TRACES

OF

LIOUVILLE M.

FUNCTIONS

IN A GENERALIZED

CLASS L.

Goltdman

U D C 517o5

INTRODUCTION T h e t h e o r y of t r a c e s f o r c l a s s e s of d i f f e r e n t i a b l e functions o r i g i n a t e d in the w o r k s of Sobolev (see [1]). M o d e r n d e v e l o p m e n t s in this t h e o r y and an e x t e n s i v e bibliogTaphy a r e p r e s e n t e d in [2] by- Nikol'skii~ In the p r e s e n t w o r k we will s t u d y the question of e x i s t e n c e of t r a c e s of functiot~s in the g e n e r a l i z e d Liouville class L~(R N) on an m-dimensional subspace R m of a Euclidean space R N (1 <- m < N)~

Definition 1. Suppose that the function ~(u) > 0 is defined for u > 0 amd has continuous derivatives up to order n'>---~; ~-~'(u) = uk4~(k)(u). We will say that ~(u) satisfies condition (Dn) (briefly, ~ ~ if)n)) if

w h e r e co, c t > 0 do not depend on u. H e r e a f t e r we will always a s s u m e that cI,(u) goes to z e r o as u ~ ~ . Now s u p p o s e that x, ~ ~ R N (N - 1).

Definition 2. We will say that a function f(x) belongs to the generalized Liouville class LI~CRN) if there exists a function v(x) ~ Lp(RN) such that 1

~,

f(x) =.: l . z , ~ D ( | ' : - ~ ) t , " ~ ( ~ )

IN

(x),

(2)

with

i~!;Lpc(Rn) ~!c'JLr,IR,~-)'

(3)

Here 1 ~p_<~, O(u)~0 ( u ~ o ) , ~ ( u ) ~ (Dn), w h e r e n > N / 2 f o r 1 < p < ~ , while n > ( N + l ) / 2 f o r p = l or~. T h e s y m b o l s ~ri - a , r e s p e c t i v e l y , denote 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 a t i o n in RN, in g e n e r a l , the g e n e r a l i z e d t r a n s f o r m a t i o n (in the s e n s e of the S c h w a r t z space S'(RN); see [2]). Definition 2 m a k e s s e n s e s i n c e the function ~ ( ~ is a m u l t i p l i e r in Lp(RN). F o r 1 < p < ~ this fact follows, f o r i n s t a n c e , f r o m (1) and the m u l t i p l i e r t h e o r e m (see [3, Chap. IV, T h e o r e m 3]), and f o r p = 1 I

---IN

o r ~ it follows f r o m i n t e g r a b i l i t y of the k e r n e l Gff(x) = 4~(v~l + I~t 2) (x) (see [1, w G~*v. Thus L ~ ( R N) ~ Lp(RN).

and the equation I ~ v =

F o r 4,(t) = t - ~ (o~ >0) we obtain the usual Liouville c l a s s L~(R N) (see [2, 3]). C l a s s e s which g e n e r a l i z e L~ in s o m e s e n s e o r o t h e r have b e e n i n v e s t i g a t e d by v a r i o u s a u t h o r s . In [4] Volevich and P a n e y a k h c o n s i d e r e d a g e n e r a l i z e d Liouville c l a s s in which ~ , ( ~ ) w a s r e p l a c e d by 1/p (~), w h e r e ~ (}) is an infinitely d i f f e r e n tiable function h a v i n g p o l y n o m i a l growth and an e s t i m a t e on its m u l t i p l i e r n o r m . In [5] H S r m a n d e r i n t r o d u c e d the c l a s s e s Bp, p, which c o i n c i d e with the c l a s s e s H~ of Volevieh and P a n e y a k h f o r p = 2. C l a s s e s whiah a r e c l o s e r to B e s o v c l a s s e s and which coincide with H2~ f o r p = 2 w e r e studied by R a m a z a n o v in [6]. In all of these w o r k s t h e r e a r e s t u d i e s of the q u e s t i o n of e x i s t e n c e of t r a c e s on s u b s p a e e s of l o w e r d i m e n s i o n f o r functions in the indicated c l a s s e s . In [4] f o r p = 2 t h e r e a r e n e c e s s a r y and sufficient conditions f o r e x i s t e n c e of t r a c e s in t e r m s of r e s t r i c t i o n s on the " s m o o t h n e s s function" #(}). In [6] t h e o r e m s on t r a c e s w e r e p r o v e d f o r the c a s e of a p o w e r " s m o o t h n e s s function" f o r an a r b i t r a r y p. We note that the study of t r a c e s f o r p ~ 2 f o r c l a s s e s of

1976.

T r a n s l a t e d f r o m Sibirskii M a t e m a t i e h e s k i i Z h u r n a l , Vol. 17, No. 6, pp. 1236-1255, N o v e m b e r - D e c e m b e r , O r i g i n a l a r t i c l e s u b m i t t e d F e b r u a r y 18, 1975.

