T-product of hypoelliptic operators

18.

19. 20.

21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.

V . P . Maslov, "The canonical operator on a Lagrangian manifold with complex germ and the r e g u l a r i z e r for pseudodifferential operators and difference schemes," Dokl. Akad. Nauk SSSR, 195, No. 3, 551-554 (1970). V . P . Maslov, Operator Methods [in Russian], Nauka, Moscow (1973}. V . P . Maslov and V. G. Danilov, "Quasiinvertibility of functions of o r d e r e d operators in the theory of pseudodifferential equations," in: Sovrem. Probl. Mat., Vol. 6 (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1976), pp. 5-132. V . P . Maslov and Yu. Yu. Sternin, "The canonical operator (the complex case}," in: Sovrem. Probl. Mat., Vol. 1 (Itogi Nauki i Tekhn. VINITI AN SSSR), Moscow (1973), pp. 169-195. V . P . Maslov and M. V. F e d o r y u k , "The canonical operator (the real case)," in: Sovrem. Probl. Mat., Vol. 1 (Itogi Nauki i Tekhn. VINITI AN SSSR}, Moscow (1973}, pp. 85-167. J. Milnor, Morse Theory [Russian translation], Mir, Moscow (1965}. A . S . Mishchenko, B. Yu. Sternin, and V. E. Shatalov, The Method of the Canonical Maslov Operator (Complex Theory}, MIEM (1974}. A . G . Prudkovskii, " T h e method of stationary phase in application to integrals depending on a p a r a m e t e r ,M Zh. Vychisl. Mat. Mat. Fiz., 13_, No. 2, 275-293 (1973). A . G . Prudkovskii, "The method of stationary phase in application to integrals depending on a p a r a m e t e r ~ " (nonanalytic c a se~, Z h Vychisl. Mat. Mat. Fiz., 1..~4, No. 2, 299-311 (1974). B. Yu. Sternin and V. E. Shatalov, "The smooth theory of the canonical Maslov operator on a complex Lagrangian germ," Usp. Mat. Nank, 29, No. 3, 229-230 (1974}. M . V . Fedoryuk, "The method of stationary phase and pseudodifferential operat ors, " Usp. Mat. Fiz., 26, No. 1, 67-112 (1971}. L. HSrmander, " F o u r i e r integral operators. I," Matematika, Period. Sb. Perev. In. Statei, 16, No. 1, 17-61 (1972}: L. HSrmander, " F o u r i e r integral operators. II," Matematika, Period. Sb. Perev. In. Statei, 16, No. 2, 67-136 (1972). V. Maslov, "The c h a r a c t e r i s t i c s of pseudodifferential operators and difference schemes," Actes Congr. Int. Mathematiciens, 1970, Vol. 2, Par i s (1971), pp. 755-769. F. T r e v e s , "Hypoelliptic partial differential equations of principal type. Sufficiency and necessary conditions," Commun. Pure Appl. Math., 24, No. 5, 631-670 (1971}. J. SjSstrand and A. Melin, " F o u r i e r integral operators with complex-valued phase functions," P r o c e e d ings of the International Conference in Nice on Fourier Integral Operators (1974).

T-PRODUCT V. P .

OF H Y P O E L L I P T I C Maslov

OPERATORS

a n d I. A. S h i s h m a r e v

UDC 517.948:513.88

Asymptotics as t ~ 0 for the solution of the Cauchy problem for hypoelliptic equations a r e obtained. Using these results, the existence of the T-product for hypoelliptic operators is proved as is a theorem on the removal of autonomous brackets in the T - p r o d u c t INTRODUCTION So-called chronological products (or T-products} of operators play a major role in modern quantum field theory as one of the most effective means of describing physical processes in the microworld (cf. [2]}. With their help, in particular, it is possible to describe the formalism of second quantization. A rigorous mathematical foundation for T-products encountered in theoretical physics is still impossible. It has been found, however, that there is a model on which it is possible to study the machinery of T-products. Namely, the theory of hypoelliptic operators, which has recently been developing intensively, has emerged as an ideal model in some sense for th e investigation of the properties of T-products. T- P r o d u cts in some form have also been applied in the theory of differential equations and the theory of semigroups in the proof of existence theorems (cf. the work of T r o t t e r [11], Daletskii [4], Kato [6], and Nelson [10]}. Namely, consider the Cauchy problem Translated from Itogi i Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 137-197, 1977.

0096-4104/80/1301-0081507.50

9

Plenum Publishing Corporation

81

{

-~-+ ~ (t) ~ if)--0,

(1)

The solution of this p r o b l e m in a n u m b e r of c a s e s c a n be r e p r e s e n t e d in the f o r m of a T - p r o d u c t of o p e r a t o r s t SB(s)ds ~?(t)-----T - e ~ r (2) H e r e the T - p r o d u c t is defined as follows: we take a p a r t i t i o n of the t i m e i n t e r v a l [0,t] by points t o = 0
max

Ats,

Ats-----tl+t--tJ.

(3)

j--0,...,n.l

This l i m i t (if it exists) is called the T - p r o d u c t of the o p e r a t o r s and is b r i e f l y denoted by (2). A c l o s e l y r e l a t e d p r o b l e m both in field theory and in differential equations is the r e p r e s e n t a t i o n of s o l u tions in the f o r m of a F e y n m a n continuum integral. In view of the a b s e n c e , in g e n e r a l , of a m e a s u r e in the F e y n m a n i n t e g r a l it is t r e a t e d as a l i m i t of f i n i t e - d i m e n s i o n a l i n t e g r a l s . The connection between this a p p r o a c h to the F e y n m a n i n t e g r a l and T - p r o d u c t s w a s a l r e a d y noted by F e y n m a n [8]. If the o p e r a t o r ~I(t) in p r o b l e m (1) is a differential or pseudodifferential o p e r a t o r with s y m b o l H(t, ~, x), then the connection mentioned is e x p r e s s e d by the following relation: n

'k~ (xk+'-x~)li~t fi eU(tl.~,k.xt+t)Litt~o (X0)a.~,a~,.

"

lira e

-

-

"""

a-.e (2~)"

(4)

:<,

In the p r e s e n t w o r k w e c l a r i f y the a l g e b r a i c s t r u c t u r e of Eq. (4) and g e n e r a l i z e it to the c a s e of T - p r o d ucts in the m i x e d H e i s e n b e r g - S c h r S d i n g e r r e p r e s e n t a t i o n (the i n t e r a c t i o n - t y p e representation}. We note that the a r g u m e n t of the l i m i t on the right side of Eq. (4") can be r e p r e s e n t e d in the f o r m of a product of the o p e r a t o r s 9

12

12

e"(r

9 9 9 e"('$'p''~)A,*,

(5)

1 2

w h e r e the o p e r a t o r e~(tT'P'x)ats acts a c c o r d i n g to the rule

e"(is.p.-~)'9 f = (_~_~- ~#,t(:-,)e"(9"")Aq ,12

3"(y>dCdy.

,

Using the concept of autonomous b r a c k e t s introduced in [7], the a r g u m e n t of the l i m i t on the left side of Eq. (4) c a n be w r i t t e n in the following manner: 12

I 2

etn(t~_~,p,xiiAt._, . . . eW.~,p,~lla~..

(6)

The limit equation (4) c a n be i n t e r p r e t e d now as r e m o v i n g the autonomous b r a c k e t s in the exponential function. At f i r s t glance the possibility of r e m o v i n g autonomous b r a c k e t s in the exponential function s e e m s unexpected. Indeed, if A and B a r e any m a t r i c e s , then the r e l a t i o n 1

2

i

2

ei A+BI~ eA+S

(7)

is equivalent to e A+B = eBe A which is t r u e only if the m a t r i c e s A and B c o m m u t e . H o w e v e r , in a T - p r o d u c t the r e m o v a l of autonomous b r a c k e t s is a l t o g e t h e r natural, since if the left side of Eq. (7) is w r i t t e n in the f o r m 1

e0

(A..bS)dt

and is r e p r e s e n t e d as a T - p r o d u c t , then since 1

2

1

2

e (A+B)at Jr-0 (At) 2 ~ enate a n t = e (A+B)a*and[A -t-BI = [A -q-BI, we obtain the f o r m u l a for the r e m o v a l of autonomous b r a c k e t s in the T - p r o d u c t : 1

2

1

2

1

2

lira e [a+ala:n-1 . . . e [A+Blat*-~ lim e ( A + s ) a t n - t A~0

82

h't~0

1

...

2

e (A+B)At..

(8)

The right side of (8) can naturally be written as the integral n

t s-0 4+0 [( A+s )de

tk+ ]-t~-~-

( A

+

B )~t t

llm e k'O

= e~

;

(9)

here we have used the o r d e r i n g notation of Feynman according to which the o p e r a t o r w i t h the l a r g e r index acts a l t e r the o p e r a t o r w i t h the s m a l l e r index. I n this notation f o r m u l a (8) has the f o r m 2

151

I

IA+alds

eo

4--0

x-{-O

I ( A + a )de --e 0

,

(10)

and Eq. (4) t a k e s the f o r m t$

I. '~

e~[/-/(s,#,.g)ld~

t $--0 a-bO fH(s, p. x )~s

-----e~

(11)

F o r bounded operators A and B there is a f o r m u l a m o r e g e n e r a l than (10), 1~

1 2

1

s-.Osd-O

~I/(~I.B)Ns Sl( A , B )~s e~ --'--e~ ,

(12)

w h e r e f is any continuous function. We s e e that the r e m o v a l of autonomous b r a c k e t s in the exponential is a c h a r a c t e r i s t i c a l g e b r a i c p r o p e r t y of the T - p r o d u c t ; one of the e x a m p l e s of this is the w e l l - k n o w n F e y n m a n f o r m u l a for the r e d u c t i o n of a T p r o d u c t of a continuum integral. In q u a n t u m e l e c t r o d y n a m i c s the T - p r o d u c t of o p e r a t o r s is o r d i n a r i l y u s e d in the i n t e r a c t i o n r e p r e s e n t a tion. If fI~ is the f r e e H a m i l t o n i a n and fI t the i n t e r a c t i o n Hamiltonian, then the T - p r o d u c t of o p e r a t o r s in t h e i n t e r a c t i o n r e p r e s e n t a t i o n has the f o r m

}[-. e~

,

(13)

w h e r e P(t) and X(t) a r e o p e r a t o r s s a t i s f y i n g the s y s t e m of H e i s e n b e r g equations 9 0/'/'o X'-----G~-

.

OH ~

I

'~

1

X(O)=x, P (0)-~D. Usually the H a m i l t o n l a n H 1 d e s c r i b i n g the i n t e r a c t i o n depends only on x and then the question of r e m o v a l of autonomous b r a c k e t s is solved t r i v i a l l y (for any m a t r i x A e [f(A)] --ef(A) by definition). The n e c e s s i t y of c o n s i d e r i n g the g e n e r a l c a s e is o c c a s i o n e d by two c i r c u m s t a n c e s . F i r s t of all, in a n u m b e r of m o d e r n physics w o r k s i n t e r a c t i o n s a r i s e which a l s o depend on the m o m e n t u m . Secondly, this r e q u i r e s writing the T - p r o d u c t in o t h e r r e p r e s e n t a t i o n s , e.g., in the s o - c a l l e d wquasiclassicalW r e p r e s e n t a t i o n . The t h e o r e m p r o v e d in Sec. 4 on the r e m o v a l of autonomous b r a c k e t s in the T - p r o d u c t , a c c o r d i n g to which the o p e r a t o r equation $ t

e~

I

,

2

.-o

.-o

I HI ~ P (s). X (s)Jds =e

~

(14)

holds, m a k e s it p o s s i b l e in this e a s e as well to w o r k with an i n t e r a c t i o n H a m i l t o n i a n j u s t as in the c a s e of no dependence on the m o m e n t u m . We note that the T h e o r e m s 3 and 4 that we p r o v e a r e s u b s t a n t i a l g e n e r a l i z a t i o n s of the w o r k of Daletskii [4], B u s l a e v [3], E v g r a f o v [5], and A l i m o v [1] on the r e p r e s e n t a t i o n of the solution of the Cauchy p r o b l e m in the f o r m of a continuum i n t e g r a l (4). In conclusion, we e m p h a s i z e the fact that the p r o o f of r e l a t i o n (11) is the f i r s t s t e p toward introducing a n o p e r a t o r m e a s u r e in the T - p r o d u c t . M o r e o v e r , the p r o b l e m naturally a r i s e s of c a r r y i n g o v e r the t h e o r e m on r e m o v a l of autonomous b r a c k e t s to the c l a s s e s of T - p r o d u c t s analogous to those widely u s e d in physics. The p a p e r c o n s i s t s of four sections. In the f i r s t s e c t i o n the Cauchy p r o b l e m (1) is studied, the s y m b o l of the solution o p e r a t o r is c o n s t r u c t e d , and its a s y m p t o t i c b e h a v i o r as t - - 0 is d e t e r m i n e d . The second s e c t i o n 83

is devoted to the p r o o f of the existence of the T - p r o d u c t for hypoelliptic o p e r a t o r s . In the third s e c t i o n a theor e m is p r o v e d on the r e m o v a l of autonomous b r a c k e t s in the T - p r o d u c t . Finally, in the fourth section we study the q u e s t i o n of r e m o v a l of autonomous b r a c k e t s in the T - p r o d u c t in the i n t e r a c t i o n r e p r e s e n t a t i o n . 1. S y m b o l

of the

Operator

U(t,

T)

1~ In N-dimensional E u c l i d e a n s p a c e R N we c o n s i d e r the Cauchy p r o b l e m

I~ - - A (x,t,

D) ~ = 0 ,

(1)

Here =~(x)E~(RN) and ~=~(RN). is the space of infinitely differenUable functions which at infinity decay with au derivatives faster than 1 / t x l k for any k~- 0, x = ( x , x ~ , . . . , X N ) , lxl = (x~ + . . . + x~)~/2; A(t) = =&, t, I

D) is a pseudodifferential o p e r a t o r with symbol a(x, t, ~). As usual Dj = - i ( 8 / S x j ) , 8j = ~/8~j, o~ = (ezl, ~2, . . . . hr

a N) is a multiindex, [=l = ~ =j, =1= =tl=21 99 .=N 1, D ~ = D ~ ' . . . "D ~ ,

x = = x ~ ' . . . x t r =~ = j > 0 . We shall a s s u m e that

j--1

a0t, t, ~) ~ Co=(RN x RI+ x R N) and for t ~ I -> C0, x ~ R N, t e [0, T] for any multiindices a and/~ {here and below Ck, Cry/3...7 a r e various constants}

l a~=~~x t, ~) I ~: c~ Ia(x, ~, ~) 1(l + I ~ 1)-~'l+~ a{~x,

t, ~)=S~

t, D , 0 -< 5 < p-< 1, 5, a n d p a r e fixed. We l a t e r imposed c e r t a i n other conditions

on the symbol a(x, t, ~} in addition to (2). We shall c o n s i d e r solutions of the Cauchy p r o b l e m (D u(x~ t) E C ' ( R N x ItS) such that f o r each fixed t (', 0 E~ ' (R~) together with d e r i v a t i v e s with r e s p e c t to t; We denote this c l a s s of functions by ~t. In the c l a s s t h e r e is a uniqueness t h e o r e m for p r o b l e m (1) [9]. Applied to functions u (x, t) E~ , the pseudodifferential o p e r a t o r in the Cauchy p r o b l e m (1) is defined by

w h e r e u(~, t} is the F o u r i e r t r a n s f o r m of the function u(x, t) in the variable x: hr

in all formulas for the d i r e c t and i n v e r s e F o u r i e r t r a n s f o r m we omit the f a c t o r (2~r)-N/2. Definition I. We define the o p e r a t o r U(t, v) by

(x, t) = U (t, ~) u0 (x),

(4)

w h e r e u(x, t) is a solution of the Cauchy p r o b l e m (1) in the c l a s s ~ (if it exists) and a0(x)65a. F o r e a c h t - r o p e r a t o r U(t, T) is a l i n e a r o p e r a t o r f r o m ] to .~. We shall show below that a solution of p r o b l e m (1) in the class ~t exists for any function u0 (x) E5~ . M o r e o v e r , we shall show that this solution can be r e p r e s e n t e d in the f o r m ~(x, t ) = f e"~'~p(x, t; ~,

~)~(~) d~,

Where the function p(x, t; ~, T) E C == in all variables. Definition 2. The function p(x, t; ~, T) in (5) is called the symbol of the o p e r a t o r U(t, r). In this section we c o n s t r u c t the symbol of the o p e r a t o r U(t, T) and study its p r o p e r t i e s . LEMMA 1. L e t

(x, t) w h e r e r{x, t; ~, r) E C =0 in all v a r i a b l e s and for any multiindices r~ and fl

9 84

(5)

I-~-~ ~x, t; ~, ,)I ~
(x) (t +l ~ 1)~'n~,

(7)

where C ~ ( x ) -< Cc~(1 + Ixl) k, k -> 0, doesnot depend on a and ~, cr(~, ~) >_ 0, Let {~n(X)}, n = 1, 2, ..... b e a sequence of functions in C~(RN) s u c h that

.'x~

fl, [x[
with d e r i v a t i v e s which a r e bounded u n i f o r m l y with r e s p e c t to n; let, i finally,

(8) w h e r e q(x, t, ~) ~C m in all v a r i a b l e s and for e a c h fixed (x, t)

]q(x, t, ~)~
(9)

where m Is any fixed constant. Then

(10) v.r(-, L ~, :)(z)dz.

