < o} = @ Vk = 1, .. ., M ,

(2.81)

WFba(U)n V n {cp < 0) = Izr,

(2.82)

then p* E wFb,(U). We note the closeness between Theorems 24 and 29. 3.2. The Wave Equation. We examine the wave equation in some more detail. More generally, we shall consider a differential operator P of second order of the form (2.65)with analytic Coefficients. Suppose that either B = I or B is a differential operator of first order satisfying conditions (2.66) and (2.67), with analytic coefficients, or, finally, B = D, + /?(x'), with fi E d ( S ) . Let us consider the problem Pu E d ( X ) ,

yBu E d ( S ) .

(2.83)

Since it is elliptic in 8,its solutions satisfy the inclusion WF,,(u) c C. Definition 12. By an analytic ray we mean a Lipschitz curve y = (0 = 0(t)}, with parameter t = xo, that lies in Z and is either a bicharacteristic or a boundary bicharacteristic in the neighbourhood of all its points.

220

11. Linear Hyperbolic Equations

V.Ya. Ivrii

The exceptional points then constitute a locally finite set on each analytic ray. For 8 E V n 2 and an open set z-lzY c V, we denote by K'(8, V ) the union of all analytic rays issuing from in the direction of increasing (decreasing)xo and lying in It can be shown that K:(8, V) possess properties (1.43)-(1.46). From Theorem 29, we have

i

Theorem 30. Let u E 9'(@satisfy (2.83), z-'zV c V be an open set, and j VnCeC.If ~ WF,-,(u)n K i ( 8 , V ) n aV = $3,

then

WFf,(u)nK;(8,

B

We discuss the propagation of singularities by assuming that either E + p. If u is a solution of (2.83), then, by Theorem 30,

ua,)nWF,-,(u) = 0=v4wF,-a(u). We recall that, by virtue of Theorem 25, (YT

y T nWFJ(u)=

K+

I-

S

I+

a+

Fig. 5

Fig. 6

In accordance with the customary terminology, y- is an incident ray, y+ a reflected ray, a+ a crawling ray, and for any p E a+ $+ = y+(p) is a glancing ray. Therefore

K : ( n = i j u Y+ u a+ u

u

or

(2.84)

@*p$

WFf(u). (2.85) As shown by a direct computation by Rauch [1977] for the wave equation and by Friedlander and Melrose [1977] for the model equation, the implication (2.85) is not valid for analytic wave fronts. We also have the following results. (7- uy+)nWF,-,(u) = $3 * i 5 $ WF,-,(U). We note that here we do not require that boundary conditions be satisfied; we also note that the assertion is not valid for COD-wavefronts.

Theorem 32 (Sjostrand [1980b, c, 19813). The problem (2.83) has a solution u such that WF,-(u) = WF,-,(u) = 7,. In particular, (a- u a+) n WF,-,(u) = 0does not imply that p $ WFf,(u). Theorem 33 (Sjostrand [1980b, c, 19811). if u is a solution of (2.83), then

(Y F u a*) u WFf,(U) = 0* i j $ WFf,(U). In conclusion, we consider precisely the wave equation nu = 0 with n space variables. Let w be an open domain in IR" with analytic boundary ao and convex complement, that is, w is an "exterior" domain. Let y' E w and u a solution of the problem

n-n I+

=i

=0 .

Theorem 31 (Kataoka [1981]). I f Pu E d ( X ) ,then

V ) = 0.

What is more, with the aid of Theorems 28 and 29 we can investigate the propagation of analytic singularities of solutions to non-classical problems. Let us discuss the distinction between the COD-raysand the analytic rays. A boundary bicharacteristic is always an analytic ray and it is a C"-ray only outside the dijiractive set '3+ = { p E '3, qx,(p) > O}. Therefore through every point of this set there passes (locally) only one COD-raybut several analytic rays, and the branching of analytic rays takes place at points of '3+ and at some points of its boundary a??+; at these points transition takes place from tangent bicharacteristic to S to boundary bicharacteristics and vice-versa. To see the effects of the fact that an analytic ray through i5 E '3+ is not unique, we let pass through this point a bicharacteristic y and a boundary bicharacteristic a. Put y * = y n { + x o > 0} and a* = a n { f x , > 0 } (we assume that x , @ ) = 0); everything is depicted in Fig. 5 for the wave equation where S is not straight. Then K + ( p ) = i j u y+ but at the same time K:(ij) has a more complicated structure (see Fig. 6). x-

22 1

Y+(P).

pea+

Instead of showing K + ( p ) and K:(ij), Fig. 6 shows their intersection with the hyperplane { x o = t } , t > 0.

nu = 0 in IR x w, ul,..=, = 0,

ulRxaw= 0

D , U ~ , , == ~ -iS(x' - y').

(2.86) (2.87)

Theorem 34 (Rauch and Sjostrand [1981]). Suppose that either n = 2 and aw is convex or n 2 3 and aw is strictly convex. If u is a solution of (2.86)-(2.87), then WF,-,(ti) is precisely the union of all the analytic rays issuing from the points (0, y', 1, <'), where 5' runs through IR"n { 15'1 = l}. At the same time, WF,-(u) is the union of all Cw-rays issuing from these very points. In particular, sing supp, u = .n,WF,-,(u) contains the frontal boundary of supp u.

Problems In conclusion, we should like to formulate a number of problems whose solution should prove, in our view, extremely useful. Among these problems there are problems of most diverse degree of difficulties.

V.Ya. Ivrii

11. Linear Hyperbolic Equations

1. Prove the Ivrii-Petkov conjecture (Q 1.4, Chap. 1) in the general case (that is, when S , u S- # 0). 2. Generalize Nishitani's results (Q 1.9, Chap. 1) to the case when S , u S- = 0 and to equations of arbitrary order. 3. Is it true that if P is an x,-hyperbolic operator, then an operator Q of order not exceeding m - 1 can be found such that the Cauchy problem for P + Q is C"-well posed? 4. Is it true that if the Cauchy problem for P is C"-well posed and I is the maximum multiplicity of characteristics (or of characteristic roots), then the Cauchy problem for P + Q is also C"-well posed provided that the order of Q does not exceed m - r? 5. Apart from the strong formulation of the Cauchy problem, which we have used, weak formulation for it is also possible under which we seek the solutions in the spaces 9' and 9i.). To prove the necessity of the well-posedness conditions, we often start from the weak formulation while to prove the sufficiency from the strong formulation, and thereby establishing their equivalence. We should like to have a more general and direct proof of the equivalence of these two formulations. 6. Give a general and direct proof of the fact that under suitable regularity conditions on the coefficients of P the following implications hold

in some spatial type lens, it will remain so in a smaller spatial type lens whose base is a piece of an arbitrary spatially similar surface, and not necessarily the hyperplane {x, = T } ? 11. Prove Theorems 36,37,38,39 and 42, Chap. 1, for all those operators for which it is known that the Cauchy problem is suitably well posed. 12. More generally, give a direct proof of the fact that the well-posedness of the Cauchy problem implies the validity of these theorems. 13. If

222

n

P=

G("I)-well-posedness (for x1 c x)

and local G(")-well-posedness G(")-well-posednessin some neighbourhood of the set X T . This problem will be solved if one solves the preceding problem. 7. Prove Theorem 2, Chap. 1, under the assumption that S , u S- = 0. 8. Let p be a critical point of the principal symbol p of a scalar x,-hyperbolic operator P . Assume that the fundamental matrix F,(p) does not have non-zero real eigenvalues and that one of the conditions a) or b) of Theorem 18, Chap. 1, is violated. Then it can easily be shown that the Cauchy problem is not G{"}-well posed for sufficiently large x. Find the (best possible) value of x. Does x depend on the structure of the fundamental matrix or on the fact which of the conditions a) or b) is violated? What, in particular, is the value of x for the operators Pl + cDn( I = 2, 3) of Example 3? 9. Is the Cauchy problem for the operator P

