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    I4 : -.\ 4 '.,Y u X Egorov M . A . Shubin (Eds.) -.1 , -Partial DifferentialEquations 11- .

    Elements of the Modern Theory.Equations with Constant Coefficients

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    ContentsI. Linear Partial Differential Equations.Elements of the Modern Theory

    Yu. V. Egorov and M. A. Shubin1

    11. Linear Partial Differential Equationswith Constant CoefficientsA . I. Komech

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    I. Linear Partial DifferentialEquations.Elements of the Modern TheoryYu.V. Egorov, M.A. Shubin

    Translated from the Russianby P.C. Sinha

    ContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Notation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

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    2 Yu.V. Egorov. M.A. Shubin 3. Linear Partial Differential Equations Elements of Modern Theory2.2. Smoo thness of Solutionsof Second-order Elliptic Equation s ............................2.3. Connection w ith Pseudodifferential Operators2.4. Diagon alization of Hyperbolic System of Equation s2.5. Calderons T heorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

    to a Problem on the Boundary ............................... 40for the Second-order Equation to a Problem on the B oundaryfor an Elliptic System to a Problem on the Boundary

    373738. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .2.6. Reduction of the Oblique Derivative Problem2.7. Reduction of the Boundary-value Problem2.8. Reduction of the Boundary-value Problem . . . . 4143. . . . . . . . . . .Q 3 Wave Front of a Distribution and Simplest Theoremson Propaga tion of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

    3.1. Definition and Examples .................................3.2. Properties of the Wave Front Set ......................... 453.3. Applications to Differential Equations ..................... 473.4. Some Generalizations ................................... 4844

    Q 4. Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1. Definition and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2. Some Properties of Fourier Integral Operators . . . . . . . . . . . . . . 50

    with Pseudodifferential Operators ......................... 524.4. Canonical T ransformations .............................. 53 .and Fourier Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.6. Lagrangian Manifolds and Phase Functions . . . . . . . . . . . . . . . . 574.7. Lagrangian Manifolds and Fourier Distributions . . . . . . . . . . . . 594.8. Global Definition of a Fourier Integral Operator . . . . . . . . . . . . 59

    4.3. Composition of Fourier Integral Operators

    4.5. Connection Between Canonical T ransformations

    Q 5 . Pseudodifferential Ope rators of Principa l Type . . . . . . . . . . . . . . . . . 605.1. Definition and Examples ................................. 605.2. Operators with R eal Principal Symbol ..................... 61

    with Real Principal Symbol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63with Complex-valued Principal Symbol .................... 64

    5.3. Solvability of E quations of Principle Type5.4. Solvability of Op erator s of Principal Typ e

    Q 6 Mixed Problems for Hyperbolic Equations . . . . . . . . . . . . . . . . . . . . . 656.1. Formulation of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 656.2. The Hersh-Kreiss Condition . . . . . . .-. .................... 666.3. The Sakamoto C onditions ............................... 686.4. Reflection of Sing ularities on the Bounda ry . . . . . . . . . . . . . . . . . 696.5. Friedlanders Example ................................... 71

    6.6. Application of Canonical Transformations .....................6.7. Classification of Boundary Po ints . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.8. Taylors Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.9. Oblique Derivative Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7573

    Q 7. Method of Stationary Phase and Short-wave Asymptotics . . . . . . . . 787.1. Method of Stationary Phase ............................. 797.2. Local Asymptotic Solutions of Hype rbolic Equations . . . . . . . . 827.3. Cauchy Problem with Rapidly Oscillating Initial Data . . . . . . . 86and P ropag ation of Singularities of Solutions . . . . . . . . . . . . . . 87

    and Global Asymptotic Solutions of the Cauchy Problem . . . . 907.4. Local Parametrix of the Cauchy Problem7.5. The Maslov Canonical Operator

    Q8. Asymptotics of Eigenvalues of Self-adjoint Differentialand Pseudodifferential Operators ............................. 96in a Euclidean Domain .................................. 99of Approximate Spectral Projection ....................... 1028.4. Tauberian Methods ...................................... 1068.5. The Hyperbolic Equation Method ........................ 110

    8.1. Variational Principles and Estimates for.Eigenvalues . . . . . . . . 968.2. Asymptotics of the Eigenvalues of the L aplace Oper ator8.3. General Formula of Weyl Asymptotics and the Method

    Bibliographical Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113References .................................................... 114

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    4 Yu.V. Egorov, M.A. Shubin I. Linear Partial Differential Equations. Elements of Modem Theory 5Preface

    In this paper we have made an at tempt to present a sketch of certain ideasand methods of the modern theory of linear partial differential equations It canbe regarded as a natural continuation of our paper (Egorov and Shubin [ 9881,EMS vol. 30)where we dealt with the classical questions, and therefore we quotethis paper for necessary definitions and results whenever possible. The presentpaper is basically devoted to those aspects of the theory that are connected withthe direction which originated in the sixties and was later called "microlocalanalysis". It contains the theory and applications of pseudodifferential operatorsand Fourier integral operators and also uses the language of wave front sets ofdistributions. But where necessary we also touch upon important topics con-nected both with the theory preceding the development of microlocal analysis,and sometimes even totally classical theories. We do not claim that the discus-sion is complete. This paper should be considered simply as an introduction toa series of more detailed papers by various other au thors which are being pub-lished in this and subsequent volumes in the present series and which will con-tain a detailed account of most of the questions raised here.

    The bibliographical references given in this paper are in no way complete.We have tried to quote mostly books o r review papers whenever possible andhave not made any at tempt t o trace original sources of described ideas or theo-rems. This will be rectified at least partially in subsequent papers of this series.

    We express our sincere gratitude to M.S. Agranovich who went through themanuscript and made a number of useful comments.

    Yu.V. Egorov M.A. Shubin

    NotationWe shall use the following standard symbols.IR is the set of all real numbers.(c is the set of all complex numbers.Z s the set of all integers.Z, is the set of all non-negative integers.IR" s the s tandard n-dimensional real vector space.(c is the standard n-dimensional complex vector space.a/ax = (a/axl, . , /ax,), where x = (xl,. . ,x,) E IR".D = i-'a/ax, where i = (c; Dj = i-'a/axj.D" = D;*. D?,where a is a multi-index, hat is, a = (al,. a,) with aj E Z,.t" = tf ..52, where 5 = (tl,. ., ) E IR" or (c" and a = (a , , . . , a,) is ax - t = x , t , + - . . +x , < , if x = ( x, ,...,x , ) ~ I R " a n d { = ( t , ..., ,,)EIR .C;(52) is the space of C"-functions having compact, support in a domaind = A , = a2/ax: + ...+ a2/ax,2 s the standard Laplacian in IR".1x1 = ( ~ : + * * * + x , 2 ) ~ / ~ f o r x = ( x ~ ,.., X,,)EIR".la1 = a1 + ... + a,, where a is a multi-index.a = a , . . a, for a multi-index a.9'(52) is the space of all distributions in 52.&(a)s the space of all distributions with compact support in 52.L&2) is the Hilbert space of all square integrable functions in 52.S(IR") s the Schwartz space of C"-functions on IR" whose derivatives decayS'(IR") s the space of all distributions with temperate growth on IR".supp u denotes the support of a function (or distribution) u .sing supp u is the singular support of a distribution u.H'(IR") denotes the Sobolev space consisting of those distributions u E S(lR")H'(52), where s EZ,, s the Hilbert space containing those functions u EH~o,,,p(52) &(a) Hs(IR").H:oc(52) s the space of those u E 9'(52)uch that rpu E H'(IR") for any functionH'(52) is the completion of the space C;(52) in the topology of H'(52).Ca(52), where a E (0,h is the space of functions continuous in 51 such that

    multi-index.

    52 c IR".

    faster than any power of 1x1as 1x1-, 0.,

    for which (1+ 1{12)"/2ii(t)L,(IR"); here ii is the Fourier transform of u.L2(52) or which D"u E L2(52)with la1 < s.

    . r p E g w ) .

    sup lu(x)- u(y)l Ix - yl-" < co for each K c c 2.x . y e K

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    6 Yu.V. Egorov, M.A. Shubin I. Linear Partial Differential Equations. Elements of Modern Theory 70 1. Pseudodifferential Operators

    1.1. Definition and Simplest Properties (Agranovich [ 9653, Egorov [1984,19851, Eskin [1973], Friedrichs [1968], Hormander [1971, 1983, 19851, Kohnand Nirenberg [1965], Kumano-go [19823, Nirenberg [ 9701, Palais [1965],Reed and Simon [1972-19783, Rempel and Schulze [1982], Shubin [1978],Taylor [1981], and Trkves [19801). The theory of pseudodifferential operators,in its present form, appeared in the mid-sixties (Kohn and Nirenberg [1965]).Its principal aim was to extend to operators with variable coefficients the stan-dard application of the Fourier transformation to operators having constantcoefficients, in which case this transformation reduces the differentiation D" omultiplication by 5 .We consider the diflerential operatorin a domain 52 c R", where a, E Cm(52), D = i- 'a/ax and a = (al, .. ,a,) is amulti-index with (a1= al + ...+ a,,. We express the function u E C;(Q) bymeans of the formula for the inverse Fourier transform

    u ( x ) = (27r)-" eix '%(

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    8 Yu.V. Egorov, M.A. Shubin I. Linear Partial Differential Equations. Elements of Modern Theory 9where x # y. Since ( - A t ) N a ( x ,5 ) E Sm-2N,his integral can be differentiated ktimes with respect to x and y provided that m - 2N + k < -n. Because N isarbitrary, it follows that K AE C" for x # y, that is, off the diagonal in 52 x 52.

    In the general case, the kernel K A of the operator A is a distribution onSZ x 52.This result follows from the Schwartz theorem on the kernel (see Egorovand Shubin [1988; 0 1.11, Chap. 21 and Hormander [1983, 1985; Chap. 51) orcan be established directly as follows. For u , u E C,"(52), we write (Au, u ) as arepeated integral:

    ( ~ u ,) = (2x)-" 111 '(X-Y)'t x3 5 ) u ( y ) u ( x )dy d5 dx.We integrate this integral by parts and use the identity

    e i ( x - y ) . t = (1 + 1

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    10 Yu.V. Egorov, M.A. Shubin I. Linear Partial Differential Equations. Elements of Modern Theory 11

    Every operator A E L" is pseudolocal, this result being proved in the sameway as for the operators A = a ( x ,0,).The expression for A E L" in the form (1.22)is not unique. For example, onintegrating by parts, we can replace the amplitude a ( x , y , 5 ) by the amplitudea , ( x , y , 5 ) = 1 + Ix - 12)-"(1 - A J N a ( x , , 5 ) without changing the operatoritself. But the expression for A in the form a(x ,0,)educes this non-uniquenesssignificantly. For example, such an expression is, in general, unique if 8 IR",that is, the symbol a ( x , 5 ) is uniquely determined by A. Therefore it is desirableto simplify (1.22)by moving over, for example, to (1.6) with a suitably. chosensymbol. It turns out that this can be done to within operators with smoothkernels, as the following theorem shows.