~§

(11)

ProOf. Since g 0 ( x ) ~ , it follows that #~(~) ~q~, i.e., u0(~) together with its derivatives decals at infinity f a s t e r than any power of 1/I ~ I. Therefore, from (6) and (7) we have for any multiindices a and/~

x~D~u(x, t)=x~(ettXa~)uo(~ ) %" G ~r =Ie""~)i ~ ~

%,~,~,~,t~ ~o~ ~j

~,)

(x. t; L ~}d~ (x, t; ~,~)#.~.

(~2)

Using (7), f r o m this we obtain

(1 § x ])i~ l # s (x, 0 t < c a (l § ] x 1)4.

(13)

Here and below different constants depending on ~ and/3 are denoted by the s a m e letter. Since in (13) k is fixed while c~ is arbitrary, for any ~ and ~ w e have

i ~ u (x, 0] < c ~ (1 + j x J)-~.

(14)

F rom (13) it follows, in particRlar, that u(x, t) is ubsolutely integrable in x on RN; therefore

f

[xl>a

(15)

for n - - 0o u n i f o r m l y with r e s p e c t to ~ E R N. F u r t h e r , for any multiindex fl

by (14) and the definition of the sequence {qn(X)}. P r o p e r t y 1) follows f r o m (15), (16), and (9) by L e b e s g u e ' s t h e o r e m , since for the integrand in (10) t h e r e exists the integrable majorant C(1 + [ ~ I)- N - t fin (16) it is n e c e s s a r y to take i~l= r e + N + iI. W e shall prove 2). W e have

l el(x.~,q(x. t.

~)?n"~(L O d ~ =

~d(x'i)q(x, t. ~)d~ ~ e-'(t.,'~n (y) du ~e~',.,}r (y. t; ~. ,) ~ (~) d~].

We have twice changed the o r d e r of integration which is justified by the absolute convergence of the c o r r e sponding double integrals. The proof of the l e m m a is complete. LEMMA 2. Let q(x, t, ~) e C'~(R N • R~+ • R N) and for some r e a l m sup lq(x. t..DI

--

t)< ~

(17)

85

for all (x, t) E R N x l ~ and suppose that for any multiindex

I q(=) (x, t, ~) I < co i q (x, t, ~) !(!+1~1-0~

(z8)

for all (x, t) E RN x R~ and I f I -> Co; let g (x) E~. Then for the integral

sfJ(~, ~, ~)=I~<:,:'q(~, ~, ~+,)~(=)a=

(19)

j (x, t, ~)= ~ ~ q~o~(x, t, ~) Dog (x)+ J0,

(20)

we have lo[.
w h e r e Jo .admits the e s t i m a t e

IA(x, t, w)l
.

s,,p

L, D==(Y)

[

(21)

yt:R,,, i~-N,,v § Ix--yl) Mr

and M >- 0 and N t > m + N a r e a r b i t r a r y . P r o o f (cs [9, p. 305]). L e t @(x) E C ~ = ~ N ) , ~(x) = 1 f o r I x l -< 1 / 4 ,

O(x) = 0 f o r I x l >

1/2. We have

j (x, if, ~4)=~et'x, ")q(x, t, ~1§ z)u(z)dz= ~ I q(O,(x' l, ~l)D'.(x) ,.

,, Iq.I < . i V t '

+i'<""[* (,-c,) --I

+=> (=) <,=.

,. i__

(22)

That is !

y (x, t, rj)--=-~ -~ q!~) (x, t, "4)Dxtt (x) +.~,

(23)

[ot
where

I0=/I§247

(24)

F r o m Taylor's formula 1

1 q (x, t, ",l-t-z)= ~.i -di q~) (x' t, ~ l ) z = §

~TN'z<~ q(C=)(x. t, ~l§

(25)

t, ~,, x - y )

(26)

it follows that

s,(x, t, ~,)=.E ~ I ~~

o-,~ (y)~y,

VxI-N=

where !

0

Integrating by parts, we obtain from this for any multiindex/3

l=~K=(x, t, ~l, w)l =

1 --a)~-tda

e '(=, ") (--Dz)ll[q(~ (x, #,, ~l§

r~l dz (27)

Since i n the d o m a i n i n i n t e g r a t i o n I ~ + am I -> I ~ I / 2, f r o m this and (17) and (18) we have

(2S)

86

F r o m (26) a n d (28) it follows that

W e t a k e I ~ I -- N + 1 + M, w h e r e M -> 0 is a r b i t r a r y .

1

ij,(x, t, "tl)i<.C...~ ,I=l-iV,,ll sup ~i (1 +1 /9%(y) xLl/I) ~ I(l+lx__t#l)iV+, •

~, ~, x--,#)Idv~ C~,.,

-~up I

o%(y)

~i~N, i~i-m I (1 +t x--! I)~

Ni, 214>0--

I~(x ' 0(i+t~l).+...w,, (29)

ale a~bitmay.

Further,

1 q(=~(x, t, ~)z=;(z)d=

,, l=l
= ~ ~ i - q(=, (x; if, ~i)[(ei<X.'ig(,--~-,lz~(z)dz--L~g(x)]. I~il
L ,J

'<:'"*(A)

(30)

'q ' i I#

(31) where

V (y) = f e.u. ~)~ (x) dx. T h u s , the e x p r e s s i o n in s q u a r e b r a c k e t s in (30) is equal to

(32) By T a y l o r ' s f o r m u l a

(33) IPI
IPl-,'Vt-lal

w h e r e x 0 lies in the i n t e r v a l joining the points x and x - y / I

ID'u(xo)l=

~1,

lO'u(Xo)l ( 1 . + l x _ x o l ) M ~ ( l + l ~ ) M s u p IDV,,(y)l (1 + 1x--x. I)a ufa.v (1 + I x - - e Da'"

S u b s t i t u t i n g (34) w i t h 13' I = N 1 into (33) and noting that I ~ (y) dy = o (0) = 1 and ~ y~.~ (y) d y = D ~ , (0) = 0 I(xl >- 1 f r o m (32) and (33) we obtain

(34)

for

IuERN h,l_N, (l + l r - - y l)M

(35) s i n c e $(y) d e c a y s a t infinity f a s t e r t h a n any p o w e r . W e s u b s t i t u t e (35) into (30):

Ij2(x,t,'~)l
sup

yERN, Iu

l~

-

(1 +~ x ~ y l ) hI '

or, using (17) and (18),

]j2(x,t,~)l < c ~ , n ,

su p

-. 1D~u(v)l , . . a q-'-'x t , t ) ( l + l ~ ] ) ~-mo+(o-u,

~ERN, i~I--Nt ~I + l X - - Y l )

(36)

M >_ 0 is a r b i t r a r y . We p r o c e e d to e s t i m a t e

,.

[ , - , (,+,)]

,..,+ ,); (,),,.

(37)

87

Since

I=I--N,

~t'""~tN--I

on m u l t i p l y i n g and dividing u n d e r the i n t e g r a l s i g n i n (37) by Iz F", (I*I>L~L), w e o b t a i n

1~ (x,

----

--

t ~1

~ (x, t, ~,

(38)

where Z~

The i n t e g r a l i n (39) exists i f N t > m + N, We l ~ v e f o r any/3 z

X

z~

lzl ~

By (17) and (18)for I z l ~ t q t / 4

{['-, (@)]

'.

I

.~ c~l r (.,. t.., + ~)j ~ ~, ~ cd(~. txl +l~ + : D--.,", ~ . =

c~(~. txt + j , t)--",,

(4 1)

F r o m (40) a n d (41) f o r a n y / 3 a~mi_N1 > m + N

! ~ , ~ . (x, t, .~, *)1<:

cd#(x,0 f ~ + l z

D"-',~z ~ c,,?(~, 0(! +1 ~ !)'+~'-",.

(42)

We note that e s t i m a t e s (41) and (42) c a n be i m p r o v e d . I n (41) w e a c t u a l l y have on the r i g h t s i d e Ch~(x , t)(1 + _ m and C,q(.,c, t ) 0 +1 ~+~i) 1 pill-'' l z t1N, i f p l f l l -> m. Substituting this into (42), we find I zl) m - p Ifll-Nt i f p Ifll < that t~

(x, t, ~, vJ) I ~ C~q(x, t)(t +J ~ j)~+N-~mI-~, if p t fll --< m and in the c a s e p i/31 - m w e have dz

l ~

00 + t ",tt)-"',

+

1

.3_.0+[~f).ml-"

~C~(x, t)(t + I '~ I)"+~-'l"~-'v', =tc~(x,

+

i.f ~ 1~ I ~;'~ + N, i~ P I # I > , , , + N .

(42')

F r o m (38) and (42) w e find

I A ( x , t, ~)1 "-<~ 9

Xl

I~t~N, "

=l ~ l (l + l x - v O ~ + ~ + ' ~ . ( x, ~, "~, x - v ) i

D%.(y) .....I.... "y )N+I~..~CM.N~(2C,,)(I.~_,~,)m+N_Ntsu~ (~ + I x - - e l ) a ~ ( t + t x - v ~6, ~, l=l-m

w h e r e M - 0 and N 1 > N + m are a r b i t r a r y .

]0 +lx--~yl) o'u(e)a I'

C o m p a r i n g (29). (36). and (43). w e obtain f i n a l l y

tjo(x, t, ~)j=lYt (X, t, "~)-}-jz(x, t, ~i)-}-]a (X, t, ~I) 1 -.
IO~u(Y)l

~t~ ~. t=t=~, (l + l x - - e I ) m '

w h e r e M >- 0 and N~ > m + N a r e a r b i t r a r y .

(43)

(44)

The p r o o f of the l e m m a is c o m p l e t e .

L E M M A 3. Suppose that q(x, t, ~) s a t i s f i e s the c o n d i t i o n s of the p r e c e d i n g l e m m a , and un{x) = ~n(x)r(x), w h e r e the s e q u e n c e ~n(X) is defined in L e m m a 1 and r(x) ~ C ~ ( R N) and g r o w s at infinity no f a s t e r than s o m e 88

ftxed power of lxf then

~e'(x,.)q(x, t, ~]q-z) u.n (z) de ~,,~rl--f q(~' (x, t, ~)D~r(x)-l-

jo (x., t, ~),

(45)

w h e r e jo(X, t, r~) is defined in L e m m a 2 [where u(x) is e v e r y w h e r e r e p l a c e d by r(x)]. The limit in (45) is unif o r m with r e s p e c t to (t, ~1) ~ [0, T] x R N and x in any compact set K ~ R ~r. Proof~ The a s s e r t i o n of the l e m m a follows i m m e d i a t e l y f r o m Eqs. (26), (32), and (38) which a r e meaningfui for any function u(y) which grows (together with its derivatives) at infinity no f a s t e r than any power ly I and the Lebesgue t h e o r e m on passing to the limit under the integral sigm We note that the e s t i m a t e (44) with r(y) in plac e of u(y) holds "for j0(x, t , ~/)in Eq. (45}. 3*. In this s e c t i o n w e c o n s t r u c t the principal p a r t of the symbol p(x, t; , , r) of the o p e r a t o r U(t, r) c o r responding t o the Cauchy p r o b l e m (D. We f o r m u l a t e the conditions that we impose on the symbol a(x, t, D of 2

1

the pseudodifferentiat o p e r a t o r a(x, t, D}. Let a(x, t, ~) satisfy the following conditions: 1) a0t, t, ~) ~ S~,6,* i.e., a(x, t, ~) e C**(RN x R t x R N) and for any multiindices a and/3 and all ~, x ER N,

t ~[o, T] I a!~, (x, t, ~,)t = I O~t~a (x, t, ~)i < C~ (1 +1 ~l)* ' - ~ + ~ , lP$ w h e r e m, a, 6 a r e r e a l n u m b e r s and 0 - 6 < p <_ 1; 2) the condition of hypoellipticity: t h e r e exists a m m h e r Co such that for I ~ I > C o and all (x, t) e R N x

lanai (x, t, i) I< C ~ l a (x, t, ~) 1(1 +1~- t ) - ~ + ~ ;

(46}

3} R e a ( x , t, E)Co; t

~la(x, 4)

,

s, Dlars

.r

p-6 ,,

-----f(x, t; ~, ~)-----0(1q-I ~ l) 2

, w h e r e e > 0 is a r b i t r a r y , 0 _< r _< t _< T and I ~ I > C o.

~lRea(x,s, ~)1 ds We set

a(x, t, ~)----b(x, t, ~)+ic (x, t, ~),

(47)

w h e r e the functions b and c a r e r e a l and b(x, t, ~) < O, I ~ I > C o. We set t

ro(x,

t;

f a(.'., s, 11Ins

q,

(48)

~)=e ~

t

and for b r e v i t y we s e t 0(x, ~)= a(x, s, and/3 have and/3we wehave

~q)ds (we

omit the dependence on r and t in 0). F o r any muttiindices ot

ro (~) Cx' t; ~1, x)= e o ~ c, [0(')]t' (Ill,

(49)

. .[O(I~I+II31)]II~H-II$1,

where 1 .i, + 2 . i ~ + . . . + ( I ~ I + I P l)~,~t+1~ = I 9 l + l P~I ~nd ~, > 0.

(so)

E a c h t e r m in the s u m (49) has c o m m o n o r d e r of derivatives equal to I ~ I + I/3 I, w h e r e I ~l of t h e m a r e taken with r e s p e c t t o the variable ~ and I~ I with r e s p e c t to the variable x. We have

la~,e(x,.~)l _-

li 9

li

a (~ ')(x,s, ~)ds ~

I a(=')(x,

s,

,Otds
and s i m i l a r l y for differentiation with r e s p e c t to the variable x. We thus obtain

m

* F o r m o r e details r e g a r d i n g the c l a s s e s sp,6 see [9, pp. 298 ffl.

89

Irol~](x,

t t;~,,01 ~C~,[eI .r~,c.~.,,',u".,[ I(t +l,] l)..,,~,+6j~fj~c,

(i

~t'+"'+q~f+fPl , I,~(x,s,'~)l ds)

(51)

where by" (50)

1 <:~,

+...

+z,.,+,,,-.
S i n c e for e, a 9 0 and w(x) >- 0

e-w~x'~n~(x)~,e-~(~)" = C(~, "),

(52)

it follows that t

I."

=.*

tk

f

l,-----

\!l,c.,

.

,, , , , , /

by condition (4) of (46). F r o m (51) and (53) we obtain t

(l-s) ..f t~ (x,s,~lds

Iro[t~(X, t; ~, ~)i
~

(1 +l "I]l) -"l'l+6j"lfl'n+tpl

(54)

W e set

vo (x, t ) = I e'r176 Cx, t; ~, ,c)ff'o'('~)d~; h e r e u0~) is any function in 9

(55)

and r 0 is defined in (48). By L e m m a 1 we have

t(x,~)dro

= Jim ~""~'.o (~) a~ ~to, -

I~"x,z)a{.~, t, '~ +,0.

~."~o(-, t; ~, .,) (=)} a=.

(56)

By L e m m a 3 and the Lebesgue t h e o r e m it is possible to pass to the limAt under the integral sign in Eq. (56), and the limit of the e x p r e s s i o n in b r a c e s is equal to i. ('~, e t(x ._~to~ llTI fdro -~-. - - ~

z)_

[ (x, L.~+z)?.ro(., t, ~, ~)(z)dz)~=a(x,t,~)

X

X ro (x. t; n, ~)-- ~.~ ~a(')(x, t, ~)D=ro(x, t; ~, ,)--jo(X, t, ~) ]c~I
= -- ~.~ if,[ &a(=)tx ~ , t, ~)D=ro(x, r ~, =)--jo(x, t, ~l; ro) -------ko(x, i; ~, =)+go(x, t, ~),

(57)

w h e r e we denote by k0 the s u m and by go the r e m a i n d e r J0- Thus,

( ff-i-- A ) % (x, t) = -- Kouo+ Gouo,

(58)

w h e r e Ko and Go a r e o p e r a t o r s with kernels k0(x, t; r/, T) and g0(x, t, r/), r e s p e c t i v e l y . We now set t

~ a(x,s,ma= rt (x, t; "l, ~) = 9I~ ko (x, a; "11,x) e ~ da

(59)

vt (x, t ) = I e~tX'~)r~ (x, t; "1, X)Uo(~)d~].