= D; - x:DI

+ uD,

G(")-wellposed? If not, establish a Theorem 33 type (Chap. 1) general result that will imply this result. 10. Is it true that all the well-posedness conditions, except the x,-hyperbolicity condition, for the Cauchy problem can be written in a symplectically invariant form? Is it true, in particular, that if the Cauchy problem is well posed

1 uj(x)Dj+ b(x) j=O

C"-well-posedness =-G(")-well-posedness=. G(")-well-posedness

223

i

is a symmetric hyperbolic matrix operator, then prove that there exists a nowhere dense closed set A in the space of C"-matrices that depends on p and is such that the inclusion (1.51) turns into an equality when b $ A . 14. Similarly, show that if P is x,-hyperbolic, we can find an operator Q of order not exceeding m - 1 such that the Cauchy problem for P + Q is C"-well posed and that for this operator equality (1.51) holds. 15. Obtain necessary and sufficient conditions for the weak &-wellposedness of the normal mixed problem for equations of an arbitrary order. 16. Define the notions of a C"-ray and an analytic ray for the general mixed problem and reformulate in terms of these Theorems 24,25 and 29, Chap. 2. 17. Investigate the propagation of singularities of solutions of the mixed problem in the Gevrey classes for the wave equation, first of all, and before this for the Freidlander's model problem [1976], and show that for x > x * the G(")-singularitiesand for x > x* the G(")-singularities are propagated in the same manner as the C"-singularities are. Show further that for other values of x, the singularities in Gevrey classes are propagated just as analytic singularities are. And that in this case x* = 3, a fact dictated by the order of decay of solution to the Helmholtz equation ( A + k 2 ) o = S(x' - y') in the shade zone as the frequency k increases.

Bibliographical Remarks

1 1

'

In preparing the bibliography the author was faced with the daunting task of selecting the titles to be included in the list. It suffices to mention that the first version of the list of references was four times as large as the final version, and yet it was far from complete. In selecting the titles, the author was guided by a number of principles. An attempt was made to include the most recent works, or those which remained undeservedly, in the author's view, less known or those whose results were used by the author or, finally, the ones which had a rich bibliography. But at the same time, some works which proved to be useful for the author in writing this paper could not be includedin the list. The author does not consider himself to be competent enough to comment on the historical themes. Therefore these remarks should not be taken either as a historical survey or as an investigation into the question of priority.

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We note, first of all, the fundamental works of Hormander [1963, 1983, 19851 devoted to the general theory of partial differential operators, the works of Duistermaat and Hormander [1972], Hormander [1971,1983,1985], Taylor [1981] and Trtves [1980] devoted to the theory of pseudodifferential and Fourier integral operators, and the works of Maslov [1965, 19833 and Maslov and Fedoryuk [1973, 19763 where the theory of canonical operators has been discussed. A large number of works are devoted to the C"-well posed Cauchy problem. Here one must mention, first of all, the classical work of Hadamard [1932]. Petrovskij [1938], Leray [1954], and Girding [19571 have established the L,-well-posedness of the Cauchy problem for strictly hyperbolic operators. The necessary conditions (but not the sufficient conditions for a matrix operator) for the L,-well-posedness have been obtained by Ivrii and Petkov [1974], and Strang [1966,1967]. That hyperbolicity is necessary for the C"-well-posedness has been proved by Lax [1957], Mizohata [1961], and Ivrii and Petkov [1974]. Necessary and sufficient conditions for the Cauchy problem to be C"-well posed for equations with characteristics of constant multiplicity have been obtained by Flaschka and Strang [1971], Ivrii and Petkov [1974], and Chazarain [1974], respectively. Operators with constant coefficients have been studied by Girding [1951] and Hormander [1963, Chap. 51. These work as well as those by Bronshtejn [1974], Atiyah, Bott and Girding [1970, 19731, John [1978], Nuij [1968], and Svensson [1970] discuss the algebraic properties of hyperbolic polynomials. The necessary conditions for the regular and completely regular hyperbolicity can be found in the work of Ivrii and Petkov [1974], while the sufficient conditions in the works of Ivrii [1976a], Iwasaki [1983, 19841, and Melrose [1983]. The necessary conditions for the C"-well-posedness of the Cauchy problem for equations with characteristics of variable multiplicity were obtained by Ivrii and Petkov [1974], Hormander [1977], and Yagdzhyan [1980]; the last work mentioned deals with degenerate equations having gluing of characteristic roots of infinite order. Sufficient conditions under which the Cauchy problem for degenerate equations is well posed have been obtained by Nuij [I9681 and Yagdzhyan [1980], and for fairly general irregular hyperbolic equations by Ivrii [I9771 and Hormander [1977] by the method of energy integral. A complete analysis of equations of second order in two independent variables with analytic coefficients in an open domain has been carried out by Nishitani [1984]; some model equations with characteristics of variable multiplicity 3 or 4 have been investigated by Zajtseva [1980]. Petkov [1975, 19783 and Demay [1977] have obtained necessary and suflicient conditions for the C"-well-posedness of the Cauchy problem for systems of first order with characteristics of constant multiplicity. A necessary condition for the regular hyperbolicity of systems in two independent variables has been obtained by Petkov and Kutev [1976]. A rich bibliography on the C"-well posed Cauchy problem can be found in the works of Ivrii and Petkov [1974], Girding [1982], and Iwasaki [1983]. Not too many works have been devoted to the well-posedness of the Cauchy problem in Gevrey classes. The necessity of the hyperbolicity has been established by Ivrii [1976b], Komatsu [1977], and Nishitani [1978], while sufficiency for the well-posedness in arbitrary Gevrey classes by Ivrii [1975], Bronshtejn [1980], and Kajitani [1983]. The necessary and sufficient conditions for wellposedness in Gevrey class with suitable exponents for equations with characteristics of constant multiplicity have been obtained by Ivrii [1976b], and Komatsu [1977], while the paper by Ivrii [1976c] contains the necessary conditions for the well-posedness of equations with characteristics of variable multiplicity. Some degenerate equations have been dealt with by Ivrii [1978], and Yagdzhyan [1978]. Ivrii [1975, 1976b, 19783 and Nishitani [1978] have obtained the energy estimates in Gevrey classes. For a discussion of operators with constant coefficients, we once again refer the reader to Hormander [1963, Chap. 51. A large number of works have been devoted to the study of the propagation of singularities of solutions to the Cauchy problem and the related propagation in the whole space. One of the earliest works in this direction is by Babich [1960], where a fundamental solution for the strictly hyperbolic equation has been constructed, and by Lax [1975], where the propagation of oscillations has been described; a still earlier work of Courant and Lax has been overshadowed by Babich [1960]. The remaining works in our list are recent. Atiyah, Bott and Girding [1970, 19733 discuss the sin@larities of the fundamental solutions of equations with constant coefficients. Duistermaat and Hormander [19723 describe the propagation of singularities of principal type equations by meam of the Fourier integral operators. Parametrices for the Cauchy problem for operators with c h a m -