    Theorem 1. Any operator A E L"(B),with amplitude a E S", can be expressedin the form A = oA(x,0,)+ R , where a E S" and R is an operator with kernelK , E C " ( 8 x 8) . his expression can be so chosen thatfor any integer N > 0.

    We shall indicate the main points of the proof later, and for the present wenote that all the terms in the summation (1.24) depend only on the values ofa ( x ,y , 5 ) and its derivatives for y = x (in particular, the principal term in thesummation is simply a ( x , y , 5 ) ) . This means that to within symbols of orderm - N (for any N) the symbol a is determined by the values of the amplitudea ( x , y , 5 ) near A x IR", where A is the diagonal in 52 x 8. In fact, K AECm(Q x 8)f a ( x , y , 5 ) = 0 in a neighbourhood of A x IR", because in this caseintegration by parts, with the aid of ( l . l l ) ,enables us to replace the amnlitudea ( x , y , 5 ) by the amplitude ( x- l -2N(- g ) Na ( x ,, 5 ) E S"-N without changingthe operator. Let us mention, by the way, that any operator R with a smoothkernel KR can be written in the form (1.22) with amplitude a R ( x , , 5 ) =( 2 n ) " e - i ( ~ - ~ ) T ~( x , ) $ ( < ) ,where the function II/ E C,"(IR") is such that $ ( 5 ) d5= 1. Clearly,a E S- , where

    S-" = n s".L - a = n L", S" = u s", L " = u L".

    m e RIn what follows, we shall also use the following notation:

    m e R me R m e RClearly,L-" is precisely the class of all operators with smooth kernels.the form of an asymptotic seriesThe set of relations (1.24),with N = 1,2, . will be written below in short in

    We recall that a = a, . . a, for any multi-index a.(1.24')

    More generally, if we have a system of functions aj = aj(x ,5 ) E SmJ, = 0, 1,2, . ,where mi + -co a s j + CQ, and a function a = a(x, 0,we shall writea - : a j

    j = Oif

    N- 1a - c aj E S""j = O

    (1.25)

    (1.26)for any integer N 2 0, where EN max mi. Instead of this last definition of EN,it is clearly sufficient to assume that EN re arbitrary numbers for which EN+-co as N + co.The function a is obviously defined uniquely up to addition ofany function from S-". Clearly a E S", where m = max mj. It can easily be seenthat for any sequenceaj E S' J here exists a function a such that (1.25)holds. Forthis it is sufficient to take

    j > N

    j > O

    (1.27)where x E C"(IRn) and ~ ( 5 ) 1 for' 1512 while ~ ( 5 ) 0 for 151 6 1 and thenumbers tj tend to +co sufficiently rapidly as j + co.The asymptotic sums for the amplitudes a(x, y , 5 ) are defined in an analo-gous manner.With the aid of asymptotic summation, it is useful to identify in S" the classS,";of classical or polyhomogeneous symbols. This class consists of symbolsa E S"that have a decomposition of the form (1.25)in which mj = m - and the func-tion aj is positive homogeneous in 5 , with 15I 2 1 , and of degree mi = m - , thatis,

    U j ( X , t 5 ) = t"-jaj(x, t), 151 2 1, t 2 1.The classical amplitudes can be defined in a similar fashion.Let ~ : - ~ ( x ,) be a positive homogeneous function in 5 (now for all 5 # 0)on52 x (IR"\O) which coincides with a j ( x , 5 ) for 1512 . Such a function is uniquelydefined, and for a E S,";we shall write

    '

    rn

    a - 2 a:-jj = O

    (1.28)in place of (1.25).This expression is well-defined because the functions a:-j alsodefine a(mod S-"). The function a = a:(x, t),which is homogeneous of degreem in 5, is called the principal symbol of the operator A .It is clear that the transit ion from the amplitude a to the symbol 0, by meansof (1.24) does not take us out of S,";,that is, 0, E S,";if a E S,";.The symbols ofdifferential operators are also classical. The results of a number of other opera-tions also remain within the class of classical symbols and amplitudes.

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    12 Yu.V. Egorov, M.A. Shubin I. Linear Partial Differential Equations. Elements of Modern Theory 13We now present the main points in the proof of Theorem 1 . 1 . First, as we

    noted earlier, by multiplying the amplitude a ( x ,y , 5 ) by a cut-off function x =~ ( x ,) E C (52 x 52) which is unity in a neighbourhood of the diagonal, we onlychange the operator A by adding an operator with a smooth kernel, and we canarrange that a ( x , y , 5 ) = 0 for ( x ,y ) 4 U , where U is an arbitrary pre-assignedneighbourhood of the diagonal in 52 x 52. We now expand a ( x , y , 5 ) in y by theTaylor formula for y = x :

    + C M y - )I'r,(x, Y , 0, (1.29)lul=Nwhere r, E S". Substituting this expansion into (1.22)and noting that

    [ i ( y- ) I"ei(x-~) t (- a g y e i ( x - Y ) 5 Ywe obtain, on integrating by parts the terms of the first sum in (1.29),operatorswith symbols which are equal to terms of the sum in (1.24).The remainder (thatis, the second sum in (1.29))can be transformed in the same manner into anoperator with amplitude in S"-N. This implies that if a symbola, is expressed asthe asymptotic series(1.24), hen the operator A - A(x,0,)will belong toL'"-Nfor any N and will therefore be an operator with a smooth kernel, as required.Theorem 1 . 1 easily yields formulae that express the symbols or,nd a, of thetranspose and formally adjoint operators in terms of a (mod S- ) . Indeed, towithin operators with smooth kernels, f l and A* can be taken to be defined bythe amplitudes

    'a = '4x7 y , 5 ) = a,(y, - 0, *(& Y , 5 ) = C A Y , 5 )(see (1.23)). t now follows from Theorem 1 . 1 that

    (1.30)(1.31)

    In particular, a - 5 EY-'.his implies that Im a, E S -' if the operatorA E L" is formally self-adjoint (that is, if it is symmetric on C, (52)). urther, ifthe operator is also classical, then the principal homogeneous part a j ( x , 5 ) (oforder m) of its symbol is real valued.

    We cite two important examples of pseudodifferential operators which arenot differential operators.

    Example 1.1 (One-dimensional singular integral operator). Let us consider onIR' an operator A of the formAu(x) = a ( x ) u( x )+ v.p.7 jrn(x' ' ) u ( y ) d y ,711 - x - y

    where a E Cm(IR),L E Cm(Rx IR) and v.p. denotes the "valeur principale" or theprincipal value of the integral, that is,v.p.7 sm-(x7 ) u ( y ) d y= lim 1 = u ( y ) d y .711 - x - y e-+O 711 l y -xl ,e x -

    By Hadamard's lemma, we write L ( x , y ) = b ( x )+ ( y - ) L , ( x ,y ) ,where b ( x )=L ( x , x ) and L , E Cm(IRx IR), and obtainAu(x) = a ( x ) u( x )+ b(x)Su(x)+ R l u ( x ) ,where R , E L - " and

    This transformation

    S is the Hilbert transformation defined by

    leads to the multiplication of a (5 ) by -sgn 5, and there-fore, to within an operator with a smooth kernel, A has the form a ( x ,&), whereQ ( X , 5 ) = a ( x )- ( x ) x ( < ) gn 5, with x E C"(IR) such that ~ ( 5 ) 1 for 1512and ~ ( 5 ) 0 for 151 < 1/2. Thus A is a classical pseudodiffirential operator oforder zero with principal symbol a ( x , 5 ) = a ( x )- ( x )sgn 5.an operator A of the formExample 1.2 (Multid imens ional singular integral operator ). We consider in IR"r m( x , x - y )

    Au(x) = a ( x ) u( x )* "

    where a E C"(IR") and L = L ( x ,o) Cm(IRn Sn- ') is such thatL ( x ,o) o = 0, x E IR".L1Here S"-' denotes the unit sphere in IR". Then the expression IzJ- L ( x , z) defines

    a homogeneous distribution of order - n on IR; which depends smoothly onx (see Egorov and Shubin [l988, 9 1 , Chap. 23 and Hormander [1983, 1985,0 .23).The Fourier transform of this function with respect to z is a distributiong ( x , 5 ) on IR: that is homogeneous in 5 of degree zero and smooth for 5 # 0,andalso depends smoothly on x E IR". Then it follows easily that, to within anoperator with a smooth kernel, A can be written in the form a ( x ,Dx), wherea ( x , 5 ) = a ( x )+ x ( r ) g ( x ,5) , with x E C"(IR") such that ~ ( 5 ) 1 if 151 2 1 and~ ( 5 ) 0 if 151< 1/2. In particular, A is a classical pseudodifferential operator oforder zero with principal symbol a ( x , 5 ) = a ( x )+ g ( x , 5).

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    14 Yu.V. Egorov, M.A. Shubin I. Linear Partial Differential Equations. Elements of Modern Theory 151.3. The Composition Theorem. The Parametrix of an Elliptic Operator

    (Egorov [1984, 19851, Hormander [1983, 19853, Kumano-go [1982], Shubin[1978], Taylor [1981], Tr6ves [1980]). Let us take two pseudodifferential oper-ators A and B: Cg(52)+ C"(Q). For the composition A o B to be defined, it isnecessary either that B maps Cg(52) into C,"(Q) or that A can be extended to acontinuous operator acting from C"(52) into C"(l.2). In fact, it is convenientto impose a slightly stronger condition that operators be properly supported.Namely, we say that an operator A E L"(52) is properly supported if both thenatural projections a,, a,: supp K , + 2 are proper maps. We recall that a mapf: + Y of two locally compact spaces is said to be proper if the inverse imagef - ' ( K ) of any compact set K c Y is compact in X. The property of A beingproperly supported is equivalent to the following two simultaneous conditions:1)for any compact set K c Q there exists a compact set K , c 2 such that Amaps Cg(K) into C;(K,); 2) the same is true for the transpose 'A.By truncating the kernel K Anear the diagonal, we can obtain the decomposi-tion A = A, + R for any pseudodifferential operator A E L"(Q), where R EL-"(Q) and A , is a properly supported operator. This remark enables us, with-out loss of generality, to confine our attention to properly supported opera-tors in a majority of cases.A properly supported operator A E L"(Q) defines the following continuousmaps:

    A : C,m(Q)+C,m(Q),A : C"(Q) + C"(Q),A : &(Q) + #(Q),A :9'(52)9'(52).

    Thus if one of the L"(52) operators A and B is properly supported, the composi-tion A 0 B is defined.

    To describe the symbol of the composition, we first examine the case ofdifferential operators A = a(x,0,) nd B = b(x,0,). et C = A o B. By theLeibniz formula, we have

    CU(X)= 4x3 0, + 0,)Cbb, D,MY)l I=a u :

    where we have used the Taylor formula to expand a(x, 0,series in 0,. his implies that C = c(x,0,) ith + 0,) as a power

    The sum here is finite because a is a polynomial in 5. This sum makes sense asan asymptotic sum if a E S 1 and b E Sm2.