I This material is protected by copyright registered in tile name o f Plenum Publishing Corporation, 227 West J 7th Street, N e w York, iV. Y. 10011. No p a r t | o f this publication may be reproduced, stored in a retrieval system or transmitted in any form or b)' any means electronic mechanical photocopying, microfilming, recording or otherwise without written pet~nission o f the publisher. A copy o[ this article is available -j@'om the publisher for $ 7150.

I 907

Liouville type is r a t h e r c o m p l i c a t e d . T h i s is m a d e c l e a r by the e x a m p l e of the o r d i n a r y Liouville c l a s s e s L~, f o r which the t r a c e s cannot be d e s c r i b e d in t e r m s of t h e s e c l a s s e s (see [2]). In the g e n e r a l c a s e t h e r e a r e no known p r e c i s e conditions f o r e x i s t e n c e of t r a c e s f o r the c l a s s e s H~ with p ~ 2 (see [4]). T h e a i m of the p r e s e n t w o r k is to obtain a n e c e s s a r y and sufficient condition f o r e x i s t e n c e of t r a c e s f o r all functions in the c l a s s L ~ ( R N) f o r 1 < p < 0o. Such a condition (and a c o r r e s p o n d i n g t h e o r e m on t r a c e s ) is found f o r the c a s e of a r a d i a l l y s y m m e t r i c and m o n o t o n e " s m o o t h n e s s function" ~)(~/1 + ]~ 12i, w h e r e 9 ~ (Dn). w

SOME

PROPERTIES

CLASSES

OF

FUNCTIONS

L ~ ( R N) A N D

IN THE

Bp~e(Rm )

1 ~ We d e s c r i b e the g e n e r a l p r o p e r t i e s of the s p a c e s L ~ ( R N ) . We will o m i t s i m p l e p r o o f s . Analogous p r o p e r t i e s f o r the c l a s s e s H~ of V o l e v i c h and P a n e y a k h w e r e ~established in [4], which contains a good i n t r o duction to the type of q u e s t i o n s which we a r e c o n s i d e r i n g (see a l s o [2] by Nikol'skii)o 1) We let A denote the s e t of s m o o t h functions

A = {/(x): / = r c v , v E S (nN)} ~ L~ (nN) n C~ ( n ~ )

(4)

H e r e S(RN) is the S c h w a r t z s p a c e of t e s t functions (see, f o r i n s t a n c e , [2]), that is, infinitely differentiable functions which d e c r e a s e r a p i d l y at infinity t o g e t h e r with all of t h e i r d e r i v a t i v e s . T h e n A is d e n s e in L~(R N) (1 -< p < ~ ) . 2) We let ~'h denote the shift o p e r a t o r ~-hf(x) = f(x - h); x, h E R N ; to an i s o m e t r y Th : L ~ ( ~ N) --* L ~ ( R N) (1 __

class ~RN_ trace on A,

f(x) ~ A.

T h e n ~'h can be extended

(h -~ 0, i ~ p ~ oo).

(5)

L~{RN). L e t x ~ R N ; x = (x', x"), w h e r e x' = m . We f i x a ~ R N _ m a n d u s e t h e notation~ a = fl X a = f(x', a). We set Taf(X') = f(x', a) f o r f E A is continuous as an o p e r a t o r f r o m Lp~(RN) into

t

D ' ( R m ) = [C o (Rm)] . We let T~: n~ (~R\.) -~ D' (B,~) denote the o p e r a t o r which is the e x t e n s i o n of T a f r o m the d e n s e s e t A onto all of

L~(RN).

4) P r o p e r t i e s of the t r a c e . oz) T a ( f 1 + f 2 ) = T a l l

+ T a f 2 (fl, f2 ~ L ~ ( R N ) ) "

fl) If T a is defined f o r s o m e a E R N - m, then it is defined f o r an a r b i t r a r y n ~ R N _ m . y ) Suppose that t h e r e e x i s t s a B a n a c h s p a c e A(R m) ~ D ' ( R m) (in g e n e r a l , c o n s i s t i n g of g e n e r a l i z e d functions) such that

T~ : L,r (RN) -+ A (R~);

IITJI[A~ c ll/J1Lr

w h e r e c does not depend on a o r f. T h e n f o r 1 -< p < :r the function Taf(X') is continuous as a function of a ~ RN_ m with v a l u e s in the B a n a e h s p a c e A (Rm), that is,

llYol--Yo-Jllh~O (h-~0, h ~ R ~ _ ~ ) In p a r t i c u l a r ,

if A(R m) ~ L p ( R m ) , then the t r a c e T a has the p r o p e r t i e s which u n d e r l i e its definition in