teristics of constant multiplicity have been constructed by Chazarain [I9741 and by Petkov [1978], respectively, for equations and for first-order systems. Alinhac [1978], Yoshikawa [1977, 1978a,b, 19801, and Kucherenko and Osipov [1983] have constructed parametrices for various classes of effectively hyperbolic operators. Nishitani [1983] describes the propagation of singularities for general effectively hyperbolic operators; close results have been obtained by Ivrii [1979a]. The propagation of the singularities of the solutions to various classes of equations has been discussed by Ivrii [1979a], R. Lascar [1981], Melrose and Uhlmann [1979a,b], and Uhlmann [1982]. For a description of the propagation of singularities of solutions to symmetric hyperbolic systems of first order, the reader is referred to Ivrii [1979c]; all these results have been reformulated by Wakabayashi [19801 in the language of generalized bicharacteristics. A detailed discussion of hyperbolic pseudodifferential systems that are principally diagonal can be found in Kucherenko [1974], R. Lascar [1981], Lax [1957] and Ralston [1976]. All these works concern the propagation of C"-singularities. The works dealing with the propagation of singularities in Gevrey classes are few, the work by Wakabayashi [1983] being the most important. From among a significant number of works devoted to the propagation of analytic singularities, we mention only Wakabayashi [1983], where the results of Sato, Kashiwara and Kawai have been reformulated in terms of the generalized bicharacteristics. Of works concerning the well-posedness of the mixed problem, we must mention, first of all, the works of Schauder, of Krzyzanski and of Ladyzhenskaya that are devoted to strictly hyperbolic equations of second order, and the works of Friedrichs [1958] and of his students Lax and Phillips [1960] that are devoted to positive (that is, strictly dissipative) problems for symmetric systems. Of the more recent works, we note Sakamoto [1970a,b, 19801, and Kreiss [1970], Agranovich [1969, 19721, Ralston [1971] and Rauch [1972], where strong L,-well-posedness of the mixed problem has been investigated, respectively, for equations and first-order systems. The weak L,-well-posedness has been dealt with by Agemi and Shirota [1970, 19713, and by Miyatake [1975,1977]. Miyatake's papers contain almost exhaustive results for second-order equations. A necessary condition for the C"-well-posedness under very general condition can be found in Wakabayashi [1980]. The C"-well-posedness of the mixed problem for the wave equation has been analysed by Hormander [1983, 19851 and Soga [1979, 19803. The general problems for strictly hyperbolic equations of second order have been studied by Eskin [1979/80, 1981a,b]. A non-strictly model hyperbolic problem has been discussed by Zajtseva [1983]. Under fairly weak assumptions, it has been shown by Duff [1959] that the mixed problem is analytically well posed. Girding's [I9821 survey paper should be mentioned once again. Povzner and Sukharevskij inagurated the study of the propagation of C"-singularities of the solutions to the mixed problep. Chazarain [1977] and Nirenberg dealt with the same question in the language of wave fronts, but they examined only transversal reflection of waves from the boundary. For this case, a parametrix has been constructed by Fedoryuk [1977]. The propagation of singularities in bicharacteristically strictly concave domains has been investigated by Friedlander [1976], Melrose [1975, 19781, Taylor [1976a,b, 19791, Eskin [1976], while in bicharacteristically convex domains by Ivrii [1979d,e, 19801, Eskin [1977]; the domains of an arbitrary nature have been dealt with by Ivrii [1980b], Melrose and Sjostrand [1978, 19823; for investigations concerning very general symmetric systems, see Ivrii [1979d,e 1980a,b, 19863. The propagation of singularities of solutions to the non-classical problems for strictly hyperbolic equations of second order has been examined by Ivrii [1981], Eskin [l979/80,1981a,b], Melrose and Sjostrand [1978,1982]. Very few works deal with the question of the propagation of analytic singularities. In this direction, one must mention, first of all, a very general work of Sjostrand [1980]. Friedlander and Melrose [1977], Kataoka [1981], Rauch and Sjostrand [1981] and Sjostrand [1980b,c, 19813 have investigated the propagation of analytic singularities for strictly hyperbolic equations of second order. The maximum attention in these works is directed to bicharacteristically concave domains where the propagation of C"-singularities and that of analytic singularities are significantly different. In this very situation, Rauch [I9771 examined the propagation of the frontal boundary of the support of the solution, the propagation being closely related to that of analytic singularities.

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225

11. Linear Hyperbolic Equations

V.Ya. Ivrii

226

A Survey of Recent Results

227

Of the works on the propagation of singularities of solutions to the Cauchy problem, we mention the one by Melrose [1986]. This paper examines the propagation of singularities for the equation u,, Au = 0, where A is a self-adjoint positive definite differential operator of the second order and is hypoelliptic with a loss of 2 - 6 derivatives, with 6 > 0. A number of works deal with the propagation of the polarization front. Namely, if u E g ( X , CN), then (x, <,w ) E T * X x C N lies in WF;,,(u) provided that for any pseudodifferential operator A of order 0 the inclusion Au E H" implies that a(x, < ) w = 0, where a is the principal symbol of A. The wave front is a projection onto T*X of the polarization front. The concept of the polarization front was introduced by Dencker [1982], and his investigation concerns the propagation of polarization fronts of solutions to systems of real principal type with characteristics of constant multiplicity. It turns out that for such systems the polarization fronts are propagated along Hamiltonian orbits, namely, curves lying in T*X x (CN and determined by a system of ordinary differential equations; their projections onto T*X are bicharacteristics. Dencker [1986] studies the propagation of the polarization fronts for systems with conical refraction, and in this study the distinction between the real polarization and the complex polarization plays a significant role. Some results also exist concerning the propagation of the polarization fronts near the boundary, namely, concerning their reflection under transversality of bicharacteristics and even diffraction. Some results regarding the analytic polarization fronts have also been obtained (see the bibliography in Esser [1986]). Many works study mixed problems. Eskin [1985] has investigated the well-posedness of the mixed problem for a strictly hyperbolic operator of second order in a bicharacteristically convex domain and has generalized his results of [1981b]. For this problem, a parametrix has been constructed by Kubota [1986] under certain conditions imposed on the boundary operator and the standard results on the propagation of singularities have been obtained. Laubin [1986] has given new proofs for Theorems 32 to 34 of Chapter 2 of the present survey. Zajtseva [1987a] has obtained a condition on (a, a) E (C u co)x C that is necessary and sufficient for the mixed problem

+

The main part of the paper was written in 1984; however, the works published during the past three years' have made this supplement necessary. We mention, first of all, the surveys by Girding [1984] and by Mizohata [1986a]. The book Mizohata [1986b], where the latter survey has been published, contains a number of important works some of which have been included in our reference list. We note that at last we have in the Russian language the main works of Petrovskij; in particular, the works on the hyperbolic equations and systems (see Petrovksij [1986]). A number of works have been devoted to the C"-well-posedness of the Cauchy problem. Nishitani [1985, 19863 has obtained the energy estimates for effectively hyperbolic as well as for some other operators with double characteristics, and has given an alternative proof for the C"-wellposedness of the Cauchy problem for effectively hyperbolic operators. Yagdzhyan [1985,1986] has found necessary and sufficient conditions for the Cauchy problem to be C"-well posed for degenerate operators having gluing of characteristic roots of infinite order, and has also constructed a parametrix for the Cauchy problem. Zajtseva [1987b] has shown that in order that the Cauchy problem for the operator

p = 0: - 0: - x2"D; + x"-' DY be C"-well posed for odd n, it is necessary and sufficient that a E R, la1 < n. It has been recently discovered that the condition a E R u iR is necessary and sufficient for the problem to be C"-well posed for even n provided that la1 is sufficiently small. Nishitani [1985] has studied the first-order systems and has shown that if P is a matrix operator of first order with no more than double characteristics, then for the strong (regular) hyperbolicity of P, it is necessary that in a neighbourhood of every point p either the principal symbol p is uniformly diagonalizable or g = det p is effectively hyperbolic. If both these conditions are violated at p E T*R"+'\O,then for the Cauchy problem to be C"-well posed, it is necessary that there exists an a E [ - 1, 13 and a matrix B such that i (pscop - ? { p , '"p)

+ a Tr'g

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to be well posed. Finally, Lebeau [1984, 19861 has shown that for the Dirichlet problem for a strictly hyperbolic operator of second order with analytic coefficients in a domain having analytic boundary, the G%ingularities, for x 2 3, are propagated in the same manner as the C"4ngularities are, while for 1 < x < 3 in the same way as the analytic singularities are (recall that analytic singularities can be propagated along crawling rays and glancing rays but C"-singularities cannot). Thus, Problem 17, formulated above, has been partially solved.