    Theorem 1.2 (the composition theorem). Let A E Lml and B E Lm2 be twopseudodifferential operators in 52 one of which is properly supported. ThenC = A 0 B E L"1+"2 and C = c(x,0,) R , where R E L-" and c(x, 5 ) has theasymptotic expansion

    (1.32)This result can be proved by arguing in the same way as for differential

    operators but by confining Taylor expansion to a finite sum and estimating theremainder. An alternative way is as follows. Using the formula B = Y'B), werepresent B by means of the amplitude b ( y , 5 ) = aJy, -

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    16 Yu.V. Egorov, M.A. Shubin 17. Linear Partial Differen tial Equatio ns. Elements of Modern Theory

    and this is consistent with the definition of ellipticity of a differential operator(see Egorov and Shubin C1988, $2, Chap. 11).

    Theorem 1.3. I f A is an elliptic pseudodifferential operator of order m in 8 ,then there exists a properly supported pseudodifferential operator B E L-"(Q )such that(1.37)

    where Rj E L-"(8 ) , j = 1,2. Such a pseudodifferential operator B is unique up toaddition of operators with smooth kernels and is an elliptic pseudodifferentialoperator of order -m. If A is a classical pseudodifferential operator of order m,then B is a classical pseudodifferential operator of order -m.The operator B satisfying the hypotheses of this theorem is called a para-metrix of A . The fact that the parametrix of an elliptic pseudodifferential opera-

    tor is also a pseudodifferential operator shows that the choice of the class ofpseudodifferential operators is reasonable. In particular, the parametrix of anelliptic differential operator is a classical pseudodifferential operator.

    In order to construct the parametrix B of the operator A , it is necessary totake for the first approximation the operator B, E L-" whose symbol is a- ' (x , 5 )for large R ( K ) or any compact set K c8). owB, can be made a properly supported operator by addition of an operator witha smooth kernel. The composition theorem then implies that

    B O A I - R , , A 0 B = I - R , ,

    B O A= I - T,, A B , = I - T,; E L - ' ( 8 ) , j = 1,2.We now construct properly supported operators Bb and B: such that

    goN I + Tl + T: + * . , B;I N I + T, + T: + a * ,by which we mean that the symbols of Bb and B: are defined by the corre-sponding asymptotic sums of symbols of the operators on the right-hand sides.We then set B , = BbB, and B, = BOB:.This yields

    B , A = I - R , , A B 2 = I - R ; ; R ~ E L - , , j = l , 2 .On multiplying the first equation by B, on the right and using the secondequation, we find that B , - B, E L - "( 8) . Thus for B we can take either Bj. Wehave also established at the same time that B is unique up to operators be-longing to L-"(Q).

    The existence of a parametrix implies that the solutions of elliptic equationswith smooth right-hand sides are regular. To see this, let u Eg (8 ) nd Au =f E g (8 ) C ( 8 , ) , where8, Q. Here we have assumed that Au is meaning-ful, and for this it is sufficient, for example, that f E #(a) r that A is properlysupported. Then u E Cm(8,) because, by applying B to both sides of the equa-tion Au = f, we obtain u = Bf + R , u . Then, since B is pseudolocal, it followsthat Bf E Cm(Q,)and R , u E Cm(Q)as R , is an operator with a smooth kernel.More precise regularity theorems can be formulated in terms of the Sobolevnorms, and this will be done below.

    We shall describe in some detail the structure of the parametrix B for aclassical elliptic pseudodifferential operator A of order m. Suppose that thesymbol a ( x , 5 ) of A has the asymptotic expansion (1.28). Let the symbol b(x , 5 )of the parametrix B have a similar expansion

    a (1.38)If we use the composition formula for finding B o A - , then all the homoge-neous components must vanish. We now group the members of the seriesdefining the symbol of B 0 A - according to the degree of homogeneity andobtain the equations

    bOma: = 1,4 (1.39)

    to determine the functions brl)r+ The first of these equations implies that b , =(a:)-'. The other equations enable us to determine by induction all the membersof the sum in (1.38), so defining the parametrix to within operators belonging toL- (Q).

    1.4. Action of Pseudodifferential Operators in Sobolev Spaces and PreciseRegularity Theorems for Solutions of Elliptic E quations(Egorov [1984, 19851,Hormander C1983, 19851, Kumano-go [1982], Shubin [1978], Taylor [198l],Trkves [1980]). A key to the discussion of pseudodifferential operators inSobolev spaces is the following theorem.

    Theorem 1.4. Suppose that the operator A = a(x ,0,)E Lo(IR") as a symbola ( x , 5 ) hat satisfies the estimates(1.40)a;a a(X, 5)l < C@(1+ 151)-la', x , 5 E IR".

    Then A can be extended to a continuous operatorA : L,(IR")+L,(IR").The distinction between the estimates (1.40) and the usual estimates (1.7),

    which define the class So(lR" IR") of symbols, consists in the following. Theconstants Cap n (1.40) do not depend on x whereas the estimates (1.7) aresatisfied for x E K with constants CaaKwhich depend on K for any compact setK c IR". n particular, the estimates (1.40)hold for any symbol a E S'"(IR" x R")such that a ( x , 5 ) = 0 for 1x1 > R or, more generally,a ( x , 5 ) = a,(t) for 1x1 > R ;that is, for a symbol that either vanishes or does not depend on x for large x . Inparticular, the singular integral operators of Examples 1.1 and 1.2 can be ex-tended to bounded operators in L,(IR )provided that the defining functionsa ( x )and L ( x , y ) are such that the operator in question reduces to a convolutionoperator in a neighbourhood of infinity. This means that a ( x )= a, for 1x1 > R

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    18 Yu.V. Egorov, M.A. Shubin 19. Linear Partial Differential Equations. Elements of Modern Theoryand L ( x , y ) = Lo for 1x1 > R or lyl > K in Example 1.1 while, in Example 1.2,L ( x , z ) = Lm(z).One of the possible proofs of Theorem 1.4 is based on the algebraic formal-ism already developed. To see this, we use the composition theorem, modifiedfor the case where the est imates of symbols are uniform in x , and choose aconstant M > 0 such that M > l i m sup la(x,

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    20 Yu.V. Egorov, M.A. ShubinHere x ;( y) denotes the Jacobian matrix of the map x , at the point y and 'x;(y)denotes the transpose of x; (y ). If A is a classical pseudodifferential operator oforder m, then A , is also a classical pseudodzflerential operator of order m. Further-more, the principal symbol a of A , is given by the formula

    a:m(Y, q) = a : ( x l ( ~ ) ,' ~ \ ( Y ) ) - ' V ) , (1.44)where a is the principal symbol of A.by the formula (1.22). This immediately yieldsTo establish this result, we expressA in terms of the amplitude a ( x , y , 5) E S"

    a(x , (x) ,Y , t ) U ( X ( Y ) ) dY d t -, ~ ( ~ )2 q n e i ( x l ( x ) - ~ ) .

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    22 Yu.V. Egorov, M.A. Shubin 23. Linear Partial Differential Equations. Elements of Modern TheoryUsing a partition of unity, a pseudodifferential operator on a manifoldM can

    be constructed by gluing. Namely, suppose that there is covering of M by thecoordinate neighbourhoods; that is, M = uQ j . Let Aj E Lm(Qj) or any j . Weconstruct a partition of unity 1 = c qj that is subordinate to the given covering,by which we mean that qj E Cm(M),the sum is locally finite and supp qi c Q j .We choose functions jE Cm(M)uch that supp jc Q j and i,hjqj= qj,and suchthat the sum1hj is also locally finite '. We denote by the operatorsof multiplication by qj and $j espectively. Then we can examine the operatorA = c Y,AjQj. t can be easily shown that A E L"(M) and that A E L,"1(M)fAj E L,"l(Qj)for any . Similarly, by using matrix pseudodifferential operators, wecan glue a pseudodifferential operator in the bundles. By this procedure we canconstruct, for example, the parametrix B E L-" of any elliptic operator of orderm on M . This gives

    and

    the space of smooth sections of F and C;(M, E ) is the space of smooth sectionsof E having compact support. Now let x be the projection of the vector (x, 5 )onto M . For any non-zero (x, 4 ) E T * M , the principal symbol a: = &(x, 4 ) ofthe operator A E L,"1(M, , F ) defines a linear map of fibres

    a:(x, 5) : E x + Fx. (1.47)Thus altogether we have a bundle map

    a,: 710E + n 8 F , (1.48)where no:T*M\O +M is the canonical projection of the cotangent bundlespace without the zero section onto the base M ; n;E and n,*F are the inducedbundles, with fibresE x and F, above each point (x, 5 ) E T*M\O. An operator Aof the form (1.46)s said to be elliptic if all its local representatives (obtained byall choices of the coordinate neighbourhood s2, the coordinates on it and thetrivializations E l , and FI are elliptic. These representatives are matrix pseu-dodifferential operators and their ellipticity means that .la-'(x, 411 < C151-", 151 2 R , x E K , (1.49)where C = C ( K ) ,R = R ( K ) and K is an arbitrary compact set in Q. We notethat, in the scalar case, these estimates are equivalent to (1.35). or a classicalpseudodifferential operator A E L ; (M; E , F ) the ellipticity means that all themaps (1.47) re invertible, that is, the map (1.48)s a bundle isomorphism.Example 1.3 ( A ingular integral operator on a smooth closed curoe). Let r bea smooth closed curve in the complex plane. Suppose that r s oriented, that is,a direction for going along the curve has been fixed. On r we consider anoperator A : C m ( r )+Cm(r)defined by the formula

    Au(z)= a(z)u(z)+ v.p.4 1 g u ( w ) w,ni r z - wwhere a E Cm(r),L E C m ( r x r ) nd dw denotes the complex differential of thefunction w : r + CC defined by the embedding of T in CC; the principal value of theintegral is understood in the same sense as in Example 1.1.By introducing localcoordinates on r whose orientation is consistent with that of r,we easily findthat in any local coordinates the operator A becomes the operator of Example1.1.ThereforeA is a classical pseudodifferential operator of order zero on f.Wecan assume that u (z ) s a vector function with N components, and that a( z )andL (z , w ) and N x N matrix functions. Then A becomes a matrix classical pseu-dodifferential operator of order zero. Its principal symbol is a matrix functionQ = Q ( Z , 5 ) on PT\O = r x ( lR \O) that is homogeneous in 5 of degree zero,and is given by

    a( z , 5 ) = a( z )- ( z )sgn 5, b( z )= L (z , z).The ellipticity condition for A in the scalar case means that a2( z )- Z ( z ) 0 forz E r,while in the matrix case it means that the matrices a( z )- (z ) and a( z )+b(z)are invertible at all points z E r.