[2]. m)~, (m >- 1), which a p p e a r s as the t r a c e space in T h e o r e m 2. Suppose that 2 ~. We define the c l a s s B~0(R the function ~(t) > 0 is defined f o r t >_ 1, and f o r s o m e n u m b e r s -r < fll -< f12 < + ~ the function ~b(t)tfll is a l m o s t d e c r e a s i n g and ~b(t)tfi2 is a l m o s t i n c r e a s i n g . By an a l m o s t d e c r e a s i n g (increasing) function we m e a n a function f o r which the c o r r e s p o n d i n g m o n o t o n i c i t y inequality is s a t i s f i e d with s o m e c o n s t a n t that is not n e c e s s a r i l y equal to one. We c o n s i d e r a function F ( u ) ~ S~)(Rm) (1 < p < ~), i . e . , in the s p a c e of g e n e r a l i z e d functions (the S c h w a r t z s p a c e S'; s e e [2, Chap. 1, 8]) that a r e r e g u l a r in the s e n s e of Lp, and we let Q s ( F , u) denote its Dirichlet series

908

Q~(F, u) = (I)~cS'~(~ ') (u) (s = 0, l, 2, ...), where K s =Ps\Ps_~ (s = 1 , 2 . . . . ), K o = P o , note the c h a r a c t e r i s t i c function of the s e t K s . Definition 3.

andPs={~'

~Rm,

(6)

I ~ j l - < 2 s, j = 1 . . . . .

m}.

Welet

(1)Ksde-

We will s a y that F(u) ~ B~p0(Rm) if F E S~)(Rm) and the following n o r m is finite: IIF!1B=

=

[~-0(2,

2/0 ( 1 < p < ~ ;

1~<0~),

(7)

T h e c l a s s B~p0(R m) is a B a n a c h s p a c e (see ~]) of, in g e n e r a l , g e n e r a l i z e d f u n c t i o n s . that f o r 1 < p <__ 2 the condition r <- c, and f o r 2 < p < ,r the condition

We

aote, however:

~ ~2v/(p-2) (l) t-tdt < 30

(8)

g u a r a n t e e s that we have the i m b e d d i n g B~pp(R m) -~ Lp(R m) (and the condition is a l s o n e c e s s a r y f o r the e x i s t ence of the i m b e d d i n g ) . We will s t a t e this f a c t without p r o o f s i n c e we u s e it only f o r obtaining s o m e c o r o l l a r i e s to the t h e o r e m on t r a c e s . F i n a l l y , if f o r s o m e i n t e g e r k >- 1 the c o n s t a n t s ill, f12, which d e t e r m i n e the b e h a v i o r of r s a t i s f y the condition 0 < fit -< f12 < k (this condition on the function ~(t) is e q u i v a l e n t to the (~) condition of L o z i n s k i i ) , then we have equality of the c l a s s e s B~0 ( ~ ) = B ~ ( ~ )

0- < p < ~ ,

i < o ~ ~),

(9)

w h e r e the c l a s s of B e s o v type BCp0(Rm) is defined as the s u b s p a c e of functions in Lp(Rm) with a finite n o r m

I'Fl]-g= ,IF[Ic~-) {i [% (F, t)v/~i (t-~)]~ t-~dt}~.

(10)

H e r e Cok(F, t)p is the k - t h modulus of continuity of F in the n o r m of L p ( R m ) . We note that the condition ~(~,) is not only s u f f i c i e n t but a l s o n e c e s s a r y f o r the e q u a l i t y (9) (see [7, t2, Chap. 1]). When ~b(t) = ~-~ (0 < a < k) both c l a s s e s c o i n c i d e with the B e s o v c l a s s Bp0(R a m) (see [2]). w

STATEMENT

OF

THE

RESULTS

We f o r m u l a t e the t h e o r e m s which will p r o v i d e us with n e c e s s a r y and s u f f i c i e n t conditions f o r e x i s t e n c e of t r a c e s of functions in the c l a s s L~(R N) on an m - d i m e n s i o n a l s u b s p a c e R m ~ We let X(u) denote the function (11)

~.(u) = @(u)u ....... :'~ (1 < p < ~ ) . THEOREMs, 1_,_ Suppose that 1 _< m -< N - 1; 1 < p < ~,; @ ~ (Dn) f o r n >N/2. f(x) ~ L ~ ( R N) has a t r a c e Taf(X') f o r x' R m , then

if an a r b i t r a r y function

t Z~(u) u-~du < x) ( i / q + lip = 1)

(12)

i We let l 0 denote the n u m b e r i~ =

( N - - m) (1/2 - - l/p) -+- ko -}- i/2,

(13)

w h e r e k 0 is the s m a l l e s t i n t e g e r s u c h that 2ko > m. T H E O R E M 2. Suppose that 1 ~ m _ < N - 1; l < p < ~ ; fled, then f o r an a r b i t r a r y a ~ RN_ m we have the i m b e d d i n g

~ , ~ (Dn) f o r n > m a x { N / 2 ,

/0}- if (12) is s a r i s (14)

To : L r, (R:~) Bpv (t?.~), w h e r e the function ~b(t) is defined by the r e l a t i o n s ~(t)=

i F~(u) u-~du }l,,1 ( 2 ~ - ~ p < ~ : ;

r (t) - ~. (t) -~2-~)'2(~-~)

~.q(u) u-~d~ I

l,'q+ l/p -- i),

(15)

(t < p < 2).