-

Conversely, if in the neighbourhood of every point either g is effectively hyperbolic or p is uniformly diagonalizable and if ,Z = {g = dg = 0) is a C"-manifold of codimension d < 3, TpZ = Ker F,(p), then P is a regularly hyperbolic operator. A number of works deal with the well-posedness of the Cauchy problem in Gevrey classes. Kajitani [1986] has established that under suitable regularity conditions on the coetlicients, the Cauchy problem for a uniformly diagonalizable system of first order is G(")-wellposed for x < 2. The results of Bronshtejn [1980] have been generalized by Ohya and Tarama [1986] to the case where the coefficients are not very smooth with respect to xo. Namely, they have shown that if the with u E (0,2], coefficients of the principal part of the scalar operator P lie in C'([O, TI, G(")(R")), and the lower terms lie in Co([O, TI, G(")(R")), then the Cauchy problem will be G(")-well posed if 1 < x < min(1 u/r, r/(r - l)),where r is the maximum multiplicity of the characteristic roots. The Italian mathematicians have obtained considerably more precise results for second-order operators, and very often they have examined operators without lower terms and with coefficients depending only on x,,. They have succeeded in discovering a large number of illuminating counter examples. In particular, it has been shown that for a strictly hyperbolic operator, with u < 1, x > 1/(1 - a), the Cauchy problem, in general, is neither G(")-wellposed nor C"-well posed. Moreover, for u < 1, even the uniqueness theorem may fail to hold. A survey and bibliography can be found in Spagnolo [1986] and Colombini, Jannelli and Spagnolo [1986].

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References*

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Agemi, R., Shirota, T. [1970] On necessary and sufficient conditions for L'-well-posedness of mixed problems for hyperbolic equations I. J. Fac. Sci., Hokkaido Univ., Ser. I, 21, No. 2, 133-151, Zbl.207,403 [1971] On necessary and sufficient conditions for L'-well-posedness of rmxed problems for hyperbolic equations 11. ibid. 22, No. 3, 137-149,Zb1.241.35055 Agranovich, M.S. [19691 Boundary-value problems for systems of first order pseudodifferential operators. Usp. Mat. Nauk 24, No. 1, 61-125. English transl.: Russ. Math. Surv. 24, No. 1, 59-126 (1969), Zb1.175,108 *For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled using the MATH database, have, as far as possible, been included in this bibliography.

228 [1972]

11. Linear Hyperbolic Equations

V.Ya. Ivrii A theorem on matrices depending on parameters and its applications to hyperbolic systems. Funkts. Anal. Prilozh. 6, No. 2, 1-11. English transl.: Funct. Anal. Appl. 6, 85-93 (1972), Zb1.253.35060

Alinhac, S . Branching of singularities for a class of hyperbolic operators. Indiana Univ. Math. J. 27, [1978] NO. 6, 1027-1037,Zb1.502.35058 Atiyah, M.F., Bott, R., Girding, L. Lacunas for hyperbolic differential operators with constant coefficients. I. Acta Math. [1970] 124, 109-189,Zb1.191,112 Lacunas for hyperbolic differential operators with constant coefficients. 11. ibid. 131, [1973] 145-206, Zbl.266.35045 Babich, V.M. Fundamental solutions of hyperbolic equations with variable coefficients. Mat. Sb. Nov. [1960] Ser. 52, No. 2, 709-738,Zb1.96,67 Bronshtejn, M.D. Smoothness of roots of polynomials depending on parameters. Sib. Mat. Zh. 20, No. 3, [1979] 493-501. English transl.: Sib. Math. J. 20, 347-352 (1980),Zb1.415.30003 The Cauchy problem for hyperbolic operators with characteristics of variable multi[1980] plicity. Tr. Mosk. Mat. 0.-va 41,83-99. English transl.: Trans. Mosc. Math. SOC.1982, NO. 1,87-103 (1982), Zbl.468.35062 Chazarain, J. [1974] Operateurs hyperboliques a caracteristiques de multiplicitt constante. Ann. Inst. Fourier. 24, No. 1, 173-202,Zb1.274.35045 [1977] Reflection of C"-singularities for a class of operators with multiple characteristics. Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, Suppl., 39-52,Zb1.365.35050 Colombini, F., Jannelli, E., Spagnolo, S. [1986] Non-uniqueness examples for second-order hyperbolic equations. Current Topics in Partial Differential Equations. Pap. dedic. S. Mizohata, 309-326,Zbl.627.35053 Demay, Y. Parametrix pour des systtmes hyperboliques du premier ordre multiplicitt constante. [1977] J. Math. Pures Appl. 56, No. 4, 393-422,Zb1.379.35068 Dencker, N. [1982] On the propagation of polarization sets for systems of real principal type. J. Funct. Anal. 46, NO. 3, 351-372,Zb1.487.58028 [1986] On the propagation of polarization in conical refraction. Preprint, Univ. Lund, 3,88 pp. Appeared in: Duke Math. J. 57, No. 1, 85-134 (1988), Zbl.669.35116 Duff, G.F.D. [1959] Mixed problems for hyperbolic equations of general order. Can. J. Math. 9, 195-221, Zbl.86,76 Duistermaat, J.J., Hormander, L. [1972] Fourier integral operators. 11. Acta Math. 128, No. 3-4, 183-269,Zb1.232,47055 Eskin, G. [1976] A parametrix for mixed problems for strictly hyperbolic equations of arbitrary order. Commun. Partial Differ. Equations 1, No. 6,521-560,Zb1.355.35053 [1977] Parametrix and propagation of singularities for the interior mixed hyperbolic problem. J. Anal. Math. 32, 17-62,Zb1.375.35037 [1979/80] Well-posedness and propagation of singularities for initial boundary value problems for second order hyperbolic equations with general boundary conditions. Semin. Goulaouic-Schwartz, Exp. No. 11, 14 pp. Initial boundary value problem for second order hyperbolic equations with general [1981a] boundary conditions. I. J. Anal. Math. 40,43-89,Zbl.492.35042 General initial-boundary problems for second order hyperbolic equations. Singularities [1981b] in boundary value problems. Proc. NATO Adv. Stud. Inst., MarateaDtaly 1980,19-54, Zb1.475.35061

[1985]

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229

Initial-boundary value problems for second order hyperbolic equations with general boundary conditions. 11. Commun. Partial Differ. Equations 10, No. 10, 1117-1212, Zbl.585.35059