(60)

and

Sinee ~ r l / ~ t = k0(x, t; ~/, r) + a(x, t, ~?)r~(x, t; ~, r), it follows by L e m m a 1 that ~ - - / i ) ,(X,/')=/-('otto+ n-~-

90

(61)

A g a i n on the b a s i s of L e m m a 3 and the L e b e s g u e t h e o r e m it is p o s s i b l e to pass to the l i m i t u n d e r the i n t e g r a l s i g n in Eq. (61). C o m p u t i n g the l i m i t of the e x p r e s s i o n in b r a c e s as a b o v e , we obtain e~(~' ") a (x, t, ~t+ z) ?~r~ (., t, n, ~) (z) dz

limla ix, t, ,~) r~ (x, t; n, ~) -

= -

~

! h-ra(=)(x, t, ~)D~r~(x, t; n, ~ ) - / o ( X , t, ,~; r~)=__-/~l(x, t; n, -.)+~ (x, t, %

(62)

Thus, (63)

From

(56) and (63) w e o b t a i n

( ~ - a)(~,o + ~,~)ix, 0 = - tC~o + (Oo + o,) ~o.

(64)

We f u r t h e r define r j ( x , t; ~, r) by induction. If rj(x, t; 11, r) has a l r e a d y b e e n d e f i n e d a n d

~s(x, t ) = ~ e ~(~, ~r~(x, t; n, ~)~0(n)dn,

(65)

then

( ~ - a ) i~o+ ~,+ ... +~,) (x, o =

-xj~+iao+a,+... + a,)=o.

(66)

Therefore, t

ri+t (x, t; n,

t

J"=(x. z, n)da

~)--'~klix, ~; "1, ")~

dn,

(671

.r

r

(X, t)-----~e~(x, ~')rl,=(x, l; ~1. ":) ~ (n) dn,

(68)

and

(69) where (70) O<[~I
and gi.z (x, t, ~) = -- ]0 (x, t, n; rl+~)-

(71)

LEMMA 4. Suppose that the s y m b o l a(x, t, 7) of the p s e u d o d i f f e r e n t i a l o p e r a t o r A(t) s a t i s f i e s conditions (46). T h e n the k e r n e l s r j ( x , t; 7, r) and kj(x, t; 7, r) (i = 0, 1 , 2 . . . . ) belong to C~(R N x R~ x R N x R~), r _< t, and s a t i s f y the inequalities: t

r(.,~-). (x, t; ~, ~)] ~ ~, ~'s~, e (~-8) ""fb(x, ~. ~)~,[it -+-In I)6-"f'V (1 -t-t n I)~l~'l~mlf I~'l+lnl, /[pJ

(72)

t

I kj(~) ('~ ( x , t; ~,~)l~
(z-e) .r b(x, .~, n)as

~

ta(x, t, n)[ [(l+[n!)S-Pfls(1-F!nl)~

(73)

P r o o f . We c a r r y out the p r o o f by induction. F o r r0(x , t; 7, r) the a s s e r t i o n of the l e m m a holds by (48) and (54). F o r k0(x, t; 7, r) we have N~

~o(x, t; ,~, ~)= ~ -~ a(~) (x, t, ~) O~ro (x, t; n, ,).

(74)

I~r

91

T h i s i m p l i e s that k o ~ C = in all v a r i a b l e s and z~t

(75)

o(p)

iYl-i ~,+a,--~ P,+P,,--P

F r o m (75), (46), and (54) (N l -->- 1 is fixed) we o b t a i n t

(l-s) S~

N,

k('~) (x, t; ~, ~)l ~< ~ C~v I a (x, t, 71)I (1 +l"z l)-"c~v[+~='l)~lp'le "o(p)

'

lYt--I

t

x p,+~,(i+! ~ !)-o,~,+~'~ c, (I+l~ I)-"~+~'~'2 ~' IYl--i.

The a s s e r t i o n of the lellln2a thus holds for k0(x ~ t; 17, 1"). Suppose now that the a s s e r t i o n of the ] e m m a hoLds for rj(x, t; ~, r). T h e n

kj(x, ~; 7], '~)-'--""~] _.I a(y ) (x, l, ~])DVri(x, t; ",1,~), I~-z yI

(77)

i . e . , kj ~ C = and f o r any m u l t i i n d i c e s ~ and O w e h a v e

,,~(~)'(~(x, t; ,~. ,) = ~~' ~. ~ -- c.,~al )E=.) (x, t, ,]) -(~,~ -~(v+~.,, (x, t; ~, '0-

(78)

h'l-I P,."I'P,--P

F r o m (78) by m e a n s of (72) and (46) we obtain, as above, L ~ } (x, t; ,~, ~) I ~< C~p~e(~-') ~ ~ #(~'"~)~" la(x, k('~)

t, ,Ol[(l §

fZl'(l +lnl)~-"/(l +lnl)-"'~"+~'"' /'='+I"'.

(79)

Now t

t

~ a(x,s,q)dz

r~+, (x, t; ~l, ~)= I kj (x, a; ~1, ~)e" Therefore,

da.

(80)

rj+ t ~ C ~ and f o r any a and fl t " l (c,,) l ~"a(..,:.~,'q)d~| r

r /+z(p) (~, (x, t; "~, ,).,=f

r, ~(=,) ,~

~

(81)

i%+P,-P F r o m (79) and (54) we o b t a i n f r o m this

! r~=+~,<~)(x, t; ~, ~)[ < .

t

x/'~'+f"'J I I a (x, ,~, "a) I e

Xe

(I--~)~L o i x , j,

q)u$

~

C~ [(l +1~ l)~-~ P|J (1 +[ ~ t)8-~ ]'(z +1 ~ l)-pl='l+Swp'l,m'l+~a'r(z +l~ ])-'~='J+~'~']'

t (l--e) ~ b(x,.r,~)ds ( l - - e ) ~"b(x,z,lq)d$

';

e

"

a,~ 4 c~ [0 +1 ~ 1)s~ PI" C1+ l ~ 1)'~-' / CI +1 'l !)-"f~v+sk6'/f~r~t t

i

'

(l--e,)5#(x.~,~)da ]a(x,a,'q)ld~.C~e,e "~ [(I -~- ]~ })6-P.f~lJ§ (1 d- [ "tl I)-PI~I+alPlfl ~l+lpr,

(S2)

0<~<,,;

in the l a s t inequality h e r e we have u s e d e s t i m a t e (53}. T h u s , e s t i m a t e (72) h a s b e e n o b t a i n e d f o r rj+l(x, t; ~7, r). Using e s t i m a t e (82) and Eq. (78) for j + 1, we obtain f o r kj+t(x, t; 7, r) e s t i m a t e (73). The p r o o f of the l e m m a is c o m p l e t e .

92

LEMMA 5. Suppose that the s y m b o l a(x, t, q) of the pseudodifferential o p e r a t o r A(t) s a t i s f i e s conditions (46). T h e n the functions g](x, t, 7) = j0(x, t; 7, r]) e C ' ( R N x Rl+ x R N) and for any multiindices a and

Iv(~, r J(l],) (x, t, ~) I~ c , , ~ , (I + l~ I)m+#-(~'~'' I N'[(l +I ~ I)~-~PF (l + I,~I)-*'~'+~'~'/'~+'~',

(s3)

w h e r e N t > m + N is any fixed number. Proof. Both a s s e r t i o n s of the l e m m a follow e a s i l y f r o m Eqs. (26), (30), (37), and e s t i m a t e (72}. We note only the following: in the differentiation of (37) with r e s p e c t to ~ and the subsequent e s t i m a t e using (38) and (42) the n u m b e r N t m a y be t a k e n a r b i t r a r i l y l a r g e independently on the Ni contained in Eqs. (26) and (30). Let n be any natural number. We c o n s i d e r the s u m t~

r(x, t; ~1,~ ) = ~ rl(x,

l; ~, ,r

(84)

and s e t

~, (x, t ) = ~ e ~ '~. ~ r (x, t; ,~, ~) ~o ('3 d~.

(85)

(~T-- A ) v(x, t)= Huo,

(86)

Then v(x, t) s a t i s f i e s the equation

w h e r e the k e r n e l of the o p e r a t o r H has the f o r m [cf. (66)] tJ

(87)

(x, t; .,. ~)= -- ~ (x, t; ,I, ~)+ ~ gj Cx,t, ~). 1-.o

By L e m m a s 4 and 5 h(x, t; 7, 1") E C ~ t N • R 1 x R N • RI+), r _< t, and

i z, Cx, t; ,;, ,)1 ~; c ((1 +1,1 I)" [0 --I-I ,~ I P - * P I n (t + I n I)~".f + (t + t n l) '~+'~-('~'~ N,f,).

(88)

Since f=O(l+{.ql) "T-8' ~ > 0 , it is evident f r o m e s t i m a t e (88) that for sufficiently l a r g e n and N i h{x, t; ~1, T) as a function of 7 tends to z e r o as 1,71 - - ~o f a s t e r than any power of 1 / I 771 u n i f o r m l y with r e s p e c t to x 6 R N andr_tC[0, T]. 4 ~ In this s e c t i o n we shall c o n s t r u c t the s y m b o l p(x, t; 7, r) of the o p e r a t o r U(t, r). Let {qn{X)} be the sequence of functions d e s c r i b e d in L e m m a 1. F o r all functions p(x, t; 77, r) and h(x, t; % 1") we denote by p , h the l i m i t

I e' ~.,,p (x, t; ~ + z , o) ~n/Z(', o; ~, ~) (Z) dz -- p (x, t. ~, o).I~ (x, ~;-~, ~)

(89)

which exists by Lemma 3. Let, further, r(x, t; 7, r) and h(x, t; 7, r) be defined by Eqs. (84) and (87), respectively. We consider the integral equation t

p (x, t; ~, ~) = r (x, t: ~, ~)--~ p (x, t; -q, a) 9 h (x, ~, ,~, ~) d~

(90)

*r

with the unknown function p(x, t; ~, v). W e iterate Eq. (90) (for convenience we omit the dependence of the functions on x and q, and w e drop the minus sign in the function h while retaining for it the former notation):

[i t

l

] ~t

(91)

LEMMA 6. Let

p (x, t; ~, ~),/~ (x, t; ,~, ~) andr (x, t; ~, ~)

(92)

be t h r e e functions for which the hypotheses of L e m m a 3 a r e satisfied. Then (associativity of the o p e r a t i o n *)

93

[p if,,t)9 h Cot,0)19 r (~,,)= P if,o,)9 [h (~,,~)* r (o,r

(93)

The a s s e r t i o n of the l e m m a follows easily f r o m the definition of the operation * ami L e m m a 3. If we now s e t

1,~ (~, ~) = I t, (o, s) t, (s, ~) as

(94)

9

and,

generally,

(,,..o= I

s)

(,. as=

s).k (s..:)as.

(95)

then the solution of the integral equation (90) [or (91)] can be r e p r e s e n t e d in the f o r m t

p (t, ~) = r (t, ~) + ~ r if, ~)* 9 (~, ~) a~,

(96)

~g

where

tlml

LEMMA 7. 1) The r e s o l v e n t @(x, a; ~, r) o f t b e i n t e g r a l e q u a t i o n (90) exists for any 0 - r -< a < *% belongs to C~(RN x R 1 x R N x R ~ , and s a t i s f i e s the e s t i m a t e l~(~))(x, ~;7, ~)l~ c ~ 0 +l ~ !)-J'*.'~'+~'rJ'/'~

(~

(98)

w h e r e ~, /3 a r e any mnltiindices and j, C a r e some fixed positive numbers. 2) The integral equation (90) has a unique solution defined by Eq. (96). Proof. As we have s e e n [cf. (87) and (88)], h(x, t; 77, r) ~ C~(R N x Rl+ x R N xRl+), r _ t, and s a t i s f i e s the estimate

]h(C~)(x,t; .q, ~.){43Cc,(1 +['q

{)-/-plc,lfl%

(99)

w h e r e j > 0 is any fixed number and a any multiindex. We have

/, O, s)*/, (s, O = llm (e"'.')/, (x, ~, ~ + z , s),,"T(., s; 7. =) (z) dz n.-,-co d

=2m f

h (y. ,:

o ,ty I

o, +

s)

(I00)

We let ("

.

l (x, **, ~, 7, s)=~ e - " z " ) h ( x, ~, "q+

z,

s) dz.

(101)

Then for any multiindex fl by (99) fo,

(102)

Choosing fl such that {fl { =/30 = N + 1, f r o m (100) and (102) we obtain

[hO, spa(s, ~)1 ..
(g, s, n, .)ldy

I x--y

[)N+t

~CoCa.(l-t-I'~ )-J

(103)

and t h e r e f o r e {h2 (~, r ) [ = [ i h (a, s)*h (s,

~)ds[~
(104)

F u r t h e r , by induction, if for hm(q , r) there is the e s t i m a t e ~ xm-I IhAo, 0 [-..,'If' ~,-o~-~.~

94

(a--~) m''

( ~ - 0 , (l +I ~ I)-~'

(105)

then

(lo6) and by (102) and (105) t$__T~m-t -

'

"(i + 1 x - - u

(107)

I) "++~"

Hence

t k=+, O, .)1--

~ (~, s)*h.(s, ~) ds

< (CoC~J"

(I +1 ql),_ ~ t~_ ~ r ' d s C.,,- t)~

~'-

"

==,(CoCit,)"~(lJr.l'qi) -]

(108)

,+.-

+0,

and the induction is c o m p l e t e . Thus,

Ir

o;~,91ffi

t

~.Cx, o;~, ~) < ( I + 1 ~ 1 ) -s

i +

(~-1)~

The e x i s t e n c e of the r e s o l v e n t +(x, +; •, +) of the i n t e g r a l equation (90) has thus been d e m o n s t r a t e d , and e s t i m a t e (109) holds f o r it; m o r e o v e r , f r o m the u n i f o r m c o n v e r g e n c e of the s e r i e s we find that the function ,~ is continuous in all v a r i a b l e s . We shall now p r o v e that ~(x, ~; +7, r) ~ C~(R N x Rl+ x R N x Rl+). The fact that @E C ~ in q follows i m m e d i a t e l y f r o m the fact that e a c h hn is infinitely differentiable in ~, the fact that the c o n v e r g e n c e of t h e s e r i e s for the r e s o l v e n t is u n i f o r m in ~, and e s t i m a t e (99) by virtue of which it is possible to differentiate with r e s p e c t to 77 u n d e r the i n t e g r a l sign in Eq. (106). T h i s , in p a r t i c u l a r , i m p l i e s the e s t i m a t e for the d e r i v a t i v e s w i t h r e s p e c t to +7 of the function @:

I~>'~(x, ~: ~, ~) t.<.C~,(t +l'~ I)-++-'*:+sfm~(+-'>

(no)

for any multiindex a ; C = CoC/30. The differentiability of 9 with r e s p e c t to x, a and 1- is s o m e w h a t m o r e c o m plicated. We c o n s i d e r in detail the differentiability of 9 with r e s p e c t to x (the c a s e of a and T is s i m i l a r ) . Since j >0 in the e s t i m a t e (99) is any fixed n u m b e r , we m a y a s s u m e that it is sufficiently l a r g e , m o r e p r e c i s e l y ,

] > N + m + 2.

(111)

Let fl be any multiindex. If lfll < m, then we may differentiate with r e s p e c t to x under the i n t e g r a l sign in (106) and r e p e a t all c o m p u t a t i o n s f r o m (101)through {109)with j r e p l a c e d by j - - ! ~ I ( 6---Ko) ~ (8.~_._s

a r i s e s h e r e due to

m a j o r i z i n g the function f which o c c u r s in the differentiation of h with r e s p e c t to x, cf. (83)!. If I ~ I >- m, then to c o m p u t e a c e r t a i n n u m b e r of f i r s t t e r m s of the f o r m h(~, s) * hn(s, r) we make use of L e m m a 3 in which we take for N t the n u m b e r N + m + 1. T h e s e t e r m s , w r i t t e n out a c c o r d i n g to Eq. (20), we can differentiate the r e q u i r e d n u m b e r of t i m e s (t fl I) with r e s p e c t to x. M o r e o v e r , in each e x p r e s s i o n h(cr, s) * hn(s, r) the power of decay in ]W[ at infinity is l e s s than (n + i) ] - - N t ( ~ -P-) n > n [ / - - ( N + m + l)l > n. F o r sufficiently l a r g e n this e x p r e s s i o n is g r e a t e r than I f l l + N + 1. Starting f r o m this n, we a g a i n u s e Eq. (106), etc. and differentiate with r e s p e c t to x u n d e r the i n t e g r a l sign. In this way f o r any multiindex ~ we obtain the e s t i m a t e

t ~ m (x, ~,; ~, ~)(< C~ (1 +1 ~ ])-s+8~m.~mec~o~).

(~ 12)

A s s e r t i o n 1) of L e m m a 7 has thus been proved. We shall p r o v e 2). We substitute (96) into (90) [in (90) we have changed the s i g n in f r o n t of the i n t e g r a l to plus], and, using (97) and the a s s o c i a t i v i t y of the o p e r a t i o n *, we s e e that (90) b e c o m e s the identity t

t

T

+

(s, ,) r

rl~2

r (t, s)*~ (s, ,) ds.