    B =1 BiQj,i (1 SO)where Bj denotes the parametrix of the operator A,,. Moreover, we have

    B o A = I - R , , A 0 B = I - R 2 , R ~ E L - " . (1.51)More precisely, if A is an elliptic operator on C;(M; E ) into C"(M; F), then Bis a properly supported pseudodifferential operator that maps C ; ( M ; F ) andCm(M;F ) into C,"(M;E ) and Cm(M; ) respectively, and

    R , E L-"(M; E, E), R , E L-"(M; F, F).In the case of a compact manifold M , it is also convenient to introduce theSobolev section spaces H s ( M ;E ) that are defined as the spaces of sections be-

    longing to HS,, in local coordinates on any coordinate neighbourhood Q Mand for any choice of trivialization of E above Q. If R E L- ' (M ; E , E ) and 1 > 0,it follows from the discussion of 9 1.4 that R defines a compact linear operatorin H "( M;E ) for any s E lR.By the well-known Riesz theorem, the operatorsI - R , and I - R , are Fredholm in the spaces H s ( M ;E ) and HS-"(M; F ) re-spectively. This result, together with (lSl),mplies that the elliptic operatorA E L"(M; E, F ) of order m defines a Fredholm operator

    A : H"M; E ) +H"-"(M; F ) (1.52)for any s E IR.The kernel Ker A of this operator belongs to C"(M; E ) and istherefore independent of s. By using a formally adjoint operator A*, constructedby means of any smooth density on M and smooth scalar products in fibres ofthe bundles E and F, we can easily show that the image of A in the space

    'The sumsjocally finite, property that can always be assumed to hold without loss of generality.(pj and1 j will automatically be locally finite if the covering M = U Zj is itself

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    24 Yu.V. Egorov, M.A. Shubin 25. Linear Partial Differential Equations. Elements of Modern TheoryH"-"(M; F) can be described by means of orthogonality relations to a finitenumber of smooth sections. In this way we find that dim Coker A is also inde-pendent of s. Thus we have

    Theorem 1.6. If A E L"(M; E, F) is an elliptic operator of order m on a com-pact manfold M, then A defines a Fredholm operator (1.52) for any s E IR suchthat both dim Ker A and dim Coker A are independent of s.

    In particular, the index defined byind A = dim Ker A - dim Coker A (1.53)

    is also independent of s.We note that the index can be understood simply as theindex of the operator A: C"(M; F) + Cm(M; ).

    We note further that if A is invertible under the hypotheses of Theorem 1.6(either as an operator from C"(M; E) into C"(M; F) or as an operator (1.52)for any s E IR), this being equivalent to the conditions that Ker A = 0 andKer A* = 0, then A-' is again a pseudodifferential operator belonging toL-'"(M;F, E), and it will be classical if the operator A is itself classical. Indeed,multiplying both sides of the second equation in (1.51) from the left by A-', weobtain

    A-' = B + A- 'R , . (1.54)Now A-' is a continuous map from Cm(M)nto Cm(M),and therefore A - ' R , isan operator whose kernel KA-LR2(x, y) = [A-'KR,(*, y)](x) lies in Cm(M M).Thus

    A-' - B E L-"(M), (1.55)which shows that A-' coincides with B to within operators with smooth kernels.We see that the calculus of pseudodifferential operators enables us to describethe structure of the operator A-' and even find it explicitly modulo L-"(M). Inparticular, if A is a classical operator, then the homogeneous components born+of the symbol of A-' are given by (1.39).

    1.6. Formulation of the Index Problem. The Simplest Index Formulae(Fedosov [ 974a1, Palais [1965]). According to well-known theorems of func-tional analysis, for any Fredholm operator A: H1 +H,,where H, nd H, reHilbert spaces, there exists an E > 0 such that the operator A + B is Fredholmand

    ind(A + B) = ind Aprovided that the operator B: H,+H, has the norm 11B(1< E. In particular,ind A remains unchanged under any deformation of A which is continuous inthe operator norm and does not take us out of the class of Fredholm operators.Furthermore, if A: H, + H, s Fredholm and T: H1 + H, s a compact opera-tor, then A + T is also Fredholm and

    ind(A + T) = ind A.

    This result and Theorem 1.6 imply that the index of an elliptic operator on acompact manifold depends only on the principal symbol of this operator andremains unchanged under continuous deformations of this principal symbol.Thus the index is a homotopy invariant of the principal symbol, and thereforewe can expect that the index can be expressed in terms of the homotopy in-variants of the principal symbol. The problem of computing the index of anelliptic opera tor was formulated in 1960 by Gel'fand and it was solved in thegeneral case in 1963 by Atiyah and Singer (see Palais [1965]). The Atiyah-Singer formula prescribes the construction of a certain differential form basedon the symbol of the elliptic operator A, and integration of this form yields theindex of A. Without writing down the general formula, we mention two of itsspecial cases which were known before the publication of the Atiyah-Singerwork.A. The Noether-Muskhelishuili formula. This formula gives the index of thematrix elliptic singular integral operator of Example 1.3 on a closed oriented *curve r which, for simplicity, we assume to be connected. The formula is of theform

    ind A =-arg det[a(z, l)-'u(z, - )]1,2n1

    2n- rg det [(a(z)- b(z))-'(a(z) + b(z))] Ir, (1.56)where the notation of Example 1.3 has been used and arg f ( z ) l r denotes theincrement in the argument o ff (z) on going round r n the chosen direction.B. The Dynin-Fedosou formula. This formula concerns a matrix ellipticoperator A = a(x,0,) f order m in IR" that coincides in the neighbourhoodof infinity with an operator a,(D,) having a constant symbol a,( 0 is sufficiently arge. Here a-l da is understood as the matrix ofdifferential 1-forms on IR: x IR , and (a-' da)'"-' denotes the power of thismatrix in the computation of which the elements of a-' da are multiplied by thewedge product. Thus (a-' da)'"-' is the matrix of (2n - 1)-forms and its traceis the usual (2n - 1)-form which is further integrated over a (2n - 1)-dimen-sional manifold SR= {(x,5 ) : 151= R}. This manifold is oriented as the bound-ary of the domain {(x,

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    26 Yu.V. Egorov, M.A. Shubin[ a - ' ( x , () d a ( x , ()]2"-' vanishes for large x. This is because the form a-1 da isexpressed only in terms of d ( , , . , d ( , if a = a(() and any exterior (2n - 1)-form of n variables vanishes. Therefore, for n 2 2, the integration in (1.57) isactually performed over a compact set and consequently makes sense. The inte-gral also makes sense when n = 1, because if the 1-form contains only d( andnot d x , then its restriction to S, is zero, that is, the integration is again per-formed over a compact set. For n = 1, the formula (1.57) is easily reduced to(1.56) in the preceding example (for the case of a classical pseudodifferentialoperator A on IR which coincides with an invertible pseudodifferential opera-tor a,(D) in a neighbourhood of infinity).Let us mention two more simple particular cases of the Atiyah-Singer for-mula.C. Suppose that A is a scalar elliptic pseudodifferential operator on a com-pact manifold M of dimension n 2 2 where, by the term scalar, we mean that Aacts on scalar functions rather than vector functions. Then ind A = 0. When Mis a sphere, this result is a consequence of (1.57) because the wedge product of ascalar 1-form with itself is always equal to zero.D. Let A be a differential (and not pseudodifferential ) operator which actsin the sections of vector bundles above an orientable compact manifold Mwhose dimension is an odd integer. Then ind A = 0. When M = S (n s odd)and the bundles are trivial, this result can be easily derived from (1.57). We dothis by reducing the problem to the case where the principal symbol of theoperator (a polynomial matrix that is homogeneous in () occurs in place of a;this is achieved by homotopy. Then we follow the action of the map (H - onthe index and the integral.

    1.7. Ellipticity with a Parameter. Resolvent and Complex Powers of EllipticOperators(Agranovich and Vishik [1964], Seeley [1967, 1969a, 1969b1, Shubin[1978]). We have already discussed (Egorov and Shubin [1988, $2, Chap. 23)for boundary-value problems the ellipticity condition with a parameter thatguarantees the unique solvability of the boundary-value problem for largevalues of the parameter. Here we deal with the similar condition for pseudo-differential operators on a manifold M .Let A be a closed angle with vertex at the origin of the complex plane (c, andwe allow the possibility that A may degenerate into a ray. Let A be a classicalpseudodifferential operator of order rn > 0 on M . Here and in what follows wetake A to be a scalar operator for simplicity, although all the results obtainedhere can be easily extended to the case of operators acting in the sections of thegiven vector bundle E . Suppose that a = a:(x, 0 for 1 A and [(I ' + 111 = 1. (1.58)This condition (perhaps with a smaller 8) continues to be satisfied if the angle Ais slightly enlarged. Therefore we can assume that the angle A does not degener-ate into a ray.Suppose that the asymptotic expansion of the symbol a ( x , 5 ) of the operatorA in Q in terms of homogeneous functions is of the form (1.28).It follows from (1.58) that it is possible to extend the function a:(x, 5 ) fromthe set 2'= ((x, 5 ) : x E Q, 151> l} to a smooth function a,(x, 5 ) on Q x IR"that satisfies

    (1.59)Also, in particular, a, E Sm(Q x IR"). Let am P j= u,-~(x, (), j = 1,2 , . similarlydenote any extension of the functions U : - ~ ( X , ) from 2' to the functions ECm(Qx IR"). Then automaticallyThe construction of the required parametrix proceeds on the same lines as ifthe function a, - 1 s the principal symbol of A - 1I.We may then expect, onaccount of the composition theorem, that a good approximation to the re-solvent in Q will be the operator B(1) whose symbol is the sum

    la,(x, 5) - 11 2 > 0 for ( x , 5 ) E 52 x IR", 1 E A .

    E S -j(Q x IR").

    (1.60)where K > 0 is sufficiently arge and the components b-m-k are found from thefollowing equations:

    (compare with (1.39)).

    (1.61)

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    28 Yu.V. Egorov, M.A. Shubin I. Linear Partial Differential Equations. Elements of Modern Theory 29NSuppose now that there is a finite covering M = u Q j of the manifold M by

    coordinate neighbourhoods Qj . Let Bj(A)be the parametrix of A - Z on Qj,constructed as above. We glue the operators Bj = Bj(A) n accordance with (1.50)to obtain B = B(A).This is the required parametrix of A - Z. Let us describe itsproperties.The fundamental fact that follows by analysing the composition B ( A ) ( A- Z)in the spirit of the proof of the composition theorem consists in the following.There exists an integer N > 0, depending on K in (1.60), such that (i) N -, +00as K -, + 0 0 ; (ii) if K , = K , (x , y, A) is the kernel of the operator R(A)=B(A)(A- Z) - , then K , E C N ( Mx M ) or each fixed A; (iii) if L is a differen-tial operator on M x M of order 0. Thus it follows from (1.65) and (1.66) that( A - z) -1 = B(A)+ T(A), (1.68)

    where the operator T(A)has a sufficiently smooth kernel that decays as O ( l A l - * )for A E A , 111 -, 00, together with any large number of derivatives (depending onK ) . In this way we have established the desired result regarding the existence ofthe resolvent ( A - AZ)-' and its approximate representation to within O ( l A l - 2 ) .From (1.66), (1.67) and (1.68), we have the following estimate for the norm ofthe resolvent:ll(A' A Z ) - l l l s , s < CNIAl-', A E A , 111 2 Ro; s E [ - N , N]. (1.69)

    This is true for any N > 0 because we can take the parametrix B(A) o containas many terms as we like.It also follows from (1.68) that ( A - Z)-' is a classical pseudodifferentialoperator of order -m. This operator is compact in L 2 ( M ) .This fact, togetherwith well-known theorems of functional analysis (see Gokhberg and Krejn[1965]), implies that the spectrum a(A) of A in L 2 ( M ) s a discrete set of pointswith finite multiplicities. Here A should be regarded as an unbounded operatorin L 2 ( M )with domain of definition H"(M). Further, only a finite number ofpoints of the spectrum can lie in A . Therefore by narrowing A we can arrangethat o ( A )n A contains no more than the single point 0. What is more, for theoperator A - Z with any fixed 6, E (C\o(A), we find that 0 4 a(A - ,Z). Onreplacing A by A - o l if necessary, we can assume in the sequel that 0 4 a(A),that is, A is an invertible operator. Now, by narrowing A if necessary, we canarrange that a(A) n A = 0,nd this will also be assumed to hold in the sequel.Under these assumptions, we can construct the complex powers A" of theoperator A . To do this, we choose a ray L = (reicp0:B 0 } lying in A in thecomplex A-plane, and construct a contour f n the following manner: f =r,u 2 v r',where

    A = reivo ( r varies from +co to p > 0) on f,,A= pe" (cp varies from cp, to cpo - 2n) on f 2

    andA = rei(cpo-2n) (r varies from p to +a) on f,.