Polarisation analytique dans les probltmes aux limites non-diffractifs de multiplicitC constante. Bull. SOC.R. Sci. Liege 55, No. 3, 371-465,Zb1.603.35089 Fedoryuk, M.V. [1977] Singularities of the kernels of Fourier integral operators and asymptotics of the solutions of mixed problems. Usp Mat. Nauk 32, No. 6,67-115. English transl.: Russ. Math. Surv. 32, No. 6,67-120 (1977), Zb1.376.35001 Flaschka, H., Strang, G. The correctness of the Cauchy problem. Adv. Math. 6, No. 3,347-379,Zbl.213,373 [1971] Friedlander, F.G. [1976] The wave front set of the solution of a simple initial-boundary value problem with glancing rays. I. Math. Proc. Camb. Philos. SOC.79, No. 1, 145-159,Zb1.319.35053 Friedlander, F.G., Melrose, R.B. [1977] The wave front set of solutions of a simple initial-boundary value problem with glancing rays. 11. Math. Proc. Camb. Philos. SOC.81, No. 1,97-120,Zb1.356.35056 Friedrichs, K.O. [1958] Symmetric positive linear differential equations. Commun. Pure Appl. Math. 11, 333418, Zb1.83,318 Girding, L. Linear hyperbolic partial differential equations with constant coefficients. Acta Math. [19511 85, NO. 1, 1-62,Zb1.45,202 Cauchy's problem for hyperbolic equations. Univ. Chicago, Zb1.91,269 [19571 Hyperbolic boundary problems. Differential Geometry and Differential Equations, [19821 Proc. 1980 Beijing Symp., Vol. 1, 371-465,Zb1.527.35051 Hyperbolic differential operators. Perspective in Math., Anniv. Obenvolfach 1984, [19841 215-247,Zb1.572.35012 Hadamard, J. Le probltme de Cauchy et les tquations aux derives partielles lineaires hyperboliques. [1932] Paris, Zb1.6,205 Hormander, L. Linear Partial Differential Operators. Berlin-Heidelberg-New York Springer-Verlag, [1963] Zb1.108,93 Fourier integral operators. I. Acta Math. 127, No. 1-2, 79-183,Zb1.212,466 [1971] The Cauchy problem for differential equations with double characteristics. J. Anal. [1977] Math. 32, 118-196,Zb1.367.35054 The Analysis of Linear Partial Differential Operators. Vols. I and 11. Berlin-Heidelberg[1983] New York Springer-Verlag, Zb1.521.35001 and Zb1.521.35002 The Analysis of Linear Partial Differential Operators. Vols. I11 and IV. Berlin[1985] Heidelberg-New York: Springer-Verlag, Zb1.601.35001 and Zb1.612.35001 Ikawa, M. Mixed problems for the wave equation. Singularities in Boundary-Value Problems. [19811 Proc. NATO Adv. Stud. Inst., MarateoJtaly 1980,97-119.Zb1.473.35047 Ivrii, V.Ya. (= Ivrij, V.Ya.) [1975] Well-posedness in Gevrey classes of the Cauchy problem for non-strictly hyperbolic operators. Mat. Sb., Nov. Ser. 96, No. 3,390-413. English transl.: Math. USSR, Sb. 25 (1975), 365-387 (1976). Zb1.319.35054 Sufficient conditions for regular and completely regular hyperbolicity. Tr. Mosk. Mat. [1976a] 0.-va 33,3-65. English transl.: Trans. Mosc. Math. SOC.33,l-65 (1978), Zbl.355.35052 [1976b] Well-posedness conditions in Gevrey classes for the Cauchy problem for non-strictly hyperbolic operators. Sib. Mat. Zh. 17, No. 3,547-563. English transl.: Sib. Math. J. I 7 (1976),422-435 (1977), Zb1.338.35088

V.Ya. Ivrii

11. Linear Hyperbolic Equations

Well-posedness conditions in Gevrey classes for the Cauchy problem for hyperbolic operators with characteristics of variable multiplicity. Sib. Mat. Zh. 17, No. 6, 12561270. English trans].: Sib. Math. J. 17,921-931 (1977), Zb1.404.35068 The well-posed Cauchy problem for non-strictly hyperbolic operators. 111. The energy [1977] integral. Tr. Mosk. Mat. 0.-va 34, 151-170. English trans].: Trans. Mosc. Math. SOC.34, 149-168 (1978), Zbl.403.35060 Well-posedness in Gevrey classes of the Cauchy problem for non-strictly hyperbolic [1978] operators. Izv. Vyssh. Uchebn. Zaved. Mat. 1978, No. 2, 26-35,Zb1.395.35055 Wave fronts of solutions of some pseudodifferential equations. Tr. Mosk. Mat. 0.-va [1979a] 39, 49-82. English trans].: Trans. Mosc. Math. SOC. 1981, No. 1, 49-86 (1981), Zbl. 433.35077 Wave fronts of solutions of some hyperbolic pseudodifferential equations. Tr. Mosk. [1979b] Mat. 0.-va 39, 83-112. English trans].: Trans. Mosc. Math. SOC.1981, No. 1, 87-119 (1981), Zb1.433.35078 Wave fronts of solutions of symmetric pseudodifferential systems. Sib. Mat. Zh. 20, [1979c] No. 3, 557-578. English trans].: Sib. Math. J. 20,390-405 (1980), Zb1.417.35080 Wave fronts of solutions of boundary-value problems for symmetric hyperbolic systems. [1979d] I. Sib. Mat. Zh. 20, No. 4, 741-755. English trans].: Sib. Math. J. 20, 516-524 (1980), Zb1.454.35055 [1979e] Wave fronts of solutions of boundary-value problems for symmetric hyperbolic systems. 11. Sib. Mat. Zh. 20, No. 5, 1022-1038. English transl.: Sib. Math. J. 20,722-734 (1980), Zb1.454.35056 [1980a] Wave fronts of solutions of boundary-value problems for symmetric hyperbolic systems. 111. Sib. Mat. Zh. 21, No. 1, 74-81. English trans].: Sib. Math. J. 21, 54-60 (1980), Zb1.454.35057 [1980b] Wave fronts of solutions of boundary-value problems for a class of symmetric hyperbolic systems. Sib. Mat. Zh. 21, No. 4,62-71. English trans].: Sib. Math. J. 21, 527-534 (1981), 2b1.447.35055 [1981] The propagation of singularities of solutions of non-classical boundary-value problems for hyperbolic equations of second order. Tr. Mosk. Mat. 0.-va 43, 81-91. English trans].: Trans. Mosc. Math. SOC.1983, No. 1, 87-99 (1983), Zb1.507.35059 [1986] The propagation of singularities of solutions of dissipative boundary-value problems for symmetric hyperbolic systems. Dokl. Akad. Nauk SSSR 287, No. 6,1299-1301. English transl.: Sov. Math., Dokl. 33, 526-529 (1986), Zb1.626.35057 Ivrii, V.Ya; Petkov, V.M. (= Ivrij, V.Ya.; Petkov, V.M.) [1974] Necessary conditions for the well-posedness of the Cauchy problem for non-strictly hyperbolic equations. Usp Mat. Nauk 29, No. 5,3-70. English trans].: Russ. Math. Surv. 29, NO. 5, 1-70 (1974), Zb1.312.35049 Iwasaki, N. [1983a] The Cauchy problem for hyperbolic equations with double characteristics. Publ. Res. Inst. Math. Sci. 19, No. 3,927-942,Zb1.571.35067 [1983b] The Cauchy problem for effectively hyperbolic equations (A special case). J. Math. Kyoto Univ. 23, 503-562,Zb1.546.35037 [1984] Cauchy problems for effectivelyhyperbolic equations (A standard type). Publ. Res. Inst. Math. Sci. 20, No. 3, 543-584, Zb1.565.35065 John, F. [1978] Algebraic conditions for hyperbolicity of systems of partial differential equations. Commun. Pure Appl. Math. 31, No. 1, 89-106,Zb1.381.35059 Kajitani, K. Cauchy problem for nonstrictly hyperbolic systems in Gevrey classes. J. Math. Kyoto [1983] Univ. 23, No. 3,599-616,Zb1.544.35063 [1986] The Cauchy problem for uniformly diagonalizable hyperbolic systems in Gevrey classes. In: Mizohata (Ed.): Hyperbolic Equations and Related Topics, Proc. Taniguchi Int. Symp. Katata and Kyoto 1984, 101-123,Zb1.669.35065