= r (t, ,) + .r

95

W e shall prove the uniqueness of the solution of Eq. (90). If there are two solutions p~(t,r) and pz(t, r), then t h e i r d i f f e r e n c e p = p~ - 1~ is a solution of the homogeneous equation; t h e r e f o r e ,

By (I05) this implies that p(x, t; ~, T) = 0: The proof of L e m m a 7 is complete. We have thus c o n s t r u c t e d a f~netion p(x, t ; ~, r) which is the solution of the i n t e g r a l equation (90):

p(x, t;.q, , ) = r ( x , t; :q, ,)+~ r(x, t; ~,.a).~(x, ~; % ,)da.

(113)

W e shall show that this function!s the symbol of the operator U(t, r), T H E O R E M 1. Suppose that the symbol a(x~ t, ~) of the pseudodifferential operator A(t) satisfies conditions (46). Then there exists (a unique) symbol of the operator U(t, r), and it is defined by Eq. (113). Proof. W e must prove that the solution of the Cauchy problem (1) in the class ~ exists and is representable in the f o r m

. (x, o--fe.x, ,.p(x, t;

(114)

F i r s t of all, it follows f r o m Eq. (114)that u(x, t) ~ C ' ( R N x Rl+) and ~Sa as a function o f x for e a c h fixed t -> z. Indeed, this follows i m m e d i a t e l y f r o m r e p r e s e n t a t i o n (113) and e s t i m a t e s (72) and (98) for the functions r(x, t; 71, r) and ,b(x,t; ~, 7") ~ C'*fft N x R i x R N x R~). We shall show, f u r t h e r that for each x ~ R N

lirag (X' Q =*,Uo(x).

(115)

We have 9 rl

r (x, i; ~, ~ ) = ~ rj (x, t; ~, ~), n m~ciently great.

(116)

F o r j --- 1 t

t

~ a(x,~,11)ds

rj(x, t; ~, * ) = I kj_~ (x, ~;~, ~)e~

da.

(117)

T

Using e s t i m a t e (73) for kj-l, we e s t i m a t e rj(x, t; 7/, r) for j _> 1: t

Irj{x; t; a

•

I< cjt(t +1 t)

1)

t

e~

Ia(x,

o,

t

o-~) 5 ~(x,s.n)~* j n(x.,.n)d*

~

,r

da~:Cj[(l +1~ t)~-,f~lJe

C1--~)! ~(x.s,n)a, i

~ Ib(x, s, ~,)tds.

"C

(118)

USing now the f i r s t inequality in (46) and rnajorizing the exponential by one, we obtain f r o m (118) the following (rough) e s t i m a t e :

1rj (x, t; ~, ~) 1< Cj [(1 + t ~ I)~-* p l i (1 + t ~1!)" (t-- ~).

(119)

The rough e s t i m a t e (119) suffices for our p u r p o s e s , but we note that (118) implies that rj - - 0 as t ~ r unif o r m l y with r e s p e c t to q E R N for alI j _ 1. F r o m (116) and (119) we h a v e t

5 a(z.s,n~ds

r(x, t;

7, ~)=e~

+ ~ rj(x, t; ~, x), 1--1

and Ir (x, t;

96

. (lZO~

F u r t h e r , f r o m L e m m a 3 f o r N i > N we h a v e

(121)

r(x, ~; % @*@(x, ~; ~, ~) -~-=~- 0 _L r(=, (x, t; 7, ~)~)(=)(6, ~; ~, x)+J0. r162 Accounting with estimates (72), (98), and (21), from Eq. (121) we obtain

I

l

i r (6, t; ~], ~),@(~, o; 7, ~)do I < C Ct - ~)(I +l+i I)'~,

(122)

I

if Nl is sufficiently large and 0 ~- t - T_ < C. From (120) and (122) we see that p(x, t; 7, T) -- 1 as t -*T at each point 77ER N and that the integrand in (114) has an integrable majorant (1 + IHl)ml~(~)I. By Lebesgue's theorem it is p o s s i b l e to p a s s to the l i m i t under the i n t e g r a l sign in (114), and equality (115) is thus proved. It is not h a r d to p r o v e that actually u(x, t) - - u b ( x ) f o r t - - T in the s e n s e of c o n v e r g e n c e in the s p a c e ~ . We shall show

(~---A)~(~, ~)=0.

that the function u(x, t) defined in (114) s a t i s f i e s the equation

Indeed,

(123) s i n c e , as is e a s i l y s e e n by a g a i n using L e m m a 3 and w r i t i n g r*@ in the f o r m (121), we have the following chain of equalities t

pCx,~; ~, ~ ) = r Cx, t; ~, ,)+IrCx, t; ~1, ~)*| (x, ~; ~, =)do and t

t

-S

,;

(124)

o;

We have thus p r o v e d that p(x, t; 7, T) is the s y m b o l of the o p e r a t o r U(t, r). By the uniqueness t h e o r e m f o r s o l u tions of the Cauchy p r o b l e m (1) this s y m b o l is unique. T h e o r e m 1 has been c o m p l e t e l y proved. R e m a r k . By m e a n s of the s y m b o l p(x, t; 7, r) the solution of the inhomogeneous Cauchy p r o b l e m

lul,-~,.=~ (x),. =o.(x), ICx,. 0~' c a n be r e p r e s e n t e d in the f o r m t

;o 2.

Existence

of the

,;

7(.,

(i26)

T-Product

We p a r t i t i o n a t i m e i n t e r v a l [c, 3"], 0 ~ : c < ~ ' < ~ by points to, t 1. . . . .

tn+ 1 into s u b i n t e r v a l s [ti, ti+l],

i = 0, 1, . . . . n, t o = ~', tn+t = ~" . L e t t~ E [ti , ti+l] , At i = ti+ 1 -- t i. We denote by e [A(t~)]Ati the o p e r a t o r U(t, ti) f r o m y to y which by Eq. (4.1) c o r r e s p o n d s to the Cauchy p r o b l e m (1.1) in which the coefficients of the o p e r 2

1

a t o r a(x, t, D) a r e " f r o z e n " at the point t i and the initial t i m e is t i . t We denote by ~ the m a x i m u m length of the s e g m e n t s of the partition: A = max{Ato . . . . , Atn}. The T - p r o d u c t (chronological product) of o p e r a t o r s is defined by the f o r m u l a J

S[A (s)Ida

r-~

~

--

Hm ~/'~ (';)]'%[~(';-,)1"~-,....[" A,-*O

('o)l.,.'

(i)

u n d e r the condition that the l i m i t e x i s t s and does not depend on the m a n n e r of partitioning the interval [~, ~-] o r on the choice of points t i E [tt, ti+1], i = 0, 1, . . . . n. t h e r e and below f o r m u l a (i, j) denotes f o r m u l a (i) in s e c t i o n j. 97

In this s e c t i o n we c l a r i f y conditions under which the T ' p r o d u c t of o p e r a t o r s exists. 1~ Suppose that the conditions i m p o s e d on the symbol a(x, t, ~) of the p s e u d o d ~ f e r e n t i a l o p e r a t o r A(t) in Sec. 1 a r e satisfied. T h e n on each s e g m e n t [}~, t~+dc[~, $'] for the Cauchy p r o b l e m

-a(x, t.,

(2)

T h e o r e m 1 of Sec. 1 holds, s o t h a t t h e o p e r a t o r s [A(t')]Ati c o r r e s p o n d i n g to p r o b l e m (2) has symbol p(x, t, t', 2 7, tj) c o n s t r u c t e d in Sec. 1; this s y m b o l p o s s e s s e s all the p r o p e r t i e s e n u m e r a t e d in Sec. 1 (the notation a(x, t ' , 1

D) i n d t e a ~ s t h a t i n the o p e r a t o r A the coefficients a r e f r o z e n at the point t' E [ti, ti+t]). We denote by L(A) the operator

=

(,;-,) l,,,-,

(,a)],,..

For each A > 0 the operator L(A), as we have s e e n i n Sec. 1, is an operator from ~ to so. We wish to prove that u n d e r p a r t i c u l a r conditions L(A) c o n v e r g e s as A - - 0 to some o p e r a t o r L 0 f r o m ~ to ~ in the s e n s e of s i m p l e c o n v e r g e n c e of o p e r a t o r s , i.e., for any ~ Hm L(A)~ffiL0~. Using the symbol of the o p e r a t o r U(t, r)

c o n s t r u c t e d in Sec. 1, the o p e r a t o r L(A) c a n b e w r i t t e n as follows: for any function u0C~ 9

2n+l

2n+2

2n

2n--1

~

!

L (A)uo----p(x, t, t;,, ~, t,).p(x, t,,, t'~_t, ~, t,_3." .p(t, t~, to, ~, ~)-~z0,

(4)

w h e r e the e ~ p r e s s i o n on the right side of Eq. (4) is r e a d f r o m r i g h t to left (according to i n c r e a s i n g time) and 2

!

the e x p r e s s i o n p ( x , t I, t~, ~, T)U0 m e a n s the following: 2

I

I

'

(p (x . . . . , ~).uo (x)) ( x ) = f e (""~p (x . . . . .

~)s

(5)

etc. We introduce the functions w(x, t) given by: 2../+2

. 2j+!

2]

.

21--1

2

1

* ( x , t ) = p ( x , t, tj, ~1, tj).p(x, t:, t:_~, ~1, t:_i)...p(x, tl, to, "~, ~).~ for tj < t _< tj+t, j = 0, 1 . . . .

(6)

, n, or, m o r e briefly, 2

~a(x, t ) = p ( x , t, t;,

I

~,

tt).~(x, t:),

(7)

w h e r e tj < t < tj+1; w(x, t 0) = w(x, r) = u0(x). F r o m the p r o p e r t i e s of the symbols p(x, t, tj, 77, tj), j = 0, 1, . . . , n, itfollows that w(x, t) e C~(RN) as a function o f x and is continuous in t~[~,3-]~ M o r e o v e r , w(x, t) as a function of t is contained in C ~~ e v e r y w h e r e on [~, $'] with the exception of the partition points tj w h e r e its d e r i v a tives, generally speaking, suffer a jump. When speaking below of derivatives with r e s p e c t to t at the points tj, we shall always m e a n the left derivative, i.e., a~] ----- lira ~v(,x'ty§ ~" t - q at--0 at :0

2

(8)

x

We apply to the function w(x, t) the o p e r a t o r 1~7--a(x, t, D)), w-here it may be applied with r e s p e c t to=x as p r e viously [cf. (3.1)], since ~v(x, t)E~(R N) in the variable x for all t. By (7) of L e m m a 1 in Sec. 1 we have f o r tj < t _ tj+: #

Q

2

1

~

d

(9)

But ~ - - a ( t ' ) " p=O, and t h e r e f o r e -M--c~ p=(a(t')--~(t))

pmlz(x, t, t , ~, t:), t'~.[t:, t/.~].

(lO)

We denote by H(t) the o p e r a t o r with k e r n e l defined in (10) and by Z the shift o p e r a t o r (11)

98

F r o m (9), (10), and (11) we see that for tj < t ~ ~+t, J = 0, ! , 9 9 9 n, w(x, t) s a t i s f i e s the equation

(t)-n {t)z).(x, 0=o. M o r e o v e r , as is evident f r o m

(7,) and

(12)

the p r o p e r t i e s of the symbol p,

(13}

(x, 01~.-,, = "o (x). On the i n t e r v a l [~, $'1 we now c o n s i d e r the p r o b l e m

(x, 0 f~-,--- ~o Cx).

(14)

We shall c o n s t r u c t the symbol of p r o b l e m (14), i.e., a function pl(x, t; ~, r) tion

6

C *~ in x and n such that the func-

~ (x, t ) = ~ e ~(x, n,px (x, ~; ~, ~) Uo (~) d'q

(15)

is a solution of p r o b l e m (14). We e m p h a s i z e that in p r o b l e m (14) we c o n s i d e r a p a r t i t i o n of the interval [~, ~ ] by fixed p a r t i t i o n points tj, ] = 0, 1 , . . . , n + 1; the k e r n e l of the o p e r a t o r Hit) on e a c h i n t e r v a l tj < t _< tj+~ is given by f o r n m l a (10), and O / a t at p a r t i t i o n points is u n d e r s t o o d as the left derivative. As in Sec. 1, we take f r0 (x, t;

Sa(x. ,. ~) ds

~, ~)~=e '~

(16)

and

r (x, t} ffi ~ # ('. n)ro (x, t; ~t, Q u0 (~) d-q.

(17)

We must now p e r f o r m r a t h e r lengthy computations. I n o r d e r to simplify them, we use the following l e m m a (cf. [9, p. 3081L k

LEMMA 1. If pj(x, ~) ~ S Z ~ , j = 0, 1, . . . . .

" "

. , a n d m j - - - ~, then t h e r e e~xists p(x, ~) ~ Sp,6 m~ such that for any

]
where ~-maxmj. j~k

The function p is uniquely d e t e r m i n e d modulo S ~ .

If 9 ~ C~(R N) is such that ~o= 0 for

I ~ I < 1 / 2 and ~0 -- 1 for I ~ i > 1, then it is possible to choose a sequence vj - - ~ sufficiently rapidly that oo

(18) 1-,0

F o r the symbol p(x, ~) s o defined on the basis of the pj(x, ~) we w r i t e p ~ 2 P]" i This convenient f o r m a l i s m enables us to drop the r e m a i n d e r s in L e m m a 3 of Sec. 1 if they belong to

s;3.

F o r the functions we c o n s i d e r on differentiation t h e r e a r i s e s , in addition to the f a c t o r (1 q-IT I)-p i~l~ I~l, also the f a c t o r fl~ I+lfll; however, since the e s t i m a t e f----O (1 +1~ I)"~-

holds for f, we may a s s u m e that the

We t u r n to the c o n s t r u c t i o n of the symbol pi(x, t; 77, r). We substitute v0(x, t) into Eq. (14); we obtain

w h e r e the o p e r a t o r K0, according to (57.1), has k e r n e l

(20) tc~l>o 99

while the operator Go has kernel go given for tj < t _< tj+~, j = O, 1 , . . . ,

n, by (21)

~o(x, t, ~'s, ~, ~)=lim ~e'(~")~(x,~,r,~ +z' t~)~n~O(', ~s, ~, ~)(Odz. According to L e m m a 3 of Sec. 1 and the r e m a r k made above,

~o(x,t,t,,

~,

~)- ~

~("(~, t, t-, ~, t~)~0,o,(~,t~, ~, ~).

(22)

ts--O

We estimate the kernels h and go- We have-

,(x,

t/)f(alx, t,.

'~

t,)

v~((,)_ p(o,,.

r

(23)

where

Z=a(x, t', ~)-~(~,t, ~ ) f f i I ~

as.

t

We shall a s s u m e that the following conditions axe satisfied:

o a~)) (X, s, I&

1)

~)l < Cr

(24)

l a{;r (x, S, ~q)l

for all (x, s) ~ R N x [0, T] and any multiindiees a and #, I~t ~ Co; 2) for any e > 0 there is a 5 > 0 such that for all tt, t2e [0, 2"] such that. ]i t - t z l < (t

.=la(x, t;. n)l ---. . . . --~1~ ta (x, t,, n)l ..~(t "t-'), .1~; I>Co,

for all x E R N. Then from condition D

tNg ~(x, s, ,9 t < C ~ 0 +l,~-I) - p ~ I ~ I

Ia (x, s, '9I a s .

(25)

F r o m Eq. (113.1t and L e m m a s 4 and 7 of See. 1 we obtain an estimate for p(x, t, V, ,7, tj):

['

Ip~ (x. t.t'. ~, tj)l< c~ (t +I ~ I)-~'~'+~'~'/.r+,~, e('-') {J"'"~)"*+(t- t~)(l +I~ I)-~'

]

(26)

,

where m0 > 0 is an arbitrarily large fixed number and ~, fl are any multiindiees. From (23), (29), and (26) we obtain for h(x, t, t ' , ~, tj)

!t"

{1--i~)tj~ #(.~,t',tlldz

la (x, s, ~)l as

l~'(")tx't't"n'thl~C,,* (l+l'tlY'~'`''~'f''+*m''(a)*

i['

+(g--tj)(l+[~[) -~

e

] "

(27)

F r o m (22) we now obtain an estimate for the kernel g0(x, t, tj, 7, r): t]

(t-e) J Hx.*.~)a* (ex) 1go(~ (x, t, t j, ~, ~)I< C~ (l + 1~ J)-'~'l+61~l.f~'l+~le

'" I[ ('-" i' ~ Xl!ta(x,s, ~)ldslLe

+(t-t,)(l +l~,)-.o],

(2s)

where ~, fl are any multiindices and tj < t _< tj+t.t t

*In Eqs. (27) and (28) and an~ogous formulas belcr~v in the integrals o~b(x, t', ~ias..(t--ts)b(x, t', n) appearing in t

Q

the exponential function it is possible to write $b (x, s. q) ds if A iS sufficiently small by condition 2) of (24). -]

100

t

By Eq. (113.1) the symbol p(x, t, t', 7, tj) is equal to a s u m of two t e r m s p = r +

fr*~da; these t e r m s are t]

e s t i m a t e d differently as is evidenced in the estimate (26). To these two t e r m s the_re correspond two t e r m s = i n the function h [by Eq. (23)] and hence also in go [cL (22)]. We shall write h = h + h, where h (respectively, h) is e s t i m a t e d by the first (respectively, the second) t e r m on the right side of Eq, (27). This also holds for go = go + go and estimate (28): We proceed f u r t h e r as in Sec. 1. We set t

t

rz (x, t; ~, x)----- I [ko (x, ~; ~, ~)-{-go (x, ~, t], ~1, x)] e ~

a(x.s.~)a,

d=

(29)

and ~t (x, t ) = I e "~'") rt (x, t; ~, ~) ~o (~) a,~.