    The direction along f s provided by f,,, nd f in that order (see Fig. 1).the spectrum of A . We now setWe must choose p > 0 so small that the disc ( A : 121 < p } does not intersect

    (1.70)where z E (c, Re z < 0 and A" is defined as a holomorphic function of 1 n C\L.Thus

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    30 Yu.V. Egorov,M.A.hubin I. Linear Partial Differential Equa tions. Elements of Modern Theory 31

    --R e AIII

    Fig. 1

    1 = zlnA = ez l n l A) +i za rg A = 2 i z a r g 114 f? 9where arg 1 s so chosen that cpo - 12< arg 1< cpo and, naturally, we takearg 1= cpo on r,and arg 1 = cpo - x on r,.In view of (1.69), when Re z < 0, the integral in (1.70)converges in the opera-tor norm in the space Hs((M)for any s E W.We substitute the expression (1.68)for ( A - Z)-' into (1.70) and use the estimates (1.67) for the norms of T(1 ) oobtain

    A , = B, + R,, B, = &jf z B ( l ) A , (1.71)where the operator R , has a kernel that is as smooth as desired (depending onK ) and depends holomorphically on z . In local coordinates the operator B, isrepresented, to within an operator with a smooth kernel that depends analyti-cally on z , in the form B, = b(')(x,Dx),where

    (1.72)We note that each of the integrals in (1.72) is an integral of rational functions,and this integral can be expanded if we use the expressions for b-m-k obtainedfrom (1.61). For instance, by the Cauchy formula, the principal term of (1.72)hasthe form

    where, in computing the powers a;, we use the same values of the argument a,as we used in computing 1 in the integral (1.70). Likewise, the remaining inte-grals contain rational functions with the only pole 1= a,(x, r ) , and they turn

    out to be smooth functions of x and ( and holomorphic functions of z. Further-more, the fact that the functions b-m-k(x , ) are homogeneous of degree -m -in (5, A' ), for 1512 1, implies tha t(1.73)

    is homogeneous in (ofdegree mz - , for I( 2 1, and depends holomorphicallyon z. Therefore B, is a classical pseudodifferential operator of order mz whosesymbol depends holomorphically on z, and hence so is the operator A,. We notethat earlier we examined classical pseudodifferential operators of real orderonly, but classical pseudodifferential operators of any complex order are definedin an analogous way.We note that with the aid of the Cauchy formula, we can easily deduce, first,the group property of the operators A, , namely, that

    A,A, = A,+,, Re z < 0, Re w < 0, (1.74)and, secondly, that A _ , = A - ' . Hence also A-k = A - k for any integer k > 0.Using these facts, we can correctly define the operators A for any z E C by

    A = A 0 k EZ, k > Re z , k 2 0. (1.75)It follows easily from the above properties of A , that A is a classical pseu-dodifferential operator of order mz whose principal symbol is [a:(x, ()I2. All thehomogeneous components of the symbol of A (in local coordinates) dependholomorphically on z and, moreover,

    where the operator T f ) has a kernel of class CN or Re z < do and N =N ( K , d o )-,+co as K -, +co for any fixed d o . The derivatives of this kernel upto order N are holomorphic in z in the half-plane { z :Re z < d o } . Thus theoperators A depend, in a natural sense, holomorphically on z.It is natural to refer to the operators A" as the complex powers of a pseu-dodifferential operator A . In fact, (1.75) easily implies that for any integer z theoperator A coincides with the usual integral power of A and, in particular,A = Z and A' = A . Further, A = A , for Re z c 0 and the group property issatisfied for all z :

    A 0 A" = A ' + W , z, w E c. (1.76)Finally, suppose that A has an eigenfunction $: A$ = A$, this implying that$ E C"(M) since A is elliptic. Then, by the Cauchy formula, we immediately seethat A& = A'$, that is, $ is an eigenfunction of A corresponding to the eigen-value A In particular, if the operator A is self-adjoint and positive (under ourassumptions, positivity of A implies that we can take the ray(-a,1 for the rayL) , then, by examining the values of A on a complete orthogonal system of

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    32 Yu.V. Egorov, M.A. Shubin 33. Linear Partial Differential Equations. Elements of Modern Theoryeigenfunctions of A , we find that our definition of A" coincides with the usualdefinition in spectral theory.

    Thus the calculus of pseudodifferential operators enables us to describe thestructure of such important objects of operator calculus as the resolvent andcomplex powers of elliptic operators on compact manifolds. We shall see laterthat it plays an equally important role in the theory of boundary-valueproblems for elliptic equations.

    1.8. Pseudodifferential Operators in IR" Hormander [ 983, 19851,Kumano-go [1982], Shubin [1978]).Although the Euclidean space I R can be regarded asa particular case of a manifold, and thus one can speak about classes Lm ofpseudodifferential operators on IR", there often arise other useful classes of pseu-dodifferential operators on IR"which are connected with the additional alge-braic structures present on IR". As mentioned above, we can, for example, con-sider operators on IR"with uniform estimates (with respect to x ) or symbols (seeTheorem 1.4,he remark following the statement of this theorem and also Theo-rem 1.4).To make this remark precise, we introduce a class of symbols S;(IR")consisting of functions a = a(x, 5) E C"(IR" x IR") which satisfy the uniform es-timates (1.42). he corresponding operators a(x ,0,) cting according to theusual formula (1.6) ap the space S(IR")and the space

    ?( ) - u : u E C"(IR"), sup la"u(x)l< 00 for all ac IR"-{ X E R"into themselves. In particular, we can form the composition of operators of thisclass without the requirement that one of the multiplying operators is properlysupported. Furthermore, the composition theorem, Theorem 1.2, emains validin this class as do the formulae (1.30) nd (1.31) hich define the symbols of thetranspose and adjoint operators, the asymptotic expansions being understoodup to symbols belonging to the classes S,-N(IRn),where N + 00. We can also usethe amplitudes a(x, y, 5 ) which satisfy the estimates

    (1.77)The operators with such amplitudes can also be defined by the symbol a, ES:(IR ), and Theorem 1.1 remains valid. If the operator A = a(x ,0,) ith sym-bol a E Sr(IR )s uni$oformly elliptic, tha t is, if there exist E > 0 and R > 0 suchthat

    la (x ,5)l > &15Irn, x E IR", 151 > R , (1.78)then we can find the parametrix of A , this being an operator B = b(x ,0,) ithsymbol b E Slm(IR") uch that B A - and A B - have symbols belonging toS;"(IR") = n S;(IR"). We note in passing that this statement does not imply inany way that the operator A is Fredholm. This is because the operators withsymbols in S,-"(lR") are not necessarily compact in L2(lR"). or example, this setcontains, among others, all those operators a(D) with constant symbols a =

    iapxslayswx, y, 511G cuslBz(iK I ) ~ - Y X , y, 5 E IR".

    m

    ~(

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    34 Yu.V. Egorov, M.A. ShubinThe connection between the Weyl symbol and the usual symbol of a givenoperator is easily established by Theorem 1.1. Thus, the usual symbol a isexpressed in terms of the Weyl symbol a by the formula

    while the inverse corresponde nce following from this result is(1.82)

    (1.83)The Weyl symbol is also often referred to as a symmetric symbol. The meaningof this terminology can be easily und erst ood the opera tor with the Weyl symbola = x j t j is 3(xjDj+ Djxj) while the operator with the usual symbol x j t j is xjDjand the operator w ith the amplitude y j c j s Djxj .The problem of establishing a correspondence between functions on thephase space lR: x lR; and operators arises in a natural way in quantum me-chanics, where such a correspo ndence is know n as quantization. The presence ofdifferent kinds of symbols reflects the fact tha t qu antization is, in principle, notunique. In particular, the correspondence a f-) aw(x,0,) etween the operatorsand the ir Weyl symbols is often referred to a s the Weyl quantization.In exam ining the transition t o classical mechanics from quantum mechanicsin the problem of quantization, the presence of a small parameter h, knownas the Planck constant, has to be taken into account. This parameter usuallyh aalso appears in the quantization under which the momentum operator - axjcorresponds to the functions t j . n view of this situatio n, one can assoc iate withthe function a(x,

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    $2. Singular Integral Operators and their Applications.Calderon's Theorem. Reduction of Boundary-valueProblems for Elliptic Equations to Problems onthe Boundary2.1. Defini tion and Boundedness Theorems (Agranovich [1965], Bers, John,

    Schechter [1964], Calderon and Zygmund [1952], Mikhlin [1962], Mizohata[1965]). By a singular integral operator we mean an operatorA : U ( X ) H K ( x ,x - ) U ( Y ) dy, x E IR", y E IR", (2.1)

    (2.2)K ( x , ) dS = 0, x E IR". (2.3)

    ss rl=l

    Au(x) = lim j

    where the function K ( x ,z ) has a singularity for z = 0 only, andK ( x , z ) = t - " K ( x , ) for t =- 0, z E IR"\O,

    In this case, the integral (2.1)can be defined in the sense of the principal value(v.P.), hat is,

    K ( x ,x - ) U ( Y ) dy.E + + O e < I x - y I < RR + m

    The existence of the integral can be established easily if, for example, K ( x , z ) isbounded for x E R", IzI = 1 u E Co(lR").

    Example 2.1. The Hilbert transformation

    The boundedness of this operator in L2(IR) s a consequenze of the Parsevalidentity, because the transition to Fourier transforms yields H u( < )= sgn < * ii( l i f

    IK(x , z) I4dz< C for x E IR",s rl=lwhere the number q i s such that p - l + q-' = 1.

    Theorem 2.2 (Holder, Korn, Lichtenshtein and Giraud; see Bers, John andSchechter [1964]). If K E C'(IR" x S"-' ), then the operator A defined by (2.1) sbounded in Cy(IR") or y E (0, 1).