(16)

909

T h e o r e m s 1 and 2 show that the r e l a t i o n (12) is a n e c e s s a r y and sufficient condition f o r the e x i s t e n c e of t r a c e of an a r b i t r a r y function fix) ~ Lp~(RN).

a

We now s t a t e the f u n d a m e n t a l c o r o l l a r i e s of T h e o r e m s 1 and 2. C O R O L L A R Y 1. > (N - m ) / p .

An a r b i t r a r y function in the Liouville c l a s s L~fR N) has a t r a c e on R m if and only if

In f a c t , h e r e ~(t) = t - a ; Mt) = t ( N - m ) / P - a ;

and condition (12) is e q u i v a l e n t to the inequality a > (N - m ) / p .

C O R O L L A R Y 2. If (~(t) = t - ( N - m ) / P l n T ( 1 + t), then a n e c e s s a r y and s u f f i c i e n t condition f o r e x i s t e n c e of a t r a c e of an a r b i t r a r y function in L~fR1q) is the inequality T < ~ l / / q . C O R O L L A R Y 3. Suppose that in T h e o r e m 2 the function kit) s a t i s f i e s the following condition: a l m o s t d e c r e a s i n g f o r s o m e ~ > 0. T h e n f o r an a r b i t r a r y a ~ R N _ m we have the i m b e d d i n g

To :L~r (RN)~ Bpp(Rm) -~

(i
~),

k(t)t c~ is

(17)

if in the definition (10) the i n t e g e r k is c h o s e n to be s u f f i c i e n t l y l a r g e and ~b(t) is r e p l a c e d by ?~(t). H e r e the condition on )~(t) i m p l i e s that (12) holds, so that f r o m T h e o r e m 2 we obtain (14). e s t i m a t e shows that the condition on Mt) g u a r a n t e e s the t w o - s i d e d inequality

q~,(t)~{is

Also, a s i m p l e

(t ~ t),

and t h e r e f o r e r ~ hit) and Bp~p(R m) = B~p(R m) (1 < p < ~o). F i n a l l y , t h e r e e x i s t s fl > 0 such that (I)(t)t/~ is i n c r e a s i n g (by v i r t u e of the r i g h t - h a n d inequality in (1)), and c o n s e q u e n t l y Mt)t "/is i n c r e a s i n g f o r s o m e T > 0. T h u s hit) ~ (~k) with k > T, and (17) follows f r o m (9) and (14). T h e i m b e d d i n g (17) w a s e s t a b l i s h e d by the author in [7]. It i m p l i e s , in p a r t i c u l a r , a t h e o r e m on t r a c e s f o r Liouville c l a s s e s with 4,it) = t - a ; ~(t) = t ( N - m ) / p - ~ ice > (N - m ) / p ) . We should m e n t i o n that (17) is p r e c i s e . T h a t is, an a r b i t r a r y function in the c l a s s B~p(R m) s a t i s f y i n g the conditions that w e r e i m p o s e d on ~.(t) c a n be e x t e n d e d onto all o f R N as a function in the c l a s s L4~(RN ) (see [7]). T h i s m e a n s that the s e t of t r a c e s of f u n c P t i o n s ' i n the c l a s s L~('RN) c o i n c i d e s with Bpkp(Rm). C O R O L L A R Y 4.

Suppose that in T h e o r e m 2 the function X(t) s a t i s f i e s the condition (with 1 / p + 1 / q = 1) (; ,)2(j,- ~),,(v-~) t-ldt ~ c~ ( 2 < p < ~ ) ~y (u) u-ldu 1

'

or

sup ~ it) -(2-p)/(~)-1) f ~,~ (u) u-idu < t>d

(1 < p ~ 2 ) .

i

T h e n f o r an a r b i t r a r y a ~ R N - m we have the i m b e d d i n g

f a : LOp(Rzr --)- Lp (Rm).

(18)

In f a c t , by v i r t u e of T h e o r e m 2 we have (14) with the function ~(t) which was defined by (15) and (16). Since f o r 1 < p <-- 2 the condition ~(t) -< c, and f o r 2 < p < oo condition (8), g u a r a n t e e s an i m b e d d i n g into L p ( R m ) , we obtain (18). C O R O L L A R Y 5. of the f o r m

If ~(t) = t - ( N - m ) / P l n 7 ( 1

~2/p--3/2

+ t), then a condition which g u a r a n t e e s the i m b e d d i n g (18) is

(2~p~

oo); ~ - - 1 / 2

(1 ~ p < 2 ) .