Kataoka, K. [1981] Microlocal analysis of boundary-value problems with application to diffraction. Singularities in Boundary-Value Problems. Proc. NATO Adv. Stud. Inst., Marateoptaly 1980, 121- 13l,Zb1.461.35088 Komatsu, H. [1977] Irregularity of characteristic elements and hyperbolicity. Publ. Res. Inst. Math. Sci. 12, Suppl., 233-245,Zb1.375.35044 Kreiss, H.O. [1970] Initial-boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23,277-298,Zb1.193,69 Kubota, K. [1986] Microlocal parametrices and propagation of singularities near gliding points for hyperbolic mixed problems. I. Hokkaido Math J. 15, No. 2,243-308,Zb1.619.35071 Kucherenko, V.V. [1974] Asymptotics, as h + 0, of solutions of the system A x, -hu = 0 in the case of

230 [1976c]

231

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i:

2

characteristics of variable multiplicity. Izv. Akad. Nauk SSSR,Ser. Mat 38, 625-662. English trans].: Math. USSR, Izvestija 8 (1974), 631-666 (1975), Zb1.308.35080 Kucherenko, V.V., Osipov, Yu.V. [1983] The Cauchy problem for non-strictly hyperbolic equations. Mat. Sb., Nov. Ser. 120, No. 1,84-111. English transl.: Math. USSR, Sb. 48, 81-109 (1984), Zb1.519.35048 Kumano-go, H. [1979] Fundamental solution for a hyperbolic system with diagonal principal part. Commun. Partial Differ. Equations 4, No. 9,959-1015,Zb1.431.35062 Kumano-go, H., Taniguchi, K. [1979] Fourier integral operators of multiphase and the fundamental solution to a hyperbolic system. Funkc. Ekvacioj, Ser. Int. 22, No. 2, 161-196,Zb1.568.35092 Lascar, R. Propagation des singularites des solutions d'equations pseudodifferentielles a carac[1981] teristiques de multiplicites variables. Lect. Notes Math. 856,Zb1.482.58031 [19861

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Ohya, Yu., Tarama, S. [1986] La probltme de Cauchy a caracteristiques multiples dans la classe de Gevreycoefficients holderiens en t. In: Mizohata (Ed.): Hyperbolic Equations and Related Topics, Proc. Taniguchi Int. Symp., Katata and Kyoto/Jap. 1984, 273-306, Zb1.665.35045 Olejnik, O.A. [1970] On the Cauchy problem for weakly hyperbolic equations. Commun. Pure Appl. Math. 23, NO. 6, 569-586, 2b1.193,386 Petkov, V.M. [1975] Necessary condition for well-posedness of the Cauchy problem for non-symmetrizable hyperbolic systems. Tr. Semin. I.G. Petrovskogo I, 21 1-236. English transl.: Transl., 11. Ser., Am. Math. SOC.118,25-50 (1982), 2b1.317.35055 [1978] A parametrix of the Cauchy problem for non-symmetrizable hyperbolic systems with characteristics of constant multiplicity. Tr. Mosk. Mat. 0.-va 37, 3-47. English transl.: Trans. Mosc. Math. SOC.1, 1-47 (1980),2b1.442.35067 [198l] Propagation des singularites sur le bord pour des systemes hyperboliques a caracteristiques de multiplicitk constante. C.R. Acad. Sci., Paris, Ser. I. 293, 637-639, Zb1.487.35059 Petkov, V.M., Kutev, N.D. [1976] On regularly hyperbolic systems of first order. God. Sofij Univ., Fak. Mat. Mekh. 67, 375-389,2b1.353.35061 Petrovskij, I.G. [1938] On the Cauchy problem for systems of linear partial differential equations. Bull. Univ. Mosk., Ser. Int. Mat. Mekh. I , No. 7, 1-74,2b1.24,37 [19861 Selected Works. Systems of Partial Differential Equations. Algebraic Geometry. Moscow: Nauka, 2bl.603.01018 Ralston, J.V. [I9711 Note on a paper of Kreiss. Commun. Pure Appl. Math. 24,759-762,2b1.215,168 [1976] On the propagation of singularities in solutions of symmetric hyperbolic partial differential equations. Commun. Partial Differ. Equations I, No. 2, 87-133,2b1.336.35066 Rauch, J. [1972] L , is a continuable initial condition for Kreiss’ mixed problems. Commun. Pure Appl. Math. 25, 265-285,2b1.226.35056 [1977] The leading wave front for hyperbolic mixed problems. Bull. SOC.R. Sci., Liege 46, 156-161,2b1.372.35051 Rauch, J., Sjostrand, J. Propagation of analytic singularities along diffracted rays. Indiana Univ. Math. J. 30, [1981] NO. 3, 389-401,2b1.424.35003 Sakamoto, R. Mixed problems for hyperbolic equations. I. J. Math. Kyoto Univ. 10, No. 2, 349-373, [1970a] Zb1.203,100 [1970b] Mixed problems for hyperbolic equations. 11. ibid, No. 3,403-417,2b1.206,401 [1980] Hyperbolic Boundary Value Problems. Cambridge: Univ. Press, Zb1.494.35001 Sjostrand, J. Analytic singularities and microhyperbolic boundary value problems. Math. Ann. 254, [1980a] 21 1-256, Zb1.459.35007 Propagation of analytic singularities for second order Dirichlet problems. I. Commun. [1980b] Partial Differ. Equations 5, No. 1,41-94,2b1.458.35026 [198Oc] Propagation of analytic singularities for second order Dirichle? problems. 11. ibid, NO. 2, 187-207,2b1.534.35030 [1981] Propagation of analytic singularities for second order Dirichlet problems. 111. ibid. 6, NO. 5,499-567,Zb1.524.35032 Soga, H. [1979] Mixed problems in a quarter space for the wave equation with a singular oblique derivative. Publ. Res. Inst. Math. Sci. 15, No. 3,631-651, Zb1.433.35044