    372.2. Smoothness of Solutions of Second-order Elliptic Equations (Bers, John

    and Schechter [1964]). We first assume that 52 is a bounded domain in IR" withsmooth boundary. Let d u = f n 52. A well-known formula (see (2.9) n Egorovand Shubin [1988]; $2,Chap. 2)enables us to express u in the fo m u = u1 + u 2 ,where

    U l ( X ) = E(x - M Y ) dy,J*

    I *and u 2 is expressed in terms of an integral over the boundary 852.For x E 52, thefunction u2 is always infinitely differentiable, and so u has the same smoothnessinside 52 as u l . If a = (al,. ,a,,) and IaJ= 2, then D"ul is given by the singularintegral

    D"ul(X) = D"E(x - )f(y) dy.Applying Theorems 2.1 and 2.2, we find that the following statements hold.1". If d u E L,(52) and p E (1, co), hen D"u E L,(sz'), where la1 = 2 and sz' C2 . If du E CY(52) nd y E (0, l), then D"u E CY(Qr),here la1 = 2 and sz' C 52.These results remain valid, and are also proved by Theorems 2.1 and 2.2,for

    general elliptic operators with smooth coefficients. Thus, if P ( x , D ) is an ellipticdifferential operator of order m with smooth coefficients, then the followingassertions hold.

    1". If P ( x , D) u E L,(O) and p E (1, GO), then D"u E L,(52'),where la( = m and52' C 52.2". If P ( x , D)u E Cy(Q) nd y E (0, ), then D"u E CY(sz ' ) ,where la1 = m andsz' C 52.

    52.

    2.3. Connection with Pseudodifferential Operators (Agranovich [ 9653,Hormander [1983, 19853, Taylor [1981]). A pseudodifferential operatorP ( x , D) , with symbol p ( x , c , in a domain 52 c IR" can be expressed as anintegral operator by the formula

    P ( x , D)u(x) = K ( x ,x - ) U ( Y )dy,sthe kernel of which is the distributionK ( x , z ) = (24- p ( x , < ) e i e 2

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    C) SlSl=, K ( x , Z ) dS, = 0, x E0.It can be shown easily that these conditions lead to the relation

    K ( x , 2) = K , ( x ) W + K , ( x , 4,where SIzI=l K , ( x , z) dS, = 0. Thus P(x , D )u (x )= K,(x)u(x) + K,u (x ) , where K ,is a singular integral operator.

    Thus Theorems 2.1 and 2.2 imply that pseudodifferential operators of orderzero are bounded in the spaces L, (1 c < co) and CY (0 < y < 1). An easyconsequence of this result is the following theorem.

    Theorem 2.3. Let P(x , D ) be a classical pseudodigerential operator of order mwhose symbol satisfies the estimates (1.7) n which the constants Ca,s re indepen-dent of x E IR". hen it is a bounded operator fro m the Sobolev space W;(IR") intoW,k-' (R )or p E (1, co), k E Z, 2 m, as well as an operator from the spaceCk+Y(IRn)nto Ck-'"+y (IR") or y E (0, ), k E Z, E Z nd k b m b 0.We note that in the general case, the kernel K ( x , z) of a pseudodifferential

    operator is a C"-function for z # 0, and for z = 0 it has a singularity which isnot greater than that of the function IzI-'"-"-', where m is the order of theoperator in question. This is because the function z a K ( x , ) is continuous in xand z for 1x12 m + n + 1.

    2.4. Diagonalization of Hyperbolic Sy stem of Equations(Mizohata [ 9653).a " aLet

    be a strictly hyperbolic system of N equations with smooth coefficients. Thusthe roots A , , . . I, of the characteristic equation

    are real and distinct if 5 E IR"\O. We writeaA k ( t , x)- = i H ( t )A ,k = i

    where H ( t ) is a singular integral operator with symboln O ) is a constant, maps the plane t = 0 onto the surfacet = 61x1'. We can therefore assume that u ( t , x) = 0 when t < 61x1, and thusu ( t , x) = 0 outside some ball for each small t > 0. Then

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    D , U ~ - A U , + ~ = O o r j=0 ,1 , ..., m - 1 ,mDrum + 1 j(t, X , Dx)A1-ju,-j+l = f,j=1

    where Q j is a differential operator of order j. It is important thatthe characteristic equation of this system coincides with the equationpo( t ,x , T,

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    k,where u1 is the trace of D,u when x , = 0. Suppose that after the above transfor-mation the operator L takes the form

    n

    Then

    Let us construct a left parametrix Q for L in IR such that Q L = I + T,where Tis a smoothing operator. Then

    In reducing the problem to one on the boundary an important role is playedby the following result of Hormander [1966].

    Theorem 2.6. Let Q be a pseudodiferential operator in a domain 52 c IR.Letits symbol q( x, 5 ) be expanded as an asymptotic sum qj( x, 5 ) each term of whichis a rational functio n of (. Let w denote the intersection of 52 and the plane x, = 0.Then for each function u in CF(w) he function Q(8()(xn)0 u ) and all its deriva-tives can be extended from 52 = { x : E 52, x , > 0 } to 52 v w . The limitingvaluesQCVu lim D:Q(8() 0 u )

    x,-+oand defined by means of pseudodiferential operators QWv ith symbols

    where for 4.we can take, for example, a disc in the half-plane Im 5, > 0 thatcontains in its interior the poles of qj lying in this half-plane.The principal symbol of the parametrix Q equals ( $l A j k t j t k ) l * For each

    5 # 0, the equationAjk 0 and Im j? < 0. Therefore the contour4.must contain the point r,, = a in its interior. I t can be seen easily that

    Thus after restriction to the plane x, = 0, the equation (2.7) assumes the form(2.8)uO = Bui + 9,

    where g = Q(LU)O~,,=~,nd A and B are pseudodifferential operators with prin-cipal symbols

    :.

    respectively. It is easy to see that A and B are elliptic operators. Thus (2.8) canbe written, for example, in the form

    u1 = couo + clg, (2.9)where Co and C , are first-order operators and the principal symbol of Co is- 1 tj - ?,and thus its imaginary part is positive if # 0.n-1 A .j = 1 A,,If we consider the boundary-value problem with the condition

    2 aj (x)Dju+ ao(x)u= h ( x ) on r,then, by excluding the derivative of u along the normal by means of (2.9), weobtain a pseudodifferential equation for uo on r.On solving this equation, wecan find the solution u of the boundary-value problem as the solution of theDirichlet problem with the condition u = uo on r.We can also use (2.9) and (2.7)to express u in an explicit form.

    j = l

    2.8. Reduction of the Boundary-value Problem for an Elliptic System to aProblem on the Boundary. We can reduce the elliptic boundary-value problemfor equations or systems of higher orders to a problem on the boundary in thesame way.

    The following construction is due to Calderon [1963]. Let 52 be a domain,with smooth boundary r, n lR. et P ( x ,0,)be a system of r differential equa-tions of order m in a neighbourhood of SZ. To make the discussion simple, weassume that the coefficients of these operators are infinitely smooth.

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    We denote by I the operator with symbol (1 + 0, 5 # 0$(~)rp(x)u(x) Crn(R").

    'Here and in a similar definition further on the extension of the operation to distributions with awave front in a fixed close cone is done by continuity with respect to a natural topology in the spaceof such distributions. This topology is defined by constants in estimates like estimates in Theorem3.1.

    Here $(D) enotes, as usual, a pseudodifferential operator with symbol $ ( 5 ) :$(D)u(x)= (2n)-" $( t )v"(5)e ixCt .s

    Sometimes it is convenient to use another equivalent definition, provided byTheorem 3.1. A point (xo,to) T*Q\O doe s not lie in WF(u) if and only if

    the next theorem.there exist a function cp E C c ( 8 ) nd a cone r c R" uch that fo r every N > 0

    cp(xO) # O, t o r, lG(t)I cN(l + It/)-" for 5 r.A subset K of T*8\0 is called conical if it contains all the points ( x , t t ) t > 0)whenever it contains the point ( x , 5 ) .Theorem 3.2. The wave front set of each distribution u E 9'(8)s a closedconical set in T*8\0 whose projection onto 8 oincides with sing sup u. Z

    cp E Cc(SZ), then WF(cpu)c WF(u).Examples. 1 . u = d(x) ;WF(u) = ((0, ) , E R"\O}.2". u = d(x,); WF(u) = ((0, ', tl, 0)E T*IR"\O, X' E IR -l, C1 E R\O}.

    2; WF(u) = {(O,O, x", tl, 0,O);X " E Rn-2, l E IR\O}.13". u = x2 + ix ,4".Let r be any closed conical subset of T*O\O. Then the functionm

    U ( X ) = C k-2cp(k(x- k))eik3,'kis continuous in IR" and WF(u) = r if cp E C;(IR"), @(O) = 1, l e k l = 1, and thesequence of points (xk, 6,) is dense in the intersection of r with the set 151 = 1(see Hormander [1983,1985]).

    k = l

    3.2. Properties of the Wave Front Set. Let X c IR" and Y c R"be open setsand let f : X --* Y be a smooth map. If cp E C;(Y ) then the function f *cp(x)=cp( f (x)) is smooth in X . An important question arises: when can the map f * beextended by continuity to distributions? The following result answers this ques-tion.

    Theorem 3.3. (Hormander [1983, 19851). The distribution f*u E g ( X ) s de-( 3 4

    fined for each u E 9 ' ( Y ) o r w hich Nfn WF(u) =0,hereFurthermore, Nf = {(f(x), r t ) E T*(Y)\O: f'(x)rt = O}.

    WF(f *u)c *WF(u)= { (x, y'(x)q) E T*X\O: (f(x), q ) E WF(u)}.

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    Clearly,Nf= $3f m = n and det f '(x) # 0, and then Theorem 3.3 is applica-ble for any u.In particular, this theorem enables us to define WF(u) for u E 9 ' ( X ) ,where Xis a smooth manifold. Moreover, WF(u) c T*X\O, and is independent of thelocal coordinates used.Let X be a smooth manifold and Y c X a smooth submanifold of smallerdimension. Let i : Y X be the embedding of Y into X . The set Ni , introducedin Theorem 3.3, coincides with the setN ( Y ) = { ( y , q ) : y E Y and q is normal to Y}.Thus we have

    Theorem 3.4. I f u E g ( X ) nd WF(u)n N ( Y ) = 0,hen the restriction uly Eg ( Y ) s defined. Furthermore,WF(ul,) c ( y , q ) E T*Y\O: 35 E N ( Y ) , y , q + 5 ) E WF(u)}.

    An important question from the point of view of applications is that ofdefining the product uu of two distributions. It is known that this product isalways defined if u E C m ( X ) . n the general case, it is important that the wavefronts WF(u) and WF(u) are compatible in the sense that, at those points whichare singular for u the function u must be smooth and vice-versa (see, for example,Hormander [1971,1983,1985], Shubin [1978]).Let u, u E 9 ' ( X ) , where X is a smooth manifold, and A: X +X x X be thediagonal map such that A(x) = (x, x). The set N,, given by (3.1) assumes theformNd= { (x, X,5, q ) E T *(X x X)\O 5 + q = O}.