(19)

F o r the p r o o f we apply the r e s u l t in C o r o l l a r y 4 to the function lit) = In T (1 + t), r e m e m b e r i n g that T < - 1 / q and oc

j ~ (u) u-adu ~ in ~1+~ (1 + t) (t > i). t

Simple c a l c u l a t i o n s then give us (19).

910

w

AUXILIARY

ASSERTIONS

In this s e c t i o n we p r e s e n t s o m e l e m m a s and s o m e known r e s u l t s which we will use l a t e r . 1~

Known r e s u l t s .

1. L e t 0 - a < b -< r Hardy inequalities

q _ 1; let cp(u) be a m e a s u r a b l e nonnegative function f o r u ~ (a, b).

uv

q~(v)dv du~.cS~'~(v) vv+~'dv

We have the

(7~/--1),

(20}

b

9

,.~

~

H e r e the c o n s t a n t c > 0 does not depend ou a, b, o r q~. T h e s e inequalities r e d u c e to the usual inequalities f o r a = 0, b = ~o (see [2, p. 423]). 2.

Suppose that }, u ~ R m (m -> 1), and the function ~o(u)~ Lp(Rm) (1 < p < ~o). We u s e the notation I

8~ (% u) = ( i ) ~ wherek

= (k 1. . . . .

kin); kj = 0 , ~1, ~2 . . . .

!m

TM

(~) (~),

(l_
(here Ak is a s e t of 2 m r e c t a n g u l a r p a r a l l e l e p i p e d s ) . IV])

We have the L i t t l e w o o d - P a l e y

,

p(m)

w h e r e A is the c o l l e c t i o n of all v e c t o r s k of the indicated f o r m ; s i d e d e s t i m a t e with c o n s t a n t s that do not depend on r 2 ~ An e s t i m a t e of p o w e r type. m e a s u r a b l e in the v a r i a b l e s .

r e t a t i o n (see [3, Chap.

Suppose that u -> 1, ~

the s y m b o l "~ indicates the e x i s t e n c e of a t w o R m (m -> 1), and the function o?(u, ~) is jointly

Definition 2. We will s a y that r (u, ~) a d m i t s an e s t i m a t e of p o w e r type with r e s p e c t to u if the condition u J u 2 ~ [~,/3], w h e r e 0 < el _< fi < ~o, i m p l i e s the r e l a t i o n r L ) / r 2, ~) ~ [7, 5], w h e r e 0 < 7 -< 6 < o0 and T and 6 depend only on a, fl (and do not depends on u l, u 2, ~)o P r o p o s i t i o n 1. Suppose that ,I,(u) ~ 0 (u * co); r ~ (I:hg. u n d e r condition (12) a l s o the function

T h e n the functions 4~(u), k(u) [see (11)], and

z (zO = I ~.~(v) v-~dv (i < q < ~) %

a d m i t an e s t i m a t e of p o w e r type (for ~ this e s t i m a t e depends only on the c o n s t a n t s c o and c 1 in [1]). F o r 4~(u) this e s t i m a t e follows f r o m m o n o t o n i c i t y and the not v e r y r a p i d d e c r e a s e at infinity; f o r Mu) it follows f r o m the v a l i d i t y of the e s t i m a t e f o r O(u); f o r • it follows f r o m m o n o t o n i c i t y and the v a l i d i t y of the e s t i m a t e f o r Mu). The e a s y details of the p r o o f a r e left to the r e a d e r . Other e x a m p l e s of functions which a d m i t an e s t i m a t e of p o w e r type will be p r e s e n t e d l a t e r . F o r an a r b i t r a r y g - 1 we let ktt denote the i n t e g e r which is d e t e r m i n e d by the r e l a t i o n

P r o p o s i t i o n 2. Theuforarbitraryt~,

Suppose that the function (p(u, ~) a d m i t s an e s t i m a t e of p o w e r type with r e s p e c t to u. v, 1 - < p - < v < o o we have 2v

l'~v

c~ f q0(u, ~ ) u - ~ d u < %~ r p~ s --h~L

2v

~, {)
~(u, ~)u-Jdu

(23)

~t

with c o n s t a n t s 0 < ci -<- c2 which do not depend on U, v, ~.

911

F o r the p r o o f we note that f o r k~ = kv we have 2~

2~

v

~t

2v

and by the e s t i m a t e of p o w e r type these i n t e g r a l s a r e of o r d e r (p(2 kv, ~). the i n e q u a l i t i e s hv

2s§

s=k~-Fl

2s

If k v - k/~ + 1, then (23) follows f r o m

2v

k~-~i 2s§

~

s=h~

2s

and the estimate of power type for (p(u, ~). 3~ Lemmas. Hereafter we always use the notation D~ D1(Dk-lf(u)) (k >- 2).