(30)

We substitute v~(x, t) into Eq. (14); we o b t a i n [ a s we noted above, we mean by 3 / ~ t the usual derivative if t ~ tj and lira ~(ts+At)-~ At-,~--o At

for points t j o r , s i m p l e r , we everywhere understand the left derivative] (d~--A (,) -- P/ (,) Z) ~1 (x, ~ ) = Kotto-~-Gotto-~-(arl)'tto--(a'ri) "it0-/'/(t) Z~I-

(31,

We set - - K t ~ = ( a r J " go-- (a*rt). tie-- -- kt (x, t;

?h x)'go

(32)

where the kernel k i is given by

~, (x,

t, ~, , ) - ~ ~a '=' (x, t, ~) r,(=~(x, t: ~,

,),

(33)

l=l=l

and

- - H (t) Zv,-- - - O , ~

2

--g~ (x,

t, tl, ~, ~).tto,

(34)

where (35)

e, (x, t, t~, ,~, ,),-, ~] ~i~ (=) (x, t, t', ,~, t~) r,(~) (x, ~s; '~, ") for tj < t <_ tj+l, j = 0, 1 . . . . .

n. We define rp, p -- 2, 3 . . . .

kp(X, t, ~, ~)~l gp(X, t, t 1, ~1, ~ ) ~ Z for tj < t - < t j + t, j = 0 ,

1,...,n,

, f u r t h e r b y induction: if rp is already defined, then

t-~ a(C~)(x, t, ~) rp(=)(x, t; ~, ~),

(36)

-dT h(=)(x' t, t', ~q, ts)rp(=)(x, tj; ~, ~),

(37)

!

and t

t

(38) ,s

We shall c a r r y out the n e c e s s a r y estimates. LEMMA 2. Suppose that the symbol a(x, t, 77) of the pseudodifferential operator A(t) satisfies conditions (46.1) and (24). Then the kernels rp, kp, andgp belong to C~(R N x RN) i n x and ~ and satisfy the estimates: t I kpl~(x,/; 7, "01~ C~e(1 --{-]~ [)-pl~l+~ll~lfl~l+lBI e(Z-e)~~(x's'n)dsIa (x, t, ~) I(1 -{- [~ DS-pf (z-}- IA)P, (39)

I~ (~)'- t, t;, ~, ~ ) l ~ C ~ d l +l~ll)-pl~l+~l~l/i~l+l~l .e

t 9

[ t.

I

If la(x, s, m) las e

t

9

(40)

101

tj < t , t ' - < t j + l , j = O , 1 , . . . , n ; t

(l-e)~ ~(x,s,~ids

Iroll](x, t; "l, ~)l < C = ~ ( l +l~ll)-~

I=l+lllle

~

( x + f h ) ~,

(41)

for any multinidiees a and #; X = (1 + I ~115-01 ~. ProoL E s t i m a t e s (39)-(411 hold for p = 0. Suppose estimate (411 is valid for rp. By (361 we then have

~or k~ t

1t,,l~7(x, t; 'l, 41":c~(i+l'l

"

l)_~i=,+o,lllfl=i+lie(,-,l~ ~(~.,.,llt, (z + f A71 a (x,

t,',i)l(l+t~ll)~-"f,

(42)

as required. B y (37) and (27) we ha-~ for gp(X, t, tj, lh ~1 with tj < t ~ tj+ 1, j = O, 1, 2 , . . . ,

<,-.,7,,'.1';

-- (~l)

le.<,. (.~. t. t,..t..) i ~ c.o

+ 1,1 I)-"=t'"f='+"'

~

m

I

9 !!i<,(x.:.~)iasl.

<'--"i. b<x't~ '

,

(431

which coincides with (40). We now e s t i m a t e rp+t(x, t; ~, r), using (381, (421, and (43). W e r e p r e s e n t rp+ i as the s u m of two t e r m s rp+ t and rp+t, and we estimate each of these:

1 lr-,,+,llRx,t; ,1,,)1=,=,.~._~ c.. L~'k,.l~:~' ~ (ox.o;~. ~0/ }=,.1<= J("', as1I i I$t+Pt"P

i t

(t-tiI~lx,s.~id* 4C=~i(l q-I1 I)-*I=i+81Plf=I+IPI(I+fA)P(l +1 ~t IP-ofe " t X I a (X, a, 7])I aa 4 C=ps,(1 +17] i)-pi='+~ (~) I rp+,~l (x, t," *,, ~)1

t

(tie,) fb(x.s.~lds

=l+'~le

7

t

I ~ =(x.~,nla,/

~ C e = (~'),x lh+{3,-l~

(44)

(Z+ f a F z ;

,)Le"

I

J(~,) do.

(45)

Suppose that t belongs to She interval tj < t _ tj+l; we substitute (431 into (45) and obtain

l-;.+,;(x.,,~.~)l?<:':'*' /(Z ~ ao+~ao~ Si~(~... ~)ia~ tkt--0

tt

t#

/

a

t

(l-e,)J ~(x.$,~)aa

~ ) l d s + a,

la(x,s,~l)tds

I] .

(46)

We estimate the last t e r m in (46)

' ' ] J--=t aolfla(x,s, ~)las : tj

Ia

11 if t -> t', then t

J ~ f f --tj) Ila(x,s,',l)]ds;

(47)

tI

2) if t < t', then J..<(t--tt

102

alas

aids 9

(48)

lnthesecondintegralia(48)wemaketheltnearchangeofvarlables-* then t"

r, sffi ~ r + B , w h e r e ~ =

t

t'-t

t_-7:77,

p =~ t,-tjt. t--O ;

t

Cf-t~)~la(x, s, ~)[ds=Ct'--t)gla(x,=~+~l,~)ld~ ~aCl+*)gla(x,~,~)t d=, t

t]

(49)

tI

since 0 < t" - t - A and if A is sufficiently s m a l l , then condition (24) is satisfied. F r o m (47)-(49) we s e e that t

J
(50)

tI

Substituting (50) into (46), we obtain ...--

t

(t-e,)S ~(x,,,~l)ds

-r(~) p+l(~)

(X, f; ~, ~)1
~

( z + f A ~ 'A

( l - $ t ) ~ b ( x , x ~)da

(51) F r o m (44) and (511 we obtain the e s t i m a t e for rp+~: r +6

~

t t l (t--~)J,~(x,s,~)ds

+

(52) which coincides with (411 f o r p + 1. The p r o o f of the l e m m a is complete.

We denote by r(x, t; ~1, ~'1 the s u m tg

(53)

r ( x , t; 7, ~ ) = Z r~(x, t; ~l, ~). p--O

Since e a c h % (x, t) ~ I et(X'~)ra (x, t; ~, ~) uo (~) d~ s a t i s f i e s the equation

(~--A(t)--H

' 6p_,uo (t ) Z)' % = ~.p_,ao -1-- T<,uo-- 6~uo. p > O,

(54)

it follows that n

(55)

p=O

s a t i s f i e s the equation "

""

r~--t

I"

(56)

The k e r n e l dn(x, t, tj, ~, r) of the o p e r a t o r D n is given by

dr(x,

t;tj, 7, ~)=

--kn(x, t; .~, :)---g~(x, t; b, 7, ~) - - ~ ~Ax, t, tj, 7, ~)--do(x,

t; 7, ~),

(57)

p-,0

w h e r e the k e r n e l s k and g a r e defined by f o r m u l a s 06) and (37), r e s p e c t i v e l y , and d o is s o m e function E Sp,,~, [it a p p e a r s in (57) b e c a u s e of dropping the r e m a i n d e r s in the s u c c e s s i v e definition of the functions kp and gp]. LEMMA 3, The k e r n e l dn(x, t, tj, ~?, r) for e a c h n ~ 0 as a function o f x and ~? belongs to C~~ N x RN). F o r any n u m b e r v > 0 the k e r n e l s d n with n - n o (v) satisfs" for m 1 ~ (p - 5) / 2 the e s t i m a t e *

t dn(~) r (x," t, tj, 7, ~)[~
(1 + [ ~ t)-~,

(58)

w h e r e ~, /3 a r e any multiindices; the e s t i m a t e (58) is u n i f o r m with r e s p e c t to (x, ~?) ER N x R N and t, ti, t ' ,

r e [0, T]. *See Eq. (64) below. 103

P r o o L The f i r s t a s s e r t i o n of the t e m m a follows i m m e d i a t e l y f r o m f o r m u l a (57) and f r o m the definition of the functions contained on the r i g h t side of this f o r m u l a . We s h a h e s t a b l i s h the e s t i m a t e (58). We _again r e t u r n to (57). F o r d o the e s t i m a t e (58) holds by definition of the c l a s s S~,,~,. We c o n s i d e r the t e r m s t, q, r). By (37) and the e s t i m a t e s for ~ and r ~ ) , we have

~p~,

lg~m(x,t, tp ~, ~)l
.f

~1.~1 ( ,s, ~)lds (t--tl)(l+l~])-~,e

~

( z d r f h ) p.

(59)

tt

L e t T ~ t <-- t 1. By (37) and (38) gp(x, t, r , 77, T) - 0, gp(x, t, r , ~, r) - 0 if p -> 1. T h u s , in the i n t e r v a l t e [1-,tl] it is n e c e s s a r y to e s t i m a t e only ~0(x, t, r , ~, r); f r o m (59) for tj = r , p = 0 we obtain

I~ ? ~ (x, t, ~, ~, ~)l ~
f l a(x, s, ~) t ds(t-- ~)(z + l ~ t)-~".

(60)

s i n c e the n u m b e r m o in the e s t i m a t e (60) m a y b e chosen a r b i t r a r i l y large, from (60) and (46.1) f o r g0(~ we obtain the estimate (ss). Since an ~p - 0 for p -- 1when t ~ [r, td, it follows, as is easily seen from Lemma Z, that

kn(x, t; n, r) for t e [r, td satisfies the estimate: t

(z-e),ra(x, ~, ,~)ds 1~(~) (x, t; ,~, ~) t < c~,~ (~ +1-~ l)-~='+~mf =~+m'e ~ Ia (x, t, ~)[ (z + f a ) z~' (~ + t ~ l)~-~.t".

(6 z)

F r o m (46.1) and the e s t i m a t e for X we s e e f r o m this that for kn e s t i m a t e (58) holds if n is sufficiently l a r g e , n _> n0(u). T h u s , e s t i m a t e (58) has been e s t a b l i s h e d for dn(x, t, tj, 77, r) for t ~ [T, t d. Suppose now that $ ' > t > t r Turning to f o r m u l a (59) f o r 1 < p - n a~d to Eqs. (39) and (40) for p = n, we s e e that they all contain a t e r m of the f o r m t

(l-,~) ~ b(x, ~,,~)ds (l-e) S/,,fx e , (Z+/A)'<~e ~ [Z~+(fa)~l. We e s t i m a t e the e x p r e s s i o n J - - e of g e n e r a l i t y we m a y s e t all Atj = A)

9

(62)

(fh)P. Since t -> tl, it follows that 0 - A <_ t - r and hence (with

l

A

m i n [b ( x , s , ~')i ~s~t

no

loss

jib(x,s,~)ds I

and t h e r e f o r e t

J=e

(l-e) S ~(x,

~

t s,

'q)ds

fPh*'~e

it

(1-,~} ~ ~gs

W

ISbds t~

u ..
p)e

j

Iks

t ~

~

nlbl

]" _

"

(63)

Starting h e r e , we s t r e n g t h e n condition 3) in the conditions (46.1); viz., we shall a s s u m e that b(x, t, e)~<--Cl~l =,, , . , > 0 , l}I>Co,

(x, t)ER~ X [0, TI.

(64)

Then p--6

!

--

< C ] ~ [ - y - , , m,; O < p - - ~ l ,

$

and therefore if m I >- (p - 5)/2 (it is sufficient that m I _> I / 2 ) it follows that

I

m i n i b[ $

~0, t~l>Co.

(65)

P r o m (63) and (65) we s e e that for J t h e r e is the estimatet ( l - e ) S bd$

J~
104

9 0 +l,~ [)-~

(66)

with any v > 0 i f p is sufficiently large. Recalling now that in Eq. (59) t h e r e is the t e r m (1 + t ~l)-rr~, w h e r e n~ > 0 may be c h o s e n a r b i t r a r i l y l a r g e and in the e s t i m a t e s (39) and (40) it is n e c e s s a r y to s e t n >- n0(v), w h e r e no(v) is sufficiently l a r g e , on applying (66) and the e s t i m a t e for X, we see that the k e r n e l s dn with n -> no(v) satisfy e s t i m a t e (58). The proof of the l e m m a is complete. LEMMA 4. The k e r n e l s rp(x, t; 7, r) and dp{x, t, tj, y, r), p -> 0, tend as & - - 0 to the kernels rp(x, t; 0 % ~'), dp(x, t; 7, r) c o r r e s p o n d i n g to the Cauchy p r o b l e m (1.D and c o n s t r u c t e d in Sec. 1. This limit is u n i f o r m with r e s p e c t to v, t ~ [0, T] and with r e s p e c t to the p a r t i t i o n points t/~[~, $'1 and the points t" 6 [t], tj+t]. P r o o L We c a r r y out the p r o o f by induction. F o r p = 0 0

ro(x, t; ~, ~)=ro(x, t; ~], ~) o

by construction. Suppose that the a s s e r t i o n of the l e m m a is t r u e for rp. T h e n by Eq. (36) kp 9kp as A - - 0 u n i f o r m l y with r e s p e c t to ~-, t, t', tj; m o r e o v e r , on the basis of e s t i m a t e s (39) and (63) for k~C~ we have a m a j o r a n t for all sufficiently s m a l l A: )

(I+1

a (x, p=O,

(l +In I) -"fz

(67)

I .....

F u r t h e r , on the basis of e s t i m a t e (40) we have, using (63), ] g~(~) I ~ C ~ (1 + [ ~1I)-~l~+~l~lf I~+I~! z p7eto max' r]_a (x ."s,

(68)

p = O , 1. . . . . -(a)

__

gp(~) 0 as A 0 u n i f o r m l y with r e s p e c t to r , t, t ' , t j e [0, T], x E R N and u n i f o r m l y with r e s p e c t to X E R N if p is sufficiently large. Considering what has been said r e g a r d i n g t h e functions kp(x, t; 7, v) and gp(x,

Hence

0

t, tj, ,7, v) and turning to Eq. (38), we see that rp+ t --- rp+ 1 as & ---0 u n i f o r m l y with r e s p e c t to r, t, t', tj. The l e m m a is thus p r o v e d for the kernels rp(x, t; 77, ~'). By means of Eqs. (57), (37), and (28) we see that gp(fl) --'v o and dp - - d p as A - - 0 u n i f o r m l y with r e s p e c t to r , t, t], t" e [0, T] and (x, 77) ER N x R N. The proof of L e m m a 4 is complete. 2 ~ In this section we c o n s t r u c t the symbol pl(x, t, '7, T, A) of p r o b l e m (14) and prove the t h e o r e m on the e x i s t e n c e of the T - p r o d u c t . Suppose some partition of the interval [=, $-]c[0, T] is fixed, and let A=0~<m max (t~+, t j ) . We c o n s i d e r the integral equation t

p,(x, t; 7, ~ ) = r ( x , t; ~, ~)+~pt(x,t; ~, ~)*m(x, ~, t l, ~, ~)do;

(69)

h e r e r(x, t; ~7, r) is defined by Eq. (53), and m = - d n i s defined by Eq. (57). The function r(x, t; ~?, ~') E C ~ x (R N x R N) i n x and ~? and is continuous in r and t E [0, T], while the k e r n e l of the integral equation (69) re(x, a, tj, 77, r) ~ C'a(R N x R N) i n x and ~ and is piecewise continuous in r and t e [0, T]. T h e s e p r o p e r t i e s and e s t i mates established in L e m m a s 2 and 3 for r and m show that L e m m a 7 of Sec. 1 holds for the integral equation (69). The integral equation (69) thus has the r e s o l v e n t

|

t, tt, .,I, , ) = ~

m, (x, t, t l, 7, ~),

(70)

where l

t

ran(x, t, tj, 7, .)=Im.-,(t, s)..z(s,

.)ds.