    Theorem 3.5. If WF(u) + WF(u)c T*X\O, then the product uu E W ( X ) sdefined, andWF(uu)c (x, 5 + q ) E T*X\O (x, 5 ) E WF(u) or 5 = 0,

    (x, q ) E WF(u) or q = O}.This result is obtained from Theorem 3.3 by applying A* to the distributionTheorems 3.3 and 3.5 enable us to examine the question of applying to theu(x)0 u( y ) E 9 ' ( X x X ) .distribution u E Q'(Y) the integral operator of the form

    = K(x, Y ) U ( Y ) dYswith kernel K lying in 9 ' ( X x Y).In the particular case where u E C;( Y), we have Au E g ( X ) nd

    WF(Au)c { (x, 5 ) E T*X\O: (x, y, 5, 0) E WF(K) for some y E supp u } .This result follows immediately from Theorem 3.5. To tackle the general casewhere u lies in &'( Y ) ,we introduce the following sets:

    WF(K)y = { ( y , q ) E T*(Y)\O 3~ E X , (x, y , 0, - q ) E WF(K)}.Theorem 3.6. Let u E sl(Y ) nd let WF(u)n WF(K), = 0.hen the distribu-tion

    A W K(x, Y ) U ( Y ) d y E 9 ' ( X )sis defined , and

    WF(Au)c WF(K), u { (x, t)E T*(X)\O:3 ( Y , E WF(u), (x, Y , 5 , q ) E W'(K)}.

    In the important particular case where A is a pseudodifferential operator, wehaveWF(K)c (x, x, 5, 5 ) E T*(X x X)\O),

    and hence WF(K), = 0 nd WF'(K), = 0,ivingWF(Au)c WF(u).

    Thus we find that a pseudodifferential operator is pseudolocal in terms ofwave front sets also (see 0 1.1).I 3.3. Applications to Differential Equations. Let P = P ( x ,D) e a differentialor a pseudodifferential operator of order m with principal symbol p o ( x , 5). Weset

    Char P = {(x,5 ) E T*X\O: p o ( x , 5 ) = O}.Theorem 3.7 (Hormander [1983,1985]). For each u E 9 ' ( X ) ,I WF(u) c Char P u WF(Pu).Corollary 3.1. I f P i s an elliptic operator (that is, Char P = a),hen WF(u) =

    WF(Pu) for all u E 9 ' ( X ) .The proof of the theorem is based on the construction of the microlocalparametrix. If the point (xo,to ) ies outside Char P, hen we can constructan operator Q(x, D) uch that the symbol of QP- will be zero in a conicalneighbourhood of the point (x,,, to). his symbol can be constructed in the sameway as in Theorem 1.3. Now if (xo, to) WF(Pu), then (xo,to) WF(QPu), andconsequently (xo, to) WF(u).Example 3.1. Let u E 9 ' ( X ) ,X c lR" nd D , u E Cm(X). f (xo,to) T*(X)\Oand (xo, to) WF(u), then each point (x('), to) f T*(X)\O lies outside WF(u)provided that the point x(') can be connected with xo by a line segment parallelto the x,-axis and lying in X .

    I

    1

    i

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    where the phase function S is determined as a solution of the non-linear equa-tion- = p t , x , r + zast ( 1

    satisfying the initial conditionS(0,x , -n2 even if u E C;(Y). We try to define this integral as thelimit

    lim+O [x(eO)a(x , y , 8 )u(y)e iS(x*Y .e)y do,where x E C;(lR") and ~ ( 0 ) 1 in a neighbourhood of 0 E IR". Assume that

    d,,eS(x, y, 8)# 0 for 8 E IRN\O, x E X , y E Y. (4.3)Setting

    i n Z a s a I N 2 a s aL = : 1 -+y 1 81 --,&=I a y k ayk k = l ae, ae,we have

    = $ ( x , y , @ i S ( x , y , e ) ,where $ ( x , y , 8)# 0 and $(x, y, to) = t 2 $ ( x , , 8) for t > 8. We choose h EC;(lR") such that h(8)= 1 if 181c 1/2 and h(8) = 0 if 181> 1 , and set

    Ll [l -- h(O)]$- ' (x , y, 8 ) L + h(8).Then L l e i S = eiS,and

    The functions a k ,bj and c satisfy estimates of the form (4.2)for m = - 1,0 and- respectively. Hence

    If k is chosen so large that k > n2 + m, then the resulting integral convergesabsolutely and uniformly in E for E E [O, 13 when u E C;( Y), and it enables us totake the limit as E + 0 under the integral sign.Since the same construction can be carried out for the integrals obtainedfrom (4.1) by differentiation with respect to x , we haveTheorem 4.1. Zf (4.3) holds, then @ is a bounded operator from C ; ( Y ) into

    Crn(X).Since the transpose of @ is of the form

    t @ u( y ) = / / e i s(x ,Yse) (x ,Y , @u (x ) x do,duality yields the following

    Theorem 4.2. Zf th e condition(4.4)

    holds, then the map @ defined by the integral (4.1)can be extended to a continuousmap

    @: S'( ) -* 9 ' ( X ) .

    d,,,J(x, y, 8)# 0 f o r 8 # 0, x E X , y E Y

    The Schwartz kernel of @ is a distribution K ( x , y) E g ( X x Y) defined bythe integral( K , w ) = / / e i s ( x * y 9 e ) a ( x ,, 8 )w(x , ) d x d y do, w E C;(X x Y ) .

    Theorem 3.6 and arguments analogous to those used earlier enable us toTheorem 4.3. Let u E &'(a)nd let @ be an operator of the for m (4.1).Assumeestablish the following result.that th e following conditions hold.1". Zf dxS(x,y , 8)= 0 and d,S(x, y, 8 ) = 0, then y , -- $ WF(u).( "a,")

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    2". Z d,S(x, y , 0) = 0 and 8 # 0, then d,S(x, y , 0) # 0.Then the distribution @u E 9 ' ( X ) s defined, and

    4.3. Composition of Fourier Integral Operators with Pseudodifferential Oper-ators. Let A be a pseudodifferential operator of order m in a domain X c IR"'and let a ( x , 5 ) be its symbol. We first examine the behaviour of the functionA( e i '$ ( x ) ) s A ++a, ssuming A to be properly supported, f E Cm(X) nd$ ( x ) E Cm(X)with $ ' ( x ) # 0 for x E X.Theorem 4.4. For any integer N 2 0 and A 2 1, we have

    + ~ m - N / 2RNV, $9 (4.5)where cp,(y)= $ ( y ) - ( x ) - ' ( x ) ( y- ) and RN is such that

    Ia,BRN(x,A)I < Cp,N,K, x E K ,in which K is a compact subset of X and the c onstants Cp,N,K are indep endent of A.We note that the summand corresponding to a in (4.5) is estimated byaCAm-lUl/', ecause p,(y)= 0 when y = x, and hence

    aYj1[I), (( y)e ' ~ x ( y ) ) ] y = x 1 = o(W2).

    Theorem 4.4 enables us to define the operator A 0 @, where @ is an operatorof order m' of the form (4.1) with amplitude b(x , y , e), that is,@u(x)= J J b(x , y , ~ ) u ( y ) e i S ( x * Y ~ ~ )y do.

    Theorem 4.5. The composition A o 0 s an operator of the form (4.1) with thesame phase function S and amplitudec(x , y , 0) = e- is (x.yle)a(x,,)[b(x, y , d)eis(x,Y.e)].

    The amplitude can be expanded in an asymptotic series

    as(% , 0)a x .s 101-, 0, where CP,(Z, y , 0) = S(Z,y , 0) - S(X, y , 0) - Z - )

    A similar result also holds for the composition @ 0 A, and this follows easilyfrom Theorem 4.5 by expressing @ 0 A as the transpose of the operator 'A o W.Theorem 4.6. The operator @ 0 A can be represented in the for m (4.1) with thesame phase function S and with an amplitude that is asymptotically equal to

    Y , 0)aY *where v,,(x, Z , 0) = S(X, Z, 0) - S ( X , , 0) - Z - )

    We now return to Example 4.3, examined in $4.1. Let us apply Theorem 4.5to compute the symbol of the operator P 0 T In view of this theorem, thesymbol a( t , x , 5 ) must satisfy the asymptotic equation

    where qox(t,z , 5 ) = s(t, , 5 ) - s(t, , 5 ) - z - )as(t' ' 'I. Moreover, a(0, x ,5 ) = 1 . If we represent a as an asymptotic sum,

    aj( t ,x , g), aj( t ,x, A 0,

    ax

    a - j = Owe find thataj(O,x , 5 ) = 0, j = 1,2, ..

    Here the functions 4 can be computed if the functions a,, a,, ..., aj-, areknown. Moreover, 4 ( t , , At) = A - j q ( t , x, 5 ) for ,I> 0. These equations arereferred to as the transport equations.The parametrix T thus constructed enables us to establish the existence of aunique solution to the Cauchy problem.

    4.4. Canonical Transformations. Let us consider a Fourier integral operatorof the special form(4.6)u(x)= (27t)-" a(x , ~ )u ' (< )e i s ( "* ')c.s

    Suppose that det (a2:::55)) # 0. The operator (4.1) takes the form (4.6) if a and

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    S are independent of y and N = n , = n, = n. The Schwartz kernel of this opera-tor has the form r

    The method of stationary phase (see, for example, Fedoryuk [1971]) enables usto show that

    Therefore Theorem 3.6 implies that

    The equations a s k ?)9 5 = -= - a s ( x , ?)all a x

    define a transformationT : x , 5 )H Y, l)

    which, in classical mechanics, is known as the canonical transformation (seeArnol'd [1979]).

    Let us recall some of the notions connected with this transformation.Let M be a smooth manifold whose dimension is even, 2n say. A symplecticstructure on M is a symplectic form on M , i.e. an exterior smooth differential

    2-form1w = 2 wi jdxi A d x j , wij = -w. i,

    which is closed, that is, dw = 0, and is non-degenerate, that is, det llwijll # 0. Amanifold on which a symplectic form w is given, is called symplectic.For example, if Q is a domain of IR", hen the manifold T *Q has thesymplectic structure, given by the form

    nj = 1o = d x A d < = C dxj A d t j ,

    where x l , . , x , are local coordinates in Q and r l , . 5, are correspondingcoordinates in fibres of the cotangent bundle. This form is independent of thecoordinates x , , , , chosen in Q.

    The transformation T: T*Q + T* Q is called canonical if it is smooth andpreserves the symplectic structure.

    Examples. 1". The change of coordinates y = F ( x ) generates a canonicaltransformation T given byy = F ( x ) , r] = 'F'(x)- ' ( .

    i

    Li

    2". A real-valued smooth function H ( x , 5 ) defines a canonical transforma-tion

    ( x , l ) H ( X ( t ) ,for each t E IR, where x ( t ) ,

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    is the image of SZ under the map defined by S , such that the operator T =P @ - @ Q is smoothing and q - o E Sm"-'(sz' x IR ) , where g o yt q ) , q ) =p ( x ,y).

    In the investigation of various a priori estimates the following related resultproves to be useful.

    Theorem 4.8 (Egorov [1984]). Let P be a properly supported pseudodiffer-ential operator with symbol p E Sm(SZ x I R ) and let @ be the Fourier integraloperator of the form (4.6) with amplitude a E So@ x I R ) and phase function S.Then the operator Q = @ *P@ is pseudodifferential with symbol q E S' (S2' x IR ),andwhere ( y , q ) is the image of (x,5 ) under the canonical transformation defined bythe generating function S.