= f(u) (fi > 0); D1f(u) = -f'(u)/u; Dkf(u) =

LEMMA 1. Suppose that q > 1; M >- 1 is an integer; the function f(u) is defined for u > 0; f(u) ~ 0(u ~oo); f ~(Dn) for n >M(I/q - 1/2) + 1/2 and i

cc

S ] (u) uM--ldu < oo; .! l (u) u(M--i)/e--~ln (U + l) du < oo. 0

(24)

1

T h e n f o r the F o u r i e r t r a n s f o r m of the function F(x) = f ( I x l ) , x ~ R M, u n d e r s t o o d in g e n e r a l in the s e n s e of S' (RM), we have oo

~7M (~) __ I~l(2-M)/2-n ~ uM/2"~-ng(.~,l__2)/2_~.r~(lt o

I~[)

Dn f (u) du,

(25)

w h e r e J v i s the B e s s e l function with index v (see [9]), ~ ~ R M, ~ # 0. Also oo

ClA

~ C1 ~ 1AM(q--1)--J] q (l~) dl~ ~ S I~M 0 RM

(~)]qd~

~ c2A,

(26)

w h e r e the c o n s t a n t s c 1, c 2 > 0 depend on f(u) only in t e r m s of the c o n s t a n t s in the definition (1) of f(u). F o r the p r o o f of the l e m m a we m a k e the following o b s e r v a t i o n s . The v a l i d i t y of (25) f o r a function f(u) which d e c r e a s e s r a p i d l y to z e r o a s u --* oo follows f r o m a f o r m u l a of B o c h n e r (see [8]) by m e a n s of an n - f o l d i n t e g r a t i o n b y p a r t s . If n > (M + 1)/2 and f(u) is b o u n d e d at z e r o , then (25) follows f r o m w in [10]. The g e n e r a l c a s e which we a r e c o n s i d e r i n g is c o m p l e t e l y a n a l o g o u s to the c a s e in [10]. T h e l e f t - h a n d e s t i m a t e in (26) holds without the condition f ~ (D n) and the s e c o n d condition in (24). It c a n be r e g a r d e d as known. One of the p o s s i b l e g e n e r a l i z a t i o n s of this e s t i m a t e is contained in [11] by Gulis a s h v i l i . T h e r i g h t - h a n d e s t i m a t e follows f r o m the inequality

(1/1~1UM--~f(U)du -F ]~,(i--M)/2--nl~l~t } ~ l$(M--l)/2--n[ ill') dl~

~M (~)] ~C. / !"

by m e a n s of the H a r d y inequalities (20) and (21) (with n > M(1/q - 1/2) + 1 / 2 ) . d e p e n d s on the following e s t i m a t e s on B e s s e l functions (see [9]):

(27)

T h e d e r i v a t i o n of (27) f r o m (25)

i]~(~)l~l) and r e q u i r e s a splitting of the i n t e g r a l (25) into two i n t e g r a l s and an a p p l i c a t i o n of (28).

(28) We a l s o use the e s t i -

mate

IDV(u) l ~c/(u)~ -2~,

(29)

which follows f r o m (1). We i n t r o d u c e the following notation ( F o u r i e r t r a n s f o r m a t i o n to be u n d e r s t o o d in the s e n s e of S'(RM)): I

IM

T (t, ~) - co ( V l + t 2 + lyl~) (~)

(y, ~ ~ RM, t ~ 0).

L E M M A 2. Suppose that q > 1; 0 < e < 2q; M >- 1 is an i n t e g e r ; O(u) t0 ( u ~ o ) ; ~ , ~ (Dn); n>- 1; n > M ( 1 / q 1/2) + 1 / 2 + e / q ;

912

(30)

the function ~(u) is defined f o r u > 0;

0

T h e n f o r an a r b i t r a r y t o > 0 t h e r e e x i s t s a c o n s t a n t c, depending only on t o and the c o n s t a n t s in (1), such that

II~ (t,.) - 9 (0,.)lJ~q(,M) ~< cA (8, r

(t ~ [0, to]).