(71)

All the m n for n - 2 a r e continuous functions of r , t E [0, T]. Convergence of the s e r i e s (70) is u n i f o r m with r e s p e c t to (x, ~7) E R N x R N and t, tj, r E [0, T] (A is a s s u m e d to be sufficiently s m a l l and fixed). The r e s o l v e n t 9 t(x, t, tj, 7, ~') E C~~ N x R N) in x and ~7 and s a t i s f i e s

[r (=~ (x, t, is, 7, ~)[~
(72)

for any multiindices e and/3; here p > 0 may be taken arbitrarily large.

105

As shown in L e m m a 7 of Sec. 1, the solution of the i n t e g r a l equation (69) is unique and c a n be r e p r e s e n t e d in the f o r m

t (73)

p1(x, t; .~, ~ ) = r (x, t; ~, ~) + ~ r (x, t; "4, ~)*r (x, ~; ~, *$

LEMaMA 5. S u p p o s e that the s y m b o l a(x, t , 77) of the pseudodifferential o p e r a t o r A(t) s a t i s f i e s conditions (46.1), (24) and ml -- ( p - 5 ) / 2 . Then t h e r e e x i s t s a unique s y m b o l of the Cauchy p r o b l e m (14). This s y m b o l is defined by Eq. (73). Proof. We m u s t p r o v e that the solution u~(x, t) of p r o b l e m (14) is s u c h that u((x, t)~5~(R ~) as a function of x and c a n be r e p r e s e n t e d in the f o r m ttl

(x, t ) = I ei(x' ~)pl](x, t, ~, ~) ~to (~) d~.

(74)

The fact that u,(x, t)Es~(R~) for e a c h fixed t >- r follows i m m e d i a t e l y f r o m (74) and (73). M o r e o v e r , f r o m (74) we o b s e r v e that ul(x, t) is a continuous function in r and t E [0, T], since this is t r u e for Pl by (73). R e p e a t i n g l i t e r a l l y the a r g u m e n t s of T h e o r e m 1, we s e e that limut(x, t)=u0(x). We v e r i f y that (74) s a t i s f i e s Eq. (14). We have t~*~

o.

=~ e,(x. .,s [here at a . = A,-.--0 llra p' (

....

t+at

. . . .

[op, ~-

a (x ,

t, ~,).p, (x, t; ,,,

) - p , ( .... t . . . . ) At

~) - - h (x, t, r ,

~, tj)*p, (x, t j; ~, ,)1 d~ =

(75)

0

], s i n c e , as in T h e o r e m 1, the following equalities hold which

follow f r o m f o r m u l a (73) and p r o p e r t i e s of the functions r and Ot (it is i m p o r t a n t that r is continuous in t, ~" E [0, T] and that Or(x, . ; ~, r) is p i e c e w i s e continuous i n a s o that in the left d e r i v a t i v e with r e s p e c t to t Ol(x, t; 77, ~') a p p e a r s ) : Or

*

t

*

Or

,-,z c)+ + i( -a.r

0o

t oo

-,,.,).*,c/o = -m, + :X

~:

n--1

X "~

0

(76)

n--I

We note that the l a s t equality is s a t i s f i e d identically in t, since all m n for n -> 2 a r e continuous functions of t and r. It has thus been PrOved that pl(x, t; ~, r) is the s y m b o l of the Cauchy p r o b l e m (14). We shall p r o v e uniqueness of the Symbol. Suppose t h e r e is another s y m b o l P2(X, t; /7, r) s u c h that

u~(x, t)=fe"'~.~)p2(x, t; ~, ~)7~(',9d~

(77)

is a l s o a solution of the Cauchy p r o b l e m (14}. We c o n s i d e r the i n t e r v a l [r, tl]. On this i n t e r v a l u 1a n d u 2 a r e solutions of the p r o b l e m

a ~ 9.I(-~'F-- A (t)) u (x, t) = H (t) Zu (x, t) = I'I (t) Uo~ F1, (uJ,-,

=

(78)

Uo.

Since F 1 is the s a m e for the functions u 1 and u 2 and the uniqueness t h e o r e m holds for the o p e r a t o r A(t), it follows that ut(x, t) -- u2(x, t), x ER N, 1- _< t _< t v We now c o n s i d e r the i n t e r v a l [tl, t2] and on it the Cauchy p r o b lem

(o)u (x, , ) = . (.)u. ( , . ) = . (.) (,.)--

ult=t, = u, (x, t,) = u: (x, t,).

(79,

J u s t as b e f o r e , we conclude that ul(x, t) - u2(x, t}, x ER N, t 1 -- t -- t 2. Continuing in this m a n n e r , we find that

Ut(X, t)~U2(X, t), xEl{ N, ~[*~, ~-].

(80)

p t ( x , t; ~, :)-----p2(x, t; ~, ,).

(81)

It follows f r o m (80) that

Indeed, we shall p r o v e that if f(x, 4) ~ Lt(RN) as a function of ~ and if

106

I e~(x'i}f (x, then fix, ~) -- 0. We have 0 =

}) u0 (~) de----0 fo~ arbitrary ~fi50,

Ie'(x,~)f(x, })Tto(})di=Itto(y)?(x, y--x)dy,

(82)

w h e r e the F o u r i e r t r a n s f o r m is taken

with r e s p e c t to the second a r g u m e n t of f. Fixing any x 0 ~R N, we set ~(x0, .)(y - x 0) - ~(y). Since

Itto(y)~(y)dy=

0 for any t~69', it follows that r = 0, i.e., f(x0, z) = 0, and t h e r e f o r e f(x0, z) = 0 for all x0, z ~R N as r e quired. The p r o o f of the l e m m a i s complete. At the beginning of Sec. 2 we introduced the function w(x, t) [cf. (6)]

(x, t)----L (a) t~o= eIa(';')b', 9 eIa('6)b'*go.

(83)

This function, as we have s e e n in (12) and (13), is a solution of p r o b l e m (14), and t h e r e f o r e by L e m m a 5 w(x, t) has, in addition to (83), another r e p r e s e n t a t i o n 2

1

(X, t) = Pl (x, i, ~], x) t~o = PlU0.

(84)

Thus L (~) = P,,

(8,5)

w h e r e Pl is the o p e r a t o r f r o m 50 to 5~ with symbol the kernel p~(x, t; ~, r). We can now prove the main theor e m of this section. THEOREM 2. Let the symbol of the pseudodifferential o p e r a t o r A(t) satisfy conditions (46.1), (24), and let m i -> (p - 5 ) / 2 . Then t h e r e e x i s t s the T - p r o d u c t of

r-e,

[aO)Ns

= nm J4'~)b,,...

el"('a)l",,

(86)

L(a)tto=Ie~(~.'l)p,(x, t, ~, ~, A)~0(~)d~,

(87)

h...~

This limit is equal to the o p e r a t o r

U(3-, x) (cL Sec.

1).

P r o o L Since

it is n e c e s s a r y to prove that there e x i s t s

~ I ~(. .) p, (x, t, ,~, x, A) ~0(~)d~,

(88)

independent of the m a n n e r of partitioning the interval ix, 3-] by points tj and of the choice of points t i ~ [tj, tj+l]. According to (73), t

pl (x, t, ~7, x, A ) = r (x, t; ~, ~)-F ! r(x, t; ~1, a) *r

(x, ~; ~l, x)dn.

0

(89) 0

By L e m m a 4 r(x, t; q, r) - - r ( x , t; ,7, r) as A - - 0 uniformly with r e s p e c t to r , t, tj, t] 6 [0, T], while m(x, t; 0

7, r) -- re(x, t; 7, r) as h - - 0 uniformly with r e s p e c t to r , t, tj, t i ~ [0, 2"] and (X, 7) ~ R N • R N. The l a t t e r fact i m m e d i a t e l y implies that the r e s o l v e n t 4~1(x, ~, q, r) defined by Eq. (70) converges to the r e s o l v e n t @(x, a; 7, r) c o r r e s p o n d i n g to p r o b l e m (1. t) and defined by (97.1). F r o m the definition of the operation * and f r o m e s t i m a t e (41) and (72) this in turn implies the possibility of passing to the limit as A - - 0 in Eq. (89). We find that lira p~(x, t, ~, x, A)----p(x, t; ~, x),

(90)

A--*0

w h e r e p(x, t; ~, T) is the symbol of the o p e r a t o r U(t, T) c o r r e s p o n d i n g to the Cauehy p r o b l e m (1.1). On the basis of e s t i m a t e s (41) and (72) and the Lebesgue t h e o r e m it is possible to pass to the limit under the integral sign in (88). It has thus been proved that t h e r e exists

lira I e~("~)m (x, t, 7, ~, a ) ~ (7) d~ =I e""~)P (x, t; ,~, 9"~o('l) d~

A--,0

(91)

and this limit does not depend on the manner of partitioning the interval Ix, 3-] by the partition points tj nor on the choice "of the points t'j ~ [tj, tj+l]. This limit is u n i f o r m with r e s p e c t to t belonging to any finite interval 107

[0, T]. We denote the i n t e g r a l on the left side of (91) by v(x, t, A) and on the r i g h t by v0(x, t). We s h a l l now p r o v e that as Z~--O, v{x,t, A) -* V0(x , t) in the s e n s e of c o n v e r g e n c e in the s p a c e S~(R~) and u n i f o r m l y with r e s p e c t to t~[~, $-]. F o r this w e need to p r o v e two facts: 1) f o r any m u l t i i n d e x a and any c o m p a c t s e t K ~ R ~r

'v(~) (X, t, h ) ~ v o (a) (x, t) uniformly on

(92)

K;

2) f o r any m u l t i i n d i c e s ~ and fl

(93)

I x % ~ ( x , t, A) I -. O and all x e R N. We p r o v e 1); w e have

I. c~' (x, t, a ) - 4 ~' (x, 01--[ Ie "x'~' ~ r

c~.~.~ ~' tp,c~,,(x, t; ~. ,. a) _p(.., (x, t; ~, ,)1 ~0(~)~ I. r

It is e a s i l y s e e n f r o m L e m m a 4 and a r g u m e n t s g i v e n a b o v e in r e g a r d to Eq. (89) that px(~,) (x, t, ~1; ~, h ) ~ p ( ~ , ) (x, t ; ~ , ~) a q h ~ 0

(94)

and h a s a m a j o r a n t bounded in all v a r i a b l e s e x c e p t ~ i n w b i e h it g r o w s at infinity no f a s t e r t h a n a p o w e r w i t h 8+p a exponent ~<7"Y--[ 2t 9 Since u0(~) d e c a y s at infinity f a s t e r than any p o w e r , by the L e b e s g u e t h e o r e m it is p o s s i b l e to p a s s to the l i m i t as & - - 0 u n d e r the i n t e g r a l sign. T h u s , (92) h a s b e e n p r o v e d (even u n i f o r m l y f o r x ~ RN). We shall p r o v e 2). By i n t e g r a t i n g by part,~, we h a v e

"-
~:+~::~

IPtl~'),)(x' t' ~' ~' h)('~"~~

(95)

s i n c e I p ~ ) 2 ) l is bounded in all v a r i a b l e s e x c e p t ~/and i n ~ it g r o w s at infinity no f a s t e r than a fixed p o w e r

[with exponent-.< ~t ~ I-- v l ,~ I+l ~ + ~ I(~-~-~ ~ -- ~) l, while G0(V) d e c a y s at infinity f a s t e r than any power. It has thus b e e n p r o v e d that

T-e ~ta(~)la~=lira e[~0h)]a'~.

9 e [A('5)]a'*- U ($', ~)

(96)

A---~0

in the s e n s e of s i m p l e o p e r a t o r c o n v e r g e n c e in S~ (R~) u n i f o r m l y with r e s p e c t to T and t belonging to any finite i n t e r v a l 0 ~< z ~ t ._<3-..< T. T h e p r o o f of T h e o r e m 2 is c o m p l e t e . 3. for

Removal the

of Autonomous

Brackets

(Another

Expression

T-Product)

In this s e c t i o n we shall p r o v e that 3"

T'e ~ [A(s)lds~lirnea( 2 n + 2

. 2n-bl

X, tn,

~

2 I )At a . . . e a ( x , t ' o . ~ l ) A t , ,

(i)

A-,0

w h e r e on the r i g h t the l i m i t is u n d e r s t o o d in the f o r m e r s e n s e , while the a r g u m e n t s of the exponential function 2

1

r e p r e s e n t the o p e r a t o r s a(x, t ' , 71) w i t h t i m e f r o z e n at t'; the i n d i c e s o v e r the e x p o n e n t s denote s u c c e s s i v e 2

( f r o m r i g h t to left) a p p l i c a t i o n of the o p e r a t i o n defined in (5.2) [e.g.,

1

ea(X'to'~)~t~

]. The

a r g u m e n t s needed f o r this p r o o f a r e a l t o g e t h e r analogous to those u s e d in See. 2, and w e s h a l l t h e r e f o r e be bries We a g a i n take an a r b i t r a r y p a r t i t i o n of the i n t e r v a l [~, 3-] by points tj, j = 0, 1 . . . . . t i E [tj, tj+ d and for t s u c h that tj < t -< tj+t, j = 0, 1 . . . . . n, we c o n s i d e r the function (x, t ) = e " ( t i)(t-'/)e ~'( ')-,)ag-, . . .

108

e"('o)atOao

n + 1, t o = T, tn.~=$-,

(2)

or 2

I

v (x, t ) = ea(X't'J'~(t-tj)v (x, t~), t] < t ~ tj+,,

(3)

where a(t.[) is abbreviated notation for the function a(x, t[, ~) in which in the interval [ti, ti+l] the time is frozen at the point t~ ~ [ti, ti+ d. F o r tj < t - tj+t, j = 0, 1 , . . . , n, we compute

- (a.e"~"-'p)

- a.:('~)"-'~)].

~, (x, t,) = [,, (t)) :('))"-'~

(x, t~) - m (t) z~,

(4)

where Z is the shift operator defined in Sec. 2, and the operator Hl(t) has kernel hi(x, t, t ' , ~, tj) given on (tj, tj+t] by ~t (x, t, t', ~, t~) = [a (t')- a (Ol e a " ' ) " - t : )

g,wo. t', ~)--a(x, t, ~)l e~(t')(t-'J) -- "~ ~T a(~) (x, t, ~) ~e~(t')(~-t ~

+[a (t) e~(t')(t-tp--a*e~'t"('-tPl-[a(x,

(5)

F r o m (68), using (24.2) and (46.1), we obtain the following estimate for the kernel h i on the interval [tj, tj+t]:

,,,~)l~
la(x, t, ~ ) l ( l + ] ~ l ) o-p:, max la(x, s, 'q)l, h-----max (tl+t--tl) .. ,2j ~ s ~ t j+ t

Or

(6)

For tE[~,gl w e now consider the problem

{ ( ~ --A(t)--Ht(t)Z)s(x, u

I,-, =

u~

t)=O,

(x),

(7)

where by a / a t we understand lim g(x' t+at)--u(x, t) 9 As in Sec. 2, w e construct the symbol p2(x, t, ~, r, A) A:-.--o At of problem (7). W e begin with t

a(x,a.rOd* ro (x, t, ~l, ~)= e ~ 2

(8)

1

~o (x, t ) = ro (x, t, ,~, ~)Uo. We substitute v0(x, t) into (7) and obtain

( ~ - - A (t)) vo--H~Z~o= -- Kouo--OoUo,

(9)

w h e r e the o p e r a t o r s K0 and C 0 have the kernels

(I0)

ko(x, t; ~, *)~. ~ =I

l~l-t and

go(x, t. tj. ~, ~)-- "~ ~ h(c~'x i, s ~, t~) ro(~(x. ti; ~. ~)

(11)

for tj < t -< t]+ v For k0 we have the old estimate, and go is estimated as follows: t

max ia (x. s, 4)1. Igo~)(~(x, t, iI, ~],":)1..
(12)

Here we have made use of condition (24.21 and the fact that A > 0 is sufficiently small. We p r o c e e d f u r t h e r by induction: if rp is a l r e a d y defined, then

kp (x, t; 4, ~) ~ "~ Za(~ Ix, t, 4) rm) (x, t: 4, ~),

(13)

I~l--t

109

~,, (~, ~, ~,, .~, ~).-. ~

~ ~,~' (~, t, r, .~, ~) ,-~(~,, (~, ~; ~, ~)

(14)

_forq
t

%+i(x,t;~,~)=I[~(x,~;~,~)+g.(x,~,t~,~,~)le~

~ a(x,$,11)da

a~.