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235

Construction of a parametrix for the Cauchy problem of some weakly hyperbolic equation. 111. ibid, 127-141,Zb1.393.35040 [1980] Parametrices for a class of effectively hyperbolic operators. Commun. Partial Differ. Equations 5, No. 11, 1073-1151,Zb1.483.35048 Zajtseva, O.V. [1980] On the well-posedness of the Cauchy problem for a model equation with quadruple characteristics. Izv. Vyssh. Uchebn. Zaved., Mat. 1980, No. 6, 20-22. English transl.: Sov. Math. 24, No. 6, 21-23 (1980), Zb1.441.35041 [1983] On the well-posedness of a model non-strictly hyperbolic mixed problem. ibid. 1983, No. 10, 22-25. English transl.: ibid. 27, No. 10, 28-32 (1983),Zb1.599.35094 [1987a] On the well-posedness of a new model non-strictly hyperbolic mixed problem. ibid. 1987, No. 11, 10-12. English transl.: ibid. 31, No. 11, 12-15 (1987),Zb1.679.35060 [1987b] The well-posedness conditions for the Cauchy problem for a certain model non-strictly hyperbolic equation. ibid. 1987, No. 12, 43-45. English transl.: ibid. 31, No. 12, 53-55 (1987),Zb1.672.35038

Author Index Agemi, R. 201,225,227 Agranovich, M.S. 15, 154,225,227 Airy, G.B. 24,203,215 Alinhac, S. 189,225,228 Arnol’d, V.I. 10, 20,46, 139 Atiyah, M.F. 32, 36, 158, 160,224,228 Babich, V.M. 224,228 Baire, R. 111, 185 Banach, S. 77 Beals, R. 21,26, 29, 88,95, !00,107, 139 Bessel, F.W. 77 Bohr,N. 49 Bokobza, J. 15,26,27, 147 Bolley, P. 100, 139 Bony, J.M. 83, 133, 139 Bott, R. 36, 158-160,224,227 Boulkhemair, A. 83, 139 Boutet de Monvel, L. 98-100, 116,118, 139 Bove, A. 81, 140 Bronshtejn, M.D. 100, 140,161,175,224,226, 228 Bros, J. 189 Calderon, A.P. 14,25, 140, 157 Campbell, J.E. 106 Camus, J. 100, 139 Cauchy,A.L. 22,34,43,44,50, 115, 119, 132, 151-158,168,169,173,177,222-224,226, 227 Chazarain, J. 83, 140, 167, 189,224,225,228 Chin-Hung-Chin 25 Colombini, F. 226,228 Courant, R. 224 Demay, Y. 173,224,228 Dencker, N. 140,227,228 Derridj, M. 114, 140 Dirac, P.A.M. 54 Dirichlet, L.P.G. 116 Doughs, A. 108, 112 Duff, G.F.D. 206,225,228 Duhamel, J.M.C. 198 Duistermaat, J.J. 76, 78, 80, 140, 180, 224, 228 Dynin, A.S. 15

Egorov, Yu. V. 7, 10, 11, 20,23, 75, 87-89, 100, 102,104,140 Eskin, G.I. 15, 141, 204-206, 215, 225, 227, 228 Esser, P. 227, 229 Farris, M. 141 Fedoryuk, M.V. 42,53,144,224,225,229, 231 Fedosov, B.V. 32,141 Fefferman, C. 26,88,100,107,139,141 Flaschka, H. 82,141,166, 167,224,229 Folland, G.B. 100,141 Fourier, J.B.J. 8, 9, 32,43-46, 50, 52, 55, 56, 58,59,61-65,71,72,102,127,128, 138 Friedlander, F.G. 141,221,223,225,229 Friedrichs, K.O. 158, 198, 229 Frobenius, F.G. 79,105 Ganzha, E.I. 100, 107,141 Girding, L. 18, 120,141,157-160,224-226, 228,229 Gerard, C. 141 Gevrey, M. 10,91,92, 114, 173, 174, 177, 189, 223,224 Giraud,G. 14 Green,G. 75 Grigis, A. 83, 99, 100, 140-142 Grubb,G. 142 Grushin, V.V. 26,84,96-100, 113,142,147 Guillemin, V. 11, 140, 142 Hadamard, J. 115,224,229 Hahn,H. 77 Hamilton, W.R. 40 Hanges, N. 81, 142 Harvey, R. 74,75, 142 Hausdorff, F. 74, 106,111 Heisenberg, W. 8,40 Helffer, B. 84,99, 100,139,140, 142 Helmholtz, H.L.F. 42,223 Hilbert, D. 33 Holmgren, A. 134,196 Hormander, L. 7, 10, 12, 13, 15, 17, 18, 24, 25, 27, 29, 31, 32, 51, 53, 54, 59, 61, 64,65, 73, 76,

238

Hormander, L. (cont.) 78,80,85,91-95,99, 100, 102, 106,109, 140, 142, 158-160, 163, 168, 175, 180, 188, 189, 224,225,228,229

Lipschitz, R. 65 Lopatinskij, Ya. B. 195, 197, 199 Ludwig, D. 215 Lychagin, V.V. 144

Iagolnitzer, D. 189 Ikawa, M. 203,204,229 Iordanov, LV. 145 Ivrii, V. Ya. 7, 39, 81, 83, 143, 155, 156, 158, 162-165, 167-169, 174, 175,177-179,181, 184, 186, 188, 192, 199, 209, 210, 212, 213, 217,224,225,229,230 Iwasaki, N. 164,168,224,230

Magenes, E. 75, 144 Mandai 169 Martineau, A. 129, 132, 144 Maslov, V.P. 20,42,46,50,53, 144, 154,188, 224,23 1 Melin, A. 71, 144 Melrose,R. 39,81,120,121,126,144,164,189, 203,208,214,215,217,221,224,225,227,229, 232 Menikoff, A. 83,99, 100, 144 Metivier, G. 100, 145 Mikhlin, S.G. 15 Mischenko, A.S. 73, 145 Miyatake, S. 202,203,205,232 Mizohata, S. 84, 153, 156,157,224,226,232

Jacobi, K.G.J. 12 Jannelli, E. 226,228 John, F. 160,224,230 Kajitani, K. 224, 226, 230 Kannai, Ya. 85,99,100,107,143 Kashiwara, M. 79,134, 143, 146,225 Kataoka, K. 221,225,231 Kawai, T. 79, 143,146,225 Kirchoff, G.R. 152 Kohn, J.J. 15,25, 143 Kolmogorov, A.N. 105 Komatsu, H. 174, 177,224,231 Korn, A. 14 Kovalevskaya, S.V. 206 Kreiss, H.O. 199, 225, 231 Krzyzanski, M. 225 Kubota, K. 227,231 Kucherenko, V.V. 73, 143,189,225,231 Kumano-go, H. 26,27, 143, 189,231 Kutev, N.D. 173,224,233 Ladyzhenskaya, O.A. 225 Landis, E.M. 115, 143 Laplace, P.S. 90, 115, 194 Lascar, B. 83, 143,144 Lascar, R. 83,143, 144, 189,225,231 Laubin, P. 227,231 Laurent, Y. 144 Lax, P. 153, 156,198,224,225,231 Lebeau, G. 227,231 Lebesgue, H.L. 40 Leibniz, G.W. 16 Leray, J. 153, 157,224,231 Levi. E.E. 81, 166, 176 Lewis, J.E. 81, 140 Lewy,H. 84 Lichtenstein, L. 14 Lie, M.S. 105, 107, 112, 114 Lions, J.L. 75, 144