(32)

P r o o f . We s e t f(u) = 4,(41 + t 2 + u2). T h e n f(u) ~ (Dn), w h e r e the c o n s t a n t s in (1) f o r f(u) do not depend on t >- 0. F r o m (31) (by the H61der inequality with n > M(1/q - 1/2) + 1/2 + e/q) and the f o r m of f(u) it follows that (24) holds, and t h e r e f o r e we c a n use the r e s u l t s in L e m m a 1. In p a r t i c u l a r , (27) holds with a c o n s t a n t that d o e s not depend on t >- 0. If we i n t e g r a t e the f i r s t t e r m in (27) to a p o w e r q > 1 o v e r the d o m a i n I ~ i -> i in s p h e r i c a l c o o r d i n a t e s and use the H a r d y inequality (20), we obtain as an u p p e r e s t i m a t e a quantity which does not e x c e e d !

c .i r

(]/-i & t2 + u~) u~'(q-t)-~du < c~A (s, q~),

(33)

0

with c o n s t a n t s that do not depend on t. In the s e c o n d t e r m in (27) we c a n isolate the i n t e g r a l o v e r (1/[ ~1, 1) f o r I ~ I -> 1. T h e p a r t of the s e c o n d t e r m which c o n t a i n s it, upon i n t e g r a t i o n to a p o w e r q > 1 o v e r the d o m a i n [ ~ I >- 1, c a n a l s o be e s t i m a t e d f r o m above by m e a n s of (33) (the H a r d y inequality (21)), u s i n g the condition n > M(1/q - 1/2) + 1 / 2 . T h e o t h e r p a r t of the s e c o n d t e r m , containing the i n t e g r a l o v e r (1, ~), c a n be e s t i m a t e d f r o m above by m e a n s of HSlderTs inequality by the quantity A~/q(e, 4~)i~ ] ( 1 - M ) / 2 - n If we i n t e g r a t e it to a p o w e r q > ! o v e r the d o m a i n I~1 -> 1, we a l s o obtain an e s t i m a t e in t e r m s of cA(e, 4,). T h u s , f o r a a a r b i t r a r y t -> 0 we have

[ IW(t, ~)l~d~ 0 does not depend on t. We still have to e s t a b l i s h the inequality which differs f r o m (32) by the r e p l a c e m e n t of Lq(R M) with Lq(B), whereB ={~R M : I ~ I - - < 1}. F r o m the identity (n>- 1) -- (I) (V-I ~- t ~ + u2)] = ( pDi~(I) (]/~ + p a + u 2) dp b

P (u, t) ~- D~- I [q) ( | / - ~ ) and (1) it follows that

[D(u, t) [ ~ct~@(~'t+u 2) ( t + u e ) - L

(34)

We a l s o need the following r e l a t i o n , which is i m p l i e d by (31) and the m o n o t o n i c i t y of 4~(u):

@(u)u-"l-l"~'-Uq=o(l) If in (25) we f i r s t s e t f(u) = r tain

(u--~ oc).

(35)

+ u2), and then f(u) = 0(~/1 + t 2 '+ u{), s u b t r a c t and i n t e g r a t e by p a r t s , we ob-

A ~ ~ (0, ~) -- T (t, ~) = I~,l~ c~-Mv~-~ ~) u ~W2'-.-~ ' a(~t_a)/2~_,(uJ~l)D(u,t)du. 0

T h e t e r m s outside the i n t e g r a l go to z e r o by v i r t u e of (28), (34), (35), and the conditions on n. (34) it then follows that ( [ { ] --< 1)

I~1 <~cr'-

"-"+z"-a( i 1-' ,,~)

-,,

r

i+

,

u"-)du+ I;I (a-'''''='-" O

F r o m (28) and

tt(M--a)/2--nd~(]/'~)du

i/Ir~l

T h e i n t e g r a l of the f i r s t t e r m to a p o w e r q > 1 o v e r 1~M can be e s t i m a t e d , as b e f o r e , by m e a n s of the H a r d y inequality (20); it does not e x c e e d cA(e, ~)t2q. The s e c o n d t e r m , by v i r t u e of the HSlder inequality, does not e x c e e d

l

uh~l

l

'

(36)

and its i n t e g r a l to a p o w e r q o v e r I~l -< 1 c a n be e s t i m a t e d f r o m above, a f t e r a change in the o r d e r of i n t e g r a tion, bY cA(e, 4,)t 2q. L e m m a 2 is p r o v e d . L E M M A 3. Suppose that M ~ 1 and k 0 -> 0 a r e i n t e g e r s ; q > 1; the function ~(u) is defined f o r u > 0; 9 (u) ~ 0 (u ~oo); 4~(u) ~ (Dn) f o r n > M ( 1 / q - 1/2) + 1 / 2 +k0; and f o r the function k(u) = ~(u)u M / p (1/p + 1 / q = 1)

913

condition (12) is s a t i s f i e d .

L e t j be the s m a l l e s t nonnegative i n t e g e r f o r which j > M(1/q - 1/2) + 1 / 2 ; l/u

o~

p (t, u) = y v~-~@ (lit + t ~ + ~ ) dv + u -j-(~-~)/~ 0

j" v(~-')/2-i(1) (I~Y-+ t~+v ~) dr.