(15)

L E M M A 1. L e t the s y m b o l a(x, t, T/) of the p s e u d o d i f f e r e n t i a l o p e r a t o r A(t) s a t i s f y conditions (46.1) and (24.2). T h e n the k e r n e l s kp, gp, a n d r p belong to C~~ N x R N) i n x and ~ and s a t i s f y the e s t i m a t e s :

f~+~"-~' {~(~'"~)"

1k,~)(x, t; ,~, ~)1~ C,~, 0 + '~ t) - ~

X [(1 -{-l~ D~'oPI ~ (1 + a max Ib (x; s, n)l)" Ia (x, t, ,~) I( 1 -t-I ~ IP-~f;

(16)

t

I ~;~.)(~) (x, t, t~, n, ~)1--< c ~ 0 +1 ~ I)- " ~ x (1 +1 ~ IP- ~ I a(x, t, n) l [(1 +1 ~ ])8-~

(l--e) J" b l x , s , ~ l d s

fl~l+l~le

p (1 -}- A max l b (x, s, ~) I)PA max I a (x, s, ~)1, '

~.~,~.~t+A

t j _ < t , t" <_tj~- 1 , j = O ,

$E[ti,tj+t]

(17)

i, 2,...,n; t ( l - - s ) ~ b(x,~,~)ds

1-o(o)" ~'~) (x, t; ~, ")l 4 C~. (1 +1 ~ I)-~

]~J+l~te

[(l + l ~ IP-oP] p (1 + A max lb (x, s, ~)IF.

(18)

ProoL Estimates (16)-(18) hold for p = 0. Suppose estimates (18) holds for rp. Then by (13), (18), and (46.1) estimate (16) holds for kp. F r o m (14) and (18) we obtain estimate (17) for gp. Finally, substituting (16) and 07) into (15) and estimating the integral as in L e m m a 2, using the fact that according to (46.1) max la(x, s, vl)[ ~ax I b(x, s, '1)1" < f ' w e obtain (18). The p r o o f of the l e m m a is c o m p l e t e . $

F u r t h e r a r g u m e n t s a r e a l m o s t w o r d - f o r - w o r d the s a m e as in Sec. 2. We have thus p r o v e d the folIowing result. T H E O R E M 3. Let the s y m b o l of p s e u d o d i f f e r e n t i a l o p e r a t o r A(t) s a t i s f y conditions (46.1), (24.2), and (64.2) with m t -> (D - 5 ) / 2 . T h e n the following o p e r a t o r e q u a t i o n holds: 12n+2 . 2 n + l ~

2

. I

lira e[a(t;)]atn.., e [A('~ )N, --lira e" ~ x. 'n. n ,at . . . . e~(x.t0 n )ato. A--~-co

(19)

A---~r

The c o m m o n l i m i t in (19) is equal to the o p e r a t o r U ($', ~). The c o n v e r g e n c e o f the o p e r a t o r s in (19) is u n d e r s t o o d as s i m p l e o p e r a t o r c o n v e r g e n c e in the s p a c e 5O(R~) w h i c h is u n i f o r m w i t h r e s p e c t to ~', t s u c h that 0 -< -r < t ___ T, w h e r e T is any finite n u m b e r . I n a s m u c h as the e x p r e s s i o n s f i g u r i n g in the l i m i t s in (19) c a n be w r i t t e n as i n t e g r a l s u m s 1

[~ 2

.

n a~'~+~ x , t j"~+'~ , O ]At]

1

eJ =~

and e

j=~

,

Eq. (19) m a y be i n t e r p r e t e d as the e x i s t e n c e and e q u a l i t y of the following i n t e g r a l s (see the I n t r o d u c t i o n ) :

T.e 9

4.

T-Product

of

Operators

in

[A(s)ld~ a ta~ x, ,. _---e~ the

Interaction

J a~ x. s, o)as =e~

(20)

Representation

The o b j e c t of this s e c t i o n is to p r o v e Eq. (14) of the i n t r o d u c t i o n . Suppose that the p s e u d o d i f f e r e n t i a l 2

1

2

1

hypoelliptic o p e r a t o r s H 0 = h0(x, D) and H 1 = h~(x, D) do not depend on the time t and that t h e i r s y m b o l s h0(x, ~) m and hi(x, 4) belong to the c l a s s e s h0(x, ~) E S~,5, hi(x, ~) E Sp,5, m > 0 (cL Sec. 1). We a l s o a s s u m e that the

110

o p e r a t o r Ht s a t i s f i e s conditions (46.1) and (64.2). We denote by U (t, r) the solution o p e r a t o r of the Cauchy problem du

/~ - ~u

(Hi + ~Ho) u=O, I~-, = ~0 (x),

(1)

and the symbol of the o p e r a t o r U(t, r) (cf. Sec. 1) we denote by p(x, t; ~7, r); by T h e o r e m 1 U(t, r) and p(x, t; ~7, r) exist, U(t, r) = e[Hl+iH0] (t-r). It follows f r o m t h e r e s u l t s o f S e c . 1 that p(x, t; tl, r) E S~ 6,, 0 -< 6' < p' -< 1 for all t - r and (x, 77) ~ R N x R N, and taxis in turn, on the basis of a t h e o r e m of H o r m a n d e r (cf. [9, p. 330]), implies that the o p e r a t o r U(t, r) can be c o n s i d e r e d as an o p e r a t o r f r o m L2(RN) to L~(R N) with

IIu (t, ~) ,~,_.,,..< Co,

(2)

w h e r e C 0 > 0 does not depend on t, ~ < t ~ < $ ' < co . We f u r t h e r introduce the o p e r a t o r f r o m L 2 to Lz, U(t, r) by m e a n s of the equality

O (t, ~)----etn, lc~-~eVn.lv-~.

(3)

The f i r s t o p e r a t o r on the right side of (3) exists again by T h e o r e m 1, while the second o p e r a t o r is bounded in L~, s i n c e h 0 ES~, 5 [cf. (49) and (53) below]. T h e r e f o r e , e s t i m a t e (2) holds for the o p e r a t o r l~(t, r) with s o m e constant CO> 0. We p a r t i t i o n the s e g m e n t i t , t] into n subintervals by partition points tj, j = 0, 1 . . . . . n; t o = r < t 1 < t 2 < . . . < i n = t , &tj=tj+ 1 - t j = A = i / a We s e t

V ~+,.

u (tl+~, t~)--U~+t. ~, O(t~+~, t~)-~O~+~. ~, ,~Vl. j-1. . .Vk+i. ,---U,,+I. ,, Dt+t. )01, ~-~ ...LI~+t. ,----- D~+~, ,.

(4)

We have the following result.* LEMMA

1. llV(~, 'O -- O,,. o ll~,.-.~,ffi o (1)

~

(5)

n~o.

Propf. By the uniqueness of the solution of the Cauchy p r o b l e m (1) we have U(t, r) = Un, 0 = Un,n_ ~. . . U~,0; t h e r e f o r e , n

v (~,

~)-0".. o = V , .

o - g . . .-, . . .0,.o = x~ v ~ . ~ [ ~ . ~_1-v~, ~-,I ~ - , . o.

.-i. . .u,,

(6)

k~l

We c o n s i d e r the d i f f e r e n c e of the o p e r a t o r s ~Ik,k_ 1 - Uk,k-v We set '9 ------U (t, x)% ~ 3 ' . whence

Then a-r~*.--Hlg+iI'I'o9

t.

9 =etn, l.-*~ + f etn'l(t-~iH~ (s, ~) ~ ds. Thus t

U (t, ~) = etn,] .-*) + f etn,] "-'~iHoU (s, ~) ds.

iT)

,r

F u r t h e r , if }---_~J (t, 3) ~--- etn,] (t-,~etm, l ('-~)?, then ~ .----Ht~ + ~] (t, ~) iH0~ ; t h e r e f o r e t

t

"~--_etn.l v-*,? + I d u d e-,,LT(s, x)iHo~ds, i.e.,

a (t, ,)ffletml ,t-~, + I etn,;,,-,,O (s, ,) iI'Iods.

(8)

F r o m (7) and (8) we obtain tk

U,, ,-1 --U,, ~-t = I

et"'] "~-'~[ 0 (s, t,_3 iHo-- iHoU (s, t,_l)] ds.

(9)

tk--1

Equality (9) enables us to e s t i m a t e the n o r m of the o p e r a t o r ~lk,k_ 1 - Uk,k_ l in the following way: t~

I1u,. ,-1 - u,. ,_, IlL:L, ..< C I

11~r (s, t,_O iHo-- $HoU (s, t,_l) llL,-,,cts,

ttl--I

(io)

*Equation (5) is an analogue of Trotter~s formula; cf. [4, 6, 11]. 111

s i n c e f o r the o p e r a t o r e [ H t l ( t k - s ) , by the s a m e c o n s i d e r a t i o n s as f o r U(t, r ) , we have the e s t i m a t e

IIetml u~-,) I1~.--,.--
(11)

W e e s t i m a t e the n o r m o f the o p e r a t o r U(s, tk_~)iH 0 - iHoU(s, tk_~). F o r c o n v e n i e n c e w e s e t ~r = s - tk_ ~ and d r o p the a u t o n o m o u s b r a c k e t in the e x p o n e n t i a l function. F o r any function 9~5a we h a v e

It) (s, t,,t) ~Ho--iHoU (s, t~-0l ~ = [e","iHo-- ~Hoe(~,+,-.) ~ ~ (o) + iT-Toe(",+".) ~ [ e " . ' " tl~,

(12)

where r = e i H ~ o and I is the i d e n t i t y o p e r a t o r . We c o n s i d e r the two t e r m s on the r i g h t side of Eq. (12) s e p a = r a r e l y ; w e b e g i n w i t h the f i r s t t e r m . T h e s y m b o l o f the o p e r a t o r e(Ht+iH0) a is p(x, a; 4, 0} -- p(x, a , ~), while the s y m b o l of the o p e r a t o r eI'I~a w e denote by q(x, a; (, 01 - q(x, a , D . I f w e denote b y k(x, a , 4) the s y m b o i of the o p e r a t o r eH~aiH~, t h e n w e h a v e

~(~, ~, ~)- ~

~ q~) (~, o; ~)~ ,~) (x, ~)--~q (~, o, ~)ho(X, ~)+m{x, ~,0,

(1~)

~{x, ~, ~)- ~~ ~q~) (~, ~, ~)~ ~)(~, ~).

(141

where lal--~

In e x a c t l y the s a m e w a y the s y m b o l l(x, ~ , 4) of the o p e r a t o r iI-I0e(ttt+iI-I0)a is

(15} where

|r

T h u s , the s y m b o l j(x, ~r, 4) of the O p e r a t o r eHl~

- iH0e(I'II+iN0)or is

] (x, ~, ~) = i/to (x, ~) [q (x, a, ~) - - p (x, ~, ~)I + m (x, a, ~) -- n (x, ,, ~).

(17)

We c o n s i d e r the s y m b o l m(x, @, s~). Since h0(x, 4) ~ Sg,6 and r e p r e s e n t a t i o n (113.1) holds f o r q(x, ~, ~) in w h i c h the functions r and r a r e e s t i m a t e d a c c o r d i n g to L e m m a s 4 and 7 of See. 1, it follows that re(x, ~r, ~) ~ S~,,6, ( p-~] f o r (x, ~)~R~XR ~, p'-~---~-e , ~ .a_ , g, 02+5 prec i s e l y , f o r re(x, ~, ~) t h e r e is the e s t i m a t e

Irnl~ (x, a, ~)l ~
(is}

w h e r e oz, /3 a r e any m u l t i i n d i c e s and e~ > 0 is any fixed n u m b e r . Since

iReh,(x, a ) l ~ < C 0 + ! a l ) ' ,

f=O(lq-lal)

e

a~d

(cf. w 1),

the inequality (18} c a n be w r i t t e n in the f o r m

I rn!~? tPl (x, a, D[ --< C ~ (1 + l ~ I)-r w h e r e ~ = m l n {I, ~ } ,

since

av,

(19)

e"-',)a'k,(x'~)(~a IRe h,(x, ~))f(I :,-I ~ l)S-o 4 Ca', (x, ~)ER~X R~ S i m i l a r a r g u m e n t s e n -

0 able us to a s s e r t that n(x, #, 4) E S~,,5, , and

I ,~(~)-~)'~t~, ~, ~) ) < c o ~ ~ (I + I t l)-p'l~+6'z~,

(2o}

w h e r e a , /3 a r e any m u l t i i n d i c e s . We c o n s i d e r the r e m a i n i n g t e r m in Eq. (17) - t h e function h0{x, D[q(x, a , 4) - p(x, a , ~)]. Since f r o m the r e s u l t s of Sec. 1 q(x, or, ~} = eh~(x,~}a + q0(x, a , ~) and p{x, ~, ~} = e(ht(x,~}+ih0(x,~)} ~ + P0(X, a , 4), this function is equal to ho (x, :4 e n'(~''t)" [1 - - e*~.(~.t)"l + ko (x, ~) [qo (x. ~, ~)--po (x, a, ~)].

112

(21)

It is obvious that

A (x, ~, ~)--- ao (x, 0 e ~'(='e)~ [l--r (x, 0ea" XR",

(22)

and

I./,~ (x, ~, 01<: c~oo +1 ~ 1)-''z~~'~"~

(23)

for any multiindices a and #. Further,. slr~e for q0{x, a, ~) [and P0(X, a, ~)] there is the estimate

l qo (x, ,~, ~.)14 Ce(~-,,)",",(-'.~,,o 1Re ax (x, ~.)l [(1 + I ~.lp-~, fl, and a similar estimate for the derivatives qo(~) w0 (fl))with any a and fl (cL L e m m a 4 of Sec. i), it follows that ,,,(~)" h0{x, $)%(x, ~) e ~ 5 ~ and l~(x, ~)P0(X, ~) e S~, 5,, {x, i)~ R N x RN; moreover, there is the inequality

I[~ (x, 0 qo(x, ~, 015:~ I < c~,, ~' 0 +l ~.l)-''~=~+~''~,

(24)

w h e r e ~ , = m i n {1,-~}, and a s i m i l a r inequality holds for hop0 (here e is the s a m e n u m b e r c o n t a i n e d in the e s t i m a t e for the function f). C o m p a r i n g E q s . (17), (19), (20), (23), and (24), we s e e that the s y m b o l j(x, ~, ~) Sg~,5~, (x, ~) 6 R N x R N and s a t i s f i e s

w h e r e a , fl a r e any m u l t i i n d i e e s and "re----rain 1, ~ ,

2~

. E s t i m a t e (25) t o g e t h e r w i t h the t h e o r e m of HBr-

m a n d e r to which we r e f e r r e d above shows that the n o r m in L~ of the o p e r a t o r eHtaiH0 - i/-I0e(Hl+iH0)a does not e x c e e d CAT0, s i n c e

II (~,"~Ho--iHoe(",§

a, IlL, ~ c,~v, t[ mI[~,, m~.

(26)

We now c o n s i d e r the o p e r a t o r iH0e(Hi+iHo)~[eiH0~ - I]. Since the o p e r a t o r H 0 is bounded in L2, setting $ = (slit0 a - I)(p, ? ~ ] , we have f o r ~b the e s t i m a t e

II'l' I!~,--
(27)

F u r t h e r , the s y m b o l of the o p e r a t o r iH0e(Hl+iH0 )~ is 1 -~T h~=) (x, ~) p(=) (x, ~, ~)

i

(28)

and t h e r e f o r e belongs to S~,,5,, (x, ~) E R N x RN. F r o m this and (27) we obtain

[IiHo e(~''+''l" [e '~~

l ] ~ [[L,4 C~ l[ ? !it,.

(29)

comparing now Eqs. (Z2), (26), and (29) and noting that II ~' (o)I!~.= ]1e~n~ IlL, ~
F r o m (30) and (9) in t u r n we obtain the estimate fh

[ cT,.~_~-- th,~_~ II=,-~-~C

I llO (s. tk_~)iHo--iHoU (s, t~.~)llL.~Js < CA'+V..

(3 i)

Ilk...I

We now p r o v e that t h e o p e r a t o r Un,0 = Un, n-1. 9 9 Ul,0 is bounded in L 2 for aU r and t, O ~ < t ~ $ ' < all n. L e t II Uj,011 - A for j = O, 1 , . . . , k - 1; w e shall p r o v e that

IIO,.oll,~A.

0% and (32)

By (6) we have k

Oj.o = th.o- ~ u~.~ iota-, - uj.j_,] ~?j-,.o. j--I

(33)

113

F o r Uj,0 f o r all j, 0 -< j -< n, e s t i m a t e (2) holds:

I! u j,0 II~< co.

(34)

I1E,'k. 0 It--< co 0 + ACkAt+Vo).

(35)

F r o m (33), (32), (34), and (31) we obtain

We m a y a s s u m e t h a t A _< 2C 0 and the r i g h t s i d e in (35) d o e s not e x c e e d (since A = 1 / n, k - rO

Co+ 2C~C~ ~<2Co, n,.