Author Index

Author Index

Nazajkinskij, V.E. 73, 145 Neumann, K.G. 98, 102,201 Newton, I. 40,171 Nirenberg, L. 15,25,87, 88, 108, 112, 145,225 Nishitani, T. 170, 174,224-226,232 Nourrigat, J. 100, 142 Nuij, W. 160,224,232 Ohya, Yu. 226,232 Olejnik, O.A. 107, 108, 111, 112, 115, 143, 145,170,233 Oshmyan, V.G. 145 Osipov, Yu. V. 189,225, 231 Parenti, C. 81, 100, 140, 145 Parseval, M. 45 Petkov,V.M. 155,156,158,162-165,168, 169, 172, 173, 189, 192, 199,224,225, 230, 233 Petrovskij, I.G. 153, 157,224, 226,233 Phillips, R.S. 198, 225, 231 Phong, D.H. 141 Planck, M. 40 Poisson, S.D. 20,40,41, 79, 107, 116, 117, 163 Polking, J. 74, 75, 141 Popivanov, P.R. 87,90,100,113,141,145 Povzner, A. Ya. 225 Puiseux, V.A. 171 Radkevich, E.V. 107, 108,111, 112, 145 Radon,J. 9 Ralston, J.V. 225,233 Rangelov, Ts. 87, 141

Rauch, J. 221,225,233 Rempel, S. 118, 145 Riemann, W. 36,74, 115 Rodino, L. 100, 145 Rothschild, L.P. 100,145 Sakamoto, R. 194,196, 197,225,233 Sato, M. 23,79,129, 146,189,225 Schapira, P. 83, 129, 135, 143, 146 Schauder, J.P. 225 Schrodinger, E. 42,49,50,83, 154 Schulze, B.W. 118, 145 Schwartz, L. 27, 146 Seeley, R.T. 15, 32, 35,37, 38, 146 Segala, F. 145 Seidenberg, A. 91 Shatalov, V.E. 145 Shirota, T. 201,225,227 Singer, I.M. 21, 32 Sjostrand, J. 23, 71, 73, 81-83, 131, 141, 144, 146,189,214,217,225,232,233 Sobolev, S.L. 26, 64,73 Soga, H. 203,204,225,233 Spagnolo, S. 226,228,234 Shananin, N.A. 146 Shubin, M.A. 7,32,37, 146 Stein, E.M. 100, 145 Sternberg, S. 11,142 Sternin, B. Yu. 23, 145, 146 Strang, G. 82, 141, 158, 166, 167,224,229, 234 Sukharevskij, LV. 225 Svensson, L. 159,224,234

239

Taniguchi, K. 26,27,189,231 Tarama, S. 226,232 Tarski, A. 91 Tartakoff, D.S. 146 Taylor, M. 7, 73, 146, 166, 180, 181, 215, 224, 225,234 Trtves, F. 7, 10,62, 72, 73,87, 88,98-100, 109, 110, 140,145,146,166,180,224,234 Tulovskij, V.N. 82, 146 Uhlmann, G. 83,142,146,188,189,225,232, 234 Unterberger, A. 15,26,27, 147 Vaillancourt, R. 25, 14 Vasil'ev, D.G. 39,147 Vishik, M.I. 15, 26, 147 Volevich, L.R. 153 Wakabayashi, S. 183, 190,200,225,234 Wasow, W. 99,147 Weierstrass, K. 167, 170, 171 Wenston, P. 147 Weyl, H. 18,29, 41 Yagdzhyan, K.A. 170,180,224,226,234 Yamaguti, K. 157 Yoshikawa, A. 189,225,234 Zajtseva, O.V. 224-227,234 Zuilly, C. 114, 140 Zygmund,A. 14

Subject Index

Subject Index

-, effectively hyperbolic 164 -, elliptic 17

quantization 40

-, elliptic at a point 218

ray, analytic 219 -,C" 214 -, crawling 220 -, generalized 216 -, glancing 220 -, incident 220 -, reflected 220 removable singularity 74

-, formally adjoint 17 -, Fourier integral 43, 150 -, Fourier integral operator with complex phase function 65

-, Girding hyperbolic in direction N

158 158 -, hyperbolic at a point x in direction N 156 -, hypoelliptic 13 -, integro-differential 15 -, Maslov canonical 46 -, microhyperbolic 210 -, microlocally hypoelliptic 13 - of constant strength 94 -, partially hypoelliptic 92 -, Poisson 117 -, principal type 22 -, pseudodifferential 15 -, real principal type 87 -, regularly hyperbolic 158 -, simplest integral 14 -, smoothing 16 -, strictly x,-hyperbolic 156 -, subelliptic 101 -, variable order 26 - with characteristic of constant multiplicity 166 -with characteristic roots of constant multiplicity 166 -, Weyl 18,29 -, x,-hyperbolic 156

-, Girding x,-hyperbolic

atlas, canonical 47 bicharacteristic billiards 21 1 212 - C"-boundary 212 -concave 212 - generalized 182 -limiting 185 - polygonal 186 -regular 185 -sheet 186 boundary, bicharacteristically non-degenerate 215 -, uniformly characteristic 194 boundary-value problem, normal 193 - boundary

Cauchy problem, Gi.1 regular 178 178 compatibility condition of order s 197 - condition of infinite order 191 composition of Fourier integral operators 59 - of pseudodifferential operators 16 condition, Ivrii-Petkov-Hormander 168 -, Levi 81 -, generalized Levi 176 -, Levi-Strang-Flaschka 166 -, Lopatinskij 200 -, uniform Lopatinskij 197, 199 cone, dual 160, 182 coordinates, local canonical 47 --,local--

-,flat 65 -, non-degenerate phase 51 - of canonical transformation, generating 20 -, operator phase 63 -phase 51, 167, 171 -, regular positive type 68 -,weight 26 Girding inequality 18 glancing surface 24 Gevrey class 91 -, inductive 174 -, projective 174 Hamiltonian field 163 hyperfunction 130 index, Maslov 47

-, well-posedness 162 integral, oscillatory 50 lacuna of distribution 119 Lagrange manifold 46 -subspace 46 lens, spatial type 156 localization at a point 160 Lopatinskij determinant 195 manifold, almost analytic 66

-, bicharacteristic 188 matrix, fundamental 163

dissipative boundary operator 209 distribution, Fourier 55 -, hypoelliptic 93 -, invertible 93 domain, bicharacteristically convex (concave) 205

-, Levi 102 -, smoothly diagonalizable 158

energy estimate 157 equation, characteristic 156 -, degenerate hyperbolic 169 -, eikonal 167, 171 -, Schrodinger 42

Neumann %problem 102 neighbourhood, conical 17

function, almost analytic 66

-, symmetrizable 158 -, uniformly diagonalizable 158 microlocality 17 microfunction 135

order of operator 15 operator, analytically hypoelliptic 108 -, classical pseudodifferential 15 -, completely characteristic 124

24 1

parametrix 18 phase, complex 65 point, critical 163 -, diffraction 126 -, elliptic 204 -, gliding 126 -, hyperbolic 204 -,tangent 204 Poisson brackets 20,40, 163 -kernel 118 proper map 11 pseudolocality 17 pullback of distribution 12 pushforward of distribution 12

set, characteristic

181, 209

-, diffractive 220 -, elliptic 204 -, stationary 187

-, tangent 204 stationary phase method 45 symbol 15 -boundary 212 -, characteristic 155 -, principal 15,155 -, real principal type 185 -, subprincipal 19, 164 symplectic form 163 transformation, boundary canonical

125

-, canonical 20 -,contact 138 -, Fourier-Bros-Iagonitr transmission 115

190

uniformly characteristic surface 194 wave front 9,180 --,boundary 208 --,boundary of order s 208 --,complete 208 -- oforders 180 well-posedness of Cauchy problem 153 __-- ,C" 155 ---- G("1 153, 174 ----,local G I x ) 174 _ - _ - , L , 156,157 well-posedness of mixed problem 190 _ _ _ - ,C" 191 _ - _ - ,L , (strong) 196 _ _ _ - ,L , (weak) 201