    Corollary 4.1. Let @ be a Fourier integral operator of the form (4.6) withamplitude a E S0(Q x I R ) and phase function S. Then for any subdomain 8' C 52there exists a constant C such that

    4(Y, q )- ( X , 5)la(x ,5)121detSXJ' E sm-l,

    II@ulloG CIlUllO, E C,m(8 ),and if la(x, )I 2 o > 0 for large 151, then

    l l 4 l o G c(ll@ullo+ I ~ I I - , ) , I4 E C,"(B').Here 11 * I I s denotes, as usual, the norm in HS(R ).

    This statement follows immediately from Theorem 4.8 with P = I, sinceI I @ ~ l l i (@*@u, u)O = (Qu,u)o = (Au,u)O + (Bu,u)o,

    where A is a pseudodifferential operator with symbol Ia(x, 5)12 ldet Sxc(x,

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    asaen Cs because S = 8.- = 0 on C,. Therefore under r the form 5 dx goes overto the form q dy, that is, the form 5 d x - d y is identically equal to zero onvectors which are tangent t o A s . Accordingly, the symplectic formd( A d x - q A d y ,

    defined on 52 x (IR"\O) x 52 x ( IRn\O), vanishes identically on A,, that is, As isa Lagrangian manifold.We now assume that X is a smooth manifold of dimension n. Let L be aconical Lagrangian manifold in T*X\O. Let us see how to associate withL localphase functions, these being real-valued smooth functions S ( x , 0) which are posi-tive and homogeneous in 8 of degree 1, and defined in an open subset T cIR" x (IRN\O) with d,,,S # 0 on r.Let S be such a function, and let the set

    be a smooth manifold. Let the rank of the differential of the mape)H( ,- ) E T*X\O as

    axe n at each point of C,. We fix a point ( x o ,0,) E ra nd let to= -(xo , Oo).Definition 4.3. A phase function S is said to be associated with a Lagrangianmanifold L in a neighbourhood of the point ( x o ,0,) E L if there exist conicalneighbourhoods ro nd r' of the points ( x o , ,) and ( x o , o)espectively, suchthat L n r' is the image of Cs n ro nder the map (4.8).The existence of such local phase functions is guaranteed by the followingProposition 4.1. Let L be a conical Lagra ngian manifold in T*X\O, and let( x o , o) L. The coordinates ( x l , , ,) may be chosen in a neighbourhood of

    xo E X so that in the neighbourhood of ( x o , o) he manifold L is defined by theequations xi = a H ( r ) / a t j j = 1, . n) where H E C" in a conical neighbourho odof to nd H(t 0. Thus the phase function S associated with L ina neighbourhood of (xo,to) an be chosen in the formS = X ( - H (

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    If A is a closed, conical Lagrangian manifold in T*(X x Y)\O, then the spaceof Fourier distributions Zm(X x A) can be regarded as a space of continuouslinear maps from Cg(Y) into 9 ' ( X ) .

    Definition 4.6. A closed conical submanifold C in T *(X x Y)\O is calleda homogeneous canonical relation from T*Y into T*X if C is contained in(T*X\O) x (T*Y\O) and is Lagrangian with respect to the form a, - a,, wherea and a,, are the canonical symplectic forms on T*X and T*Y respectively.

    As in $4.6, we can show that a canonical homogeneous relation C can bedefined locally by means of a non-degenerate phase function S(x , y, 8). Thus,with

    the map

    is a local homogeneous diffeomorphism of Cs onto C.Since C is closed, there is a local finite covering of T*(X x Y)\O by

    open conical sets 4 ( j= 1 , 2 , ...) which are coordinate neighbourhoods inT*(X x Y)\O and have the following property: for each j the set C n G isdefined by a non-degenerate phase function Si(x, y, O), 8 E RNr.Let ( g j } ( j=1 , 2 , . ) be a smooth partition of unity in P ( X x Y)\O subordinated to thiscovering, where the functions gi are positively homogeneous in 8 of degree 0.

    The operatorP P

    is called a global Fourier integral operator if ai E S m + ( n - 2 N j ) / 4 j = 1 , 2 , .. ).Example 4.4. If X = Y and C = A: is the diagonal in (T*X\O) x (T*Y\O),

    then we can take (x - )8 as the phase function, where8 E R"\O. In this case theoperator @ is a pseudodifferential operator.

    8 5. Pseudodifferential Operators of Principal Type5.1. D efinition and Examples. Let P ( D )be a differential operator of order mDefinition 5.1. The operator P ( D ) is called an operator of principal type ifThe terminology is explained by the following

    with constant coefficients, and let p o ( < ) be its principal symbol.

    d,po(5) # 0 or 5 # 0, 5 E R".

    Theorem 5.1 (see Hormander [1963]) . Z P( D) s an operator of principal type,then for each operator Q(D)of order m - with constant coeficien ts, there existsa constant C > 0 such thatIICP(D)+ Q(D)lull < C(llP(D)ull+ Ilull), E c,"(R"),

    where 1 1 . 1 1 denotes the L2(R")-norm .pal symbol and they do not depend on the lower terms.

    This theorem shows that many properties of P are determined by the princi-

    Example 5.1. The operators A and 0 re operators of principal type but thea aat - at -perators- + A and- id are not.Definition 5.1 can be extended to general pseudodifferential operators in the

    following way.Definition 5.2. A pseudodifferential operator P ( x , D), with principal symbolpo(x , t), s called an operator of principal type in a domain SZ c R" if, for all( x , 5 ) E T*SZ\O, the form d,,,p,(x, 5 ) s not proportional to the form 5 dx.We observe that if the form dx, tpo(x ,

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    It can be seen easily that, in o,S(x, t r ) = t S (x , r ) for t > 0 and det lld2S/dxid5jll# 0.

    This last inequality is a consequence of the fact that det l la2S(xo,5) /ax ia r j l l =azS(xo, t 0 ) / a x , t , , and

    by (5.1). But in view of the initial conditions, dZS / ax j d< ,= 0 ( j = 2, . , ) whenx 1 = xy. Hence i3'S(x0, ~ O ) / d x , ~ ~ ,0. Consequently, S is a generating func-tion for some canonical transformation. We set

    Qiu (x)= (274-" f i ( 0) and

    On extendingS to T*0, we find that the operatorQiP - D1Qi

    has order zero if @ is the operator defined by (5.2).Thus in this case too the equation Pu = f leads to the equation D,v =g + Rv in a conical neighbourhood of the point (xo ,to), here the symbol of R

    is zero in this neighbourhood.5.3. Solvability of Equations of Principle Type with Real Principal Symbol.

    Theorems 3.7-3.10 on singularities of solutions of the equation Pu = f,ogetherwith Theorem 5.2, enable us to tackle the question of local solvability of thisequation.

    Definition 5.4. A pseudodifferential operator P ( x , D) is said to be solvable ata point xo E 0 f there exist neighbourhoods U and V of this point in0 uch thatU C V and if for every functionf Cm(V) here exists a distribution u E & ( V )such that Pu = f i n U .globally solvable in a domain 0 R" if for each compact set K in 0 nd foreach functionf in the subspaceS with a finite codimension in Cm(0) here existsa distribution u E &(a)uch that Pu = f n K.

    The solvability conditions for P are determined by the behaviour of itsbicharacteristics.

    Examp le 5.2. The operator x 2D, - 1D2 is insolvable at the origin althoughThe following condition on P in 0 epends on the choice of the compact set

    I Definition 5.5. A pseudodifferential operator P ( x , D) is said to be semi-

    it is of principal type.K C 0.

    Condition A. Let K C 0. or each point (xo ,g o ) T*0\0 such thatpo(xo ,5 ' ) = 0, the bicharacteristic, that is, the integral curve of the Hamiltonian

    passing through the point (xo ,to), ontains the points ( x , 5 ) E T*B\O whoseprojections onto 0 ie outside K.

    Theorem 5.3 (Duistermaat and Hormander [1972]). The operator P(x,D) issolvable at a point xo if there exists a neighbourhood K of this point wherecondition A holds. In particular, P is solvable at xo if dc po(xO , ) # 0 for allTheorem 5.4 (Duistermaat and Hormander [1972]). The operator P( x, D ) issemi-globally solvable in a domain 0 IR" f condition A holds for each compact

    .

    5 E R"\O.

    64 Yu.V. Egorov, M.A. Shubin 65. Linear Partial Differential Equations. Elements ofModem Theory ,

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    set K C 51.Moreover, iff E HS(Q)n , then there exists a function u E @' -'(51)satisfying the equation Pu = f in K andI I u l l s + m - l G cllf 11s)

    where the constant C = C(s, K ) s independent off.5.4. Solvabilityof Operators of Principal Type with Complex-valued PrincipalSymbol. If the pseudodifferential operator P (x ,D) f principal type has complex-

    valued symbol po = a + ib, then the equation Pu = f, where f E Cm(51),cangenerally be solved only when f belongs to a subspace having infinitecodimension.

    Example 53. From the historical point of view, H. Lewy gave the firstexample

    au au aua x , ax , 8x3- i- + 2i (x1+ i x z ) - = ( x , , x,, x3)

    with f E C'O(IR3), which has no solution in any open set o c R3 (seeHormander [ 9633).

    Example 5.4. An even simpler example is furnished by the equation

    There exists a second category set of functions f in C'O(IR") or which thisequation is insolvable in any open set on the plane which contains points of theOxz-axis see Hormander [ 9633).

    Example 5.5. The oblique derivative problem for a second-order ellipticequation leads, after reduction to a problem on the boundary, to an insolvablepseudodifferential equation if there a re "repelling" submanifolds of dimensionn - . For example, in the case of the Laplace equation in the half-spacex , > 0,the problem with the condition

    au au- ax:- = f for x , = 0ax 1 ax,can be transformed to the equation

    ava x ,_ - x:Av = f,

    where v is the trace of the function u on the boundary and A is an operator withsymbol 151. If k is odd and a > 0, then the boundary-value problem has nosolution u in any neighbourhood of the point lying on the manifold x , = 0,X I = 0.

    At present, the solvability theory for equations of principal type has ad-vanced considerably. Let us list some of the results.

    Theorem 5.5 (Hormander [1963]). If po(xo , o ) 0 for some (xo, o)T*Q\O, and cy (x o, to) 0, where cy(x, t) = Im C aPo(x, t) PO(x9 t ) , hen in

    j=1 atj axievery neighbourhood o of xo there exists a function f ;C,"(o) sich that theequation Pu = f has no solution belonging to 9 ' ( w ) .Theorem 5.6 (Beals and Fefferman [1973]). Let P be an operator of principaltype satisfyingCondition (9').On each bicharacteristic of the function Re po along which

    Re po = 0, the function Im po does not change sign.Then for each point xo E 51 there exists a neighbourhood o uch that for anyfunction f in H S ( o ) here exists a function u E Hs+m -1 51)atisfying the equationPu = i n 8.Moreover, IIuIIS+m-l< C(I Is where C s independent off.Theorem 5.7 (Egorov [1975]). Let P be an operator of principal type thatsatisfies the following two conditions.Condition (Y). On each bicharacteristic of the function Re po along which

    Re p o = 0, the function Im p o does not change sign


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