(36)

if n -~ (2CoC) t/To. It h a s thus b e e n s h o w n that

llE]~.oll~2Co

f~all

(37)

k, 0..
T u r n i n g a g a i n to Eq. (6), w e o b t a i n on the b a s i s of (31), (34), and (37)

II u(t, =)-Er., o11~<~ tlU., ~ I! IIErk. ~ _ , - - U , , ~-, It II&~-,, o II < nCoCa'+,'.2Co = C. i .

(38)

k--I

and the p r o o f of L e m m a 1 is c o m p l e t e . F r o m the a b o v e p r o o f w e obtain the following r e s u l t . L E M M A 2. F o r any t __ s w e denote by U{t, s) the o p e r a t o r 2 1

t](t. s ) = e~,( ~. m.-.~et,ml(,-;J and w e s e t l~(tj+ 1, tj) = 0 j + l , j , j = 0, 1 . . . . .

(39)

n, and 0 j + l , j 0 j , j _ t . . . 0 k + t , k = 0 j + l , k . T h e n

II U(t, ~)-~..

a, n.-,. ~ .

otlL,-.,,=o(1)

(40)

We s h a l l now p r o v e the m a i n r e s u l t of this s e c t i o n . 2

1

2

1

.THEOREM 4~. L e t H 0 = h0(x, D) and H 1 = hi(x, D) be hypoelliptic p s e u d o d i f f e r e n t i a l o p e r a t o r s w i t h s y m b o l s belonging, r e s p e c t i v e l y , to the c l a s s e s h0(x, ~) ~ S~, 5, hi(x, ~) t sp,ms, and s u p p o s e that h i s a t i s f i e s conditions (46.1) and (64.2). The following f o r m u l a f o r r e m o v a l of a u t o n o m o u s b r a c k e t s then holds: "-Y s

2

1

; [H,(X(s), P(s))lds

et

~"

s+O

s--O

; Ht(X($), P(s))da

=e ~

,

(41)

w h e r e the o p e r a t o r s P(t) and X(t) c o n s t i t u t e a s o l u t i o n of the H e i s e n b e r g equations

I( p ( 0 =

OHo ,~( I ---~-( P), ,

(42)

P (0) = D, X (0) = x. P r o o f . L e t ~ (x, t)E5~ be a s o l u t i o n o f the C a u c h y p r o b l e m

{

~t -- (Hi + iHo) ~ = 0,

We s e t ~J= e-tin0jtr we then have

(43)

r,=, = %, .~0~r

On the b a s i s of L e m m a 1 and the fact that e[iH0]t is a g r o u p of bounded o p e r a t o r s in L 2 n--1

~=e-titC'lt~--~-e-tec*ttU (t, ~) ?o = e -tug'It lim H ett~'l~t~e~ln*tAtk% A~O/z=O

= lira {(e-tin'ltnettt']atn-'e t~Hdt") (e -[m~176176

e-V~~

(44)

A~0

We c o n s i d e r the o p e r a t o r J ~ e-Iil~,,ltjeIH,laty-teIiHolt];

(45)

It is e a s y to s e e that it is equal to the o p e r a t o r

j ~ e[e--[tH'MjH,e T M Q]Ae]_,. 114

(46)

I n d e e d , w e denote by L = L(tj) the o p e r a t o r e [iH~ and f o r any $o~5~ w e s e t $ = L - l e [ H l l t I ~ 0 and $~ = L$. The f u n c t i o n ~t is the (unique) s o l u t i o n of the p r o b l e m

{

~ t ~=/-/x~bx,

(47)

'~'tIt-o= L,~0.

In the s a m e w a y , if we s e t ~ = e [ L - ~ H t L ] t % , ~ = L$, t h e n $1 is the s o l u t i o n of the s a m e p r o b l e m (47)~ and t h e r e f o r e $~ = St and $ = $. We now c o n s i d e r the o p e r a t o r 2

e-t~n~

1

L-~kt (x, D) L.

(48)

T h e d o m a i n of o p e r a t o r (48) c o n t a i n s the s p a c e ~ . I n d e e d , l e t ~o be any function in 5a, and l e t a 0 and fl0 be any n a t u r a l n u m b e r s . Since H 0 is a bounded o p e r a t o r in I ~ , it follows that

t W e set r = H0k~, k = 1, 2, . . . . ___~

=

-~

t*H~.

(49)

F o r any m u l t i i n d i c e s ~ and ~, 0 ~

I~1

-< ~0, 0 -<- I~1

-
h a v e ?~ - - H 0 ~ _ t

d(x,~)hoCx.~)?,_x (~)d~ , and x p ~ ? , (x) = ( - ip I e,~'.~ '~ C~,.n ~-.-,,tz.!o.). 7,(~,~(r d~ PlPsPs U(.~ J (x, n " ! T/~--I

(50)

P,-~.+P,-i~ 0

Since l~(x, 61 E S~,5, s u m m i n g (50) on a a n d N s u c h that 0 -< l a l -< ~0, 0 - I~1 -< I~01 a n d e s t i m a t i n g the r i g h t s i d e o f (50) in L s by m e a n s o f H S r m a n d e r ' s t h e o r e m , w e obtain

It ~, I!=.,~..~ ~ I1x~O~,~, (x)lk, < c~ 11~.-, ll=..~.-

(5~)

0~1~1<~ O
Hence

II% I1=.,,.--
(52)

F r o m (49) a n d (52) we h a v e f o r $, = L~ the e s t i m a t e

(53) 2

Therefore,

1

2

1

o p e r a t o r ht(x , D)L is d e f i n e d as is the o p e r a t o r L - l h i ( x , D)L as a s s e r t e d .

By the d e f i n i t i o n of a function of o r d e r e d o p e r a t o r s (cf. [7]) we h a v e for any ?~5~ and any function g(xl, x 2) E C ~ w h i c h g r o w s at infinity in x t and x 2 no f a s t e r 2

1

2

1

g(x, D) ?--(I x]~ + l)Mg0(X, D)(--h -1"1)x%

(54)

where 2

!

go (x, 1:))~ I e~tleWt'g~ (tb t2) dtidt2,

(55)

and g0 (tl, t2) ~ I e-t(t'x'+t~')g~ (xt, x2) dxldxa, g (xl, x,) go (Xt, X2)------(ix, 1' + 1)M ([ x, 1' + I)x '

(56) (57)

h e r e M and K a r e n a t u r a l n u m b e r s s u f f i c i e n t l y l a r g e s o that all the definitions a r e m e a n i n g f u l ; it is s h o w n in 2

1

[7, p. 327] that the o p e r a t o r g(x, D) does not depend, on the c h o i c e of n u m b e r s M and K. F o r the p s e u d o d i f f e r e n 2

1

tial o p e r a t o r hi(x , D) the definition (54) c o i n c i d e s with the definition we have u s e d a b o v e , s i n c e if 9E5~, then

115

ra + N'+ 1

(58)

F u r t h e r , .we have 2

1

L-th, (x, 19) L = L -i (] x I~+ 1)m L f {L-tei~"'L} {L-'e~Ot'L} {L-' (-- A + 1)x L} h(t,, t=)dttdt~.

(59)

Since L-I(A ~ + 1)kL = [(L-1AL) 2 + 1] k for any o p e r a t o r A (when these e x p r e s s i o n s a r e m e a n i n g f u l ) a n d the equality (46) holds for the o p e r a t o r s in b r a c e s under the integral sign in (59), f o r m u l a (59) r e d u c e s to the equality 2

1

L-th~ (x, 19) L=h~ (L-~xL, L-tDL).

(60)

We now c o n s i d e r the o p e r a t o r s

X (t)=e-~#xeiHr and P (t)=e-~m'D#Ho~;

(61)

as noted above, t h e i r domains contain 5~ for all t - 0. By d i r e c t differentiation with r e s p e c t to t, we see that o p e r a t o r s X(t) and P(t) satisfy the s y s t e m of H e i s e n b e r g equations

X=--iIHo(X,

I

P), X(01, (62)

P = - ~ trio --(k, },), p (01, X(O)=x, P (0)=D

(the s q u a r e b r a c k e t s [ - , -] denote the c o m m u t a t o r of o p e r a t o r s ) . O p e r a t o r s (6D a l s o s a t i s f y s y s t e m (42), since we have the following chain o f o p e r a t o r equalities: 2

I

2

I

2

1

[Ho(x, P), x (191= t-/o(x (0, P (0) x ( 0 - x (0 ~/o (x (0, P (t)) 2

I

= Ho ( ~ , 2

I

2

2

I

Z-:rbX)x it)- x (t)H0( ~ , I

2

1

Z:rD-r)=

"-lOb*-

2 /~)L-"

= L-tho(x, D) LL-txL-- L-' xLL-'ho(x, D) L = t-'[ho(x, D), xl L = --~L- -5~-~x,

9 0//o..~.

1

--L-~--(A , P),

(63)

[here we have used Eq. (60) in t r a n s f o r m i n g the o p e r a t o r s H 0 and aH 0/ap]. EqualiW (63) together with the f i r s t equation o f (62) gives 2 =

0H, 1.~ /~)

0He t. ~ , /~). We shall show that S i m i l a r l y , we obtain the equation p =-~--x

the s y s t e m s of H e i s e n b e r g equations (42) and (62) have only one solution which is analytic in t in a neighborhood of the point t = 0. We c o n s i d e r s y s t e m (42) [ s y s t e m (62) is t r e a t e d similarly]. A solution of s y s t e m (42) which is analytic in t has .the f o r m co

c~

x ( o = ~ ' -~ x~k)(0)t*, p(0--'Z ~T ' P(k)(0)tk' ~=0

(64)

k=0

and it suffices for us to show that system (42) uniquely determines the operators X (k) (0) and p(k)(0). We shall show, moreover, that X (k) (0) and p(1O(0), k = 0, I, 2 . . . . . have domains containing the space ~ and that they are bounded operators from ~x,t to L2 [the class ~{,., consists of functions ~pfor which the norm }Ir 1]1,1 < ~ (cf. (51)]. We find the operators X (k) (0) and P(k) (0) by successively differentiating the equations of system (42) with r e s p e c t to t and then setting t = 0. We have ~-,.^.

X (0)= x, P (0)= D,

0Ho

.2

1

~ tu}= - ~ p (x, D), 2

(65)

1

0He C~, D~ P'(0)=---~-~

,.

To find f u r t h e r d e r i v a t i v e s we e x p r e s s the functions of o r d e r e d o p e r a t o r s by means of Eqs. (54) and (55); viz.,

2 t 0H, 2 * (X (t), P (t)) and .-~--(X (19, P (19)

0H, 2 (t), P (19)-~ (l X (19!~+ I) s e~x:('~u,elp(0y, OH.o(y,y,)OV, dy,dY2(lP (012+I) ~:, dp (X

116

(66)

where

OH,,(gt,dy,y') ----(I Yt 12+ 1)-~r ~H

and a s i m i l a r f o r m u l a holds for ~

2

,

[ 1+ 1)-~- O'H'*~': g,)";

0"~,.

(67)

"~0 0ft.

(X (t), P (t)) with "~-~t replaced by ...-~-. We now differentiate the right

side of (66) with r e s p e c t to t and consider the following operator equalities: d_ a t (IA(t)l ~+ 1)a~-- ~ ([ A (t) I 2+ 1)~[(A, )t) +(~,, A)I ([ A (t)l ~ + 1)M-~-t,

(68)

l.,O

1

d east}----.~e(l_~}a{t}A(t) e*a~t)d~l

(6 9}

at

11

h e r e A(t) is an operator depending on the time t; (A, A) is the s c a l a r product (A, A)-~XAI(0)~(0. Setting t = 0 a f t e r differentiating, we obtain ~-~

( OH" t.~. t~ ,

~

=

~a-t

(x

' '

' '

• I ~'~'ov, ~'' ~') dV~aV~e"'e'~ (-- a + 1p" + (1x p + 1)~ I Yg..ov.(v,.V,) dg,dg~ { iltt i +etx~'iY~

\

Ox ~ ' D)ea~~

1)'~-'-t

~ D) t e"~,xd~e ' ~ e '~-'~'",x ~OH (x,

ld:( - - h + l ) ~-

0

+(lxl2+ 1)"

(--a+ ty [L-- o o.,

"-'

av,

1

2

I Ot ( - a + o)

(70)

and a s i m i l a r formula holds for P" (0). It is evident f r o m Eq. (70) that the operator X" (0) is uniquely d e t e r mined, and the domain of X"(0) contains the space ~ . Recalling that l~(x, ~) E S~,6, it is not hard to conclude f r o m (70) that X" (0) is bounded as an operator f r o m ~t.1 to I.~. These a s s e r t i o n s also hold for the operator P" (0). We prove them by induction for the remaining derivatives X(k) (0) and P(k)(0), k= 3, 4 .... The uniqueness of a solution of the Heisenberg s y s t e m (42) which is analytic in t has thus been proved [this also holds for s y s t e m (62)|. We now show that this solution is given by operators (61). Indeed, by estimate (52) the o p e r a t o r s X(t) = e-[iH0]txe[itI0] t and P(t) = e-[iH0]tDe[iH0] t can be r e p r e s e n t e d in the f o r m of converging power s e r i e s in t: tt

(u),, ~ C~(_Ho)~Xt4~_~, ~=o OCt

~o tt

n....0

k~0

(71)

(itp %~ C~(--Ho)~DH~-~. p ( t ) = ~ --~,~., Here operators X(t) and P(t) are defined on functions ~ t , ~ and are bounded as operators from Mm to L2, while the convergence of the s e r i e s in (71) is understood in the sense of the uniform operator topology. We turn to equality (44). Using Eqs. (46), (60), and (61), equality (44) can be t r a n s f o r m e d to the f o r m I

2

~s 2 1 ~ [//d,X(*),P(s)}ld*

I

9 = lime IH'(x(tn)'P(tD}lntn-t 99 9 eindX(t')'P(td)lat~o -~-e *

~o,

(72)

A-*-9

where $0 = e-[iH~176

and X(L), P(t) is the solution of the Heisenberg s y s t e m (42) or (62).

On the other hand, by L e m m a 2 we can write the function 9=e-Vn'lg"~? as the following limit: rt--I

2 I

9 ~ e-Vn'l?~ltm 1~ en'~'~'~ k,,-O [lttolt

2 t Itt(X D)At

[iH,I t

[iPItlttt

2 ~ ht(x D)Aftt I [//'/*lit: t

[/Iftltt

2 t t(x D)Ato [lHo]tt

(73)

A--,O

117

Each expression in braces in product (73) can be transformed by means of Eq. (60) which is valid for a n y / u n c tion of ordered operators which grows in its ~rguments at infinity no faster than some power; as a result, we obtain 2 I

r --[tl'IgltJP~'(x'DIAtj-tP~itH']tl

9

1

~- eht(X(tJ)' P(tJl)a~J't'

(74)

"

j = 1 , . . . , n - 1, where the operators X(tj) and P(tj) form a solution of the s y s ~ m (42). Substituting (74) into (73), we obtain t,,+~

~=limeh,(x(~,,),

tn-~

,+~- t,-~-

~ ,~,( x (,)..o (,))d,

P(t~,))at..,. ;. e,~,(x(t,), p (t,))at,~o. ~_e.C

~o, ~'o=e-[~a'~

(75)

A-,,O

Comparing Eqs. (72) and (75), we see that equality (41) has been established. LITERATURE 1. 2. 3. 4. 5. 6.

7. 8. 9. 10. 11.

118

CITED

A. L. Alimov, "On the connection between continuum integrals and differential equations," Teor. Mat. Fiz., 11, No. 2, 182-189 (1972). N . N . Bogolyubov and Yu. V. Shirkov, Introduction to Quantum Field Theory [in Russian], Gostekhizdat (1957). V . S . Buslaev, "Continuum integrals and the asymptotics of solutions of parabolic equations as t -- 0. Applications to diffraction," in: ProbL Mat. Fiz., No. 2, Leningrad Univ. (1967). Yu. L. Daletskii, "Continuum integrals connected with operator evaluation equations," Usp. Mat. Nauk, 17, No. 5, 3-115 (1962). M . A . Evgrafov, "On a formula for the representation of the fundamental solution of a differential equation by a continuum integral," Dold. Akad. Nauk SSSR, 191, No. 5, 979-982 (1970). T. Kato, Perturbation Theory f o r Linear Operators, Springer-Verlag (1966). V . P . Maslov, Operator Methods [in Russian], Nauka, Moscow (1973). R. Feynman, "On an operator calculus having applications in quantum electrodyimmics," in: ProM. Sovrem. Fiz., No. 3 (1955), pp. 37-59. L. HSrmander, "Pseudodifferential operators and hypoeUiptic equations," in: Pseudodifferential Operators [Russian translation], Mir, Moscow (1967), pp. 297-367. E. Nelson, "Feynman integrals and the Schrodinger equation," J. Math. Phys., 5, No. 3, 332-343 (1964). H . F . Trotter, "On the product of semigroups of operators," Proe. Am. Math. Soc., 10, No. 4, 545551 (1959).