(37)

11~

T h e n f o r the function ,I,(t, ~) in (30) we have a t W (t, t t o7

~)i<

(3s)

cp (t, I~I) (0 < 1 4 ko).

In t h e s e i n e q u a l i t i e s e > 0 does not depend on t o r ~, P r o o f . U n i f o r m c o n v e r g e n c e with r e s p e c t to t of the s e c o n d i n t e g r a l in (37), which defines p(t, u), f o l lows f r o m (12) by m e a n s of the H~ilder inequality 0 > M ( 1 / q - 1/2) + 1/2). We note that f o r 9 ~ (Dj+k0) we have the identities (0 - l - k 0) [1121

t

+

=

+t +uO,

Z r~0

from which follows, by virtue of (i), the estimate (0 -< l -< k 0) t

"

If we use this e s t i m a t e and (28), we obtain (38) f r o m (25) with f(u) r e p l a c e d by ~(,/1 + t ~ + u ~) and n by j. The l e g a l i t y of d i f f e r e n t i a t i n g u n d e r the i n t e g r a l sign follows 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 i n t e g r a l s in (37) with r e s p e c t to t. The [ e m m a is p r o v e d . L E M M A 4. Suppose that the conditions of L e m m a 3 a r e s a t i s f i e d with k 0 = 0. we use the notation

I (Ix, t) =

j"pq (t, u) u~-~du;

F o r t -> 1 and e ~ (0, l / t ]

X (t) = . ~ (v) v-~dv.

0

(39)

t

T h e n we have the e s t i m a t e

wherec>0does

not depend o n t o r t ~ .

For 1
(q>2) wehave

D--~ 0<~
(41)

P r o o f . F o r the e s t i m a t e on I(t~, t) we begin by c o n s i d e r i n g the s e c o n d t e r m in (37). Its i n t e g r a l by a p o w e r q > 1 with weight u M-1 o v e r (0, t~) c a n be e s t i m a t e d f r o m above by m e a n s of the H a r d y inequality (21) f o r j > M ( 1 / q - 1/2) + 1 / 2 by the quantity X(g-1). In the f i r s t t e r m in (37) with u < p we isolate the i n t e g r a l o v e r (0, l/ix). T h e p a r t of the f i r s t t e r m which c o n t a i n s it r e d u c e s to the s e c o n d t e r m in (40). The o t h e r p a r t of the f i r s t t e r m in (37), c o n t a i n i n g the i n t e g r a l o v e r (1/t~, l / u ) , a f t e r i n t e g r a t i o n of it to a p o w e r q > 1 with weight u M-I o v e r (0, p) c a n be e s t i m a t e d b y m e a n s of the H a r d y inequality (20) and gives us the f i r s t t e r m in (40). We p r o c e e d to the p r o o f of (41) f o r 1 < p < 2. It s u f f i c e s to c a r r y out the p r o o f s e p a r a t e l y f o r e a c h t e r m in the e s t i m a t e (40) on I(t~, t). If in (41) we r e p l a c e I(t~, t) by X(t~-l), we obtain 2/tt

z ~-l~21q(2-p) sup %kl ) ~ ",z(t)-1-2/~(2-~) (-- X' (t)) dt ~ oc, 0
i

since the integral equals X (t)-2/q(2-p) ll/_-I ~. In order to estimate the contribution in (41) m a d e by the second term in (40) it suffices to establish uniform boundedness with respect to ~ ~ (0, I) of the following quantities: D1 =

2Mlq(2--p) 2/~S I )~q (t) X 2/~

(t)-'-2/q(2--~)(i / 2/a

ttM--l(I )

(V1 + t 2 + u2) du]21(2-P)t--idt, .---~.

~2/(2--p) _ t d t

,t)q.7~,, It~--t--'-/q(Z--p) k[ J~ uM--t ~ (V-t + t2 + u ' ) d u )i D 2 = ~9 2M/q(2--p) ~~ ~"~.

t

9

For the estimate on D I w e note that the inner integral does not exceed c~" 2/(~-p)(t)t2M/(2-p), and x(t)-t-
2/~7~(t) ~'~(~ P) i? u~ ~*(u) du )

*(t) t~-~dt

(42)

a n d the t e r m

(4~i E s t i m a t e s f o r t h e s e f o l l o w f r o m the i n e q u a l i t y 2/~

J~" w~i-lq~(u)du~

c~ --~/~ Z (t)i/q,

(43v)

w h i c h in t u r n f o l l o w s f r o m the H S l d e r i n e q u a l i t y a n d the r e l a t i o n X(u) = 4,(u)u ~ / p . S u b s t i t u t i o ~ of (43 ~) into (43) (for t -- 1) p r o d u c e s a f i n i t e c o n t r i b u t i o n in D 2. If we s u b s t i t u t e (43) into (42) and m a j o r i z e x ( t ) - l / q -< X(t) -1, we o b t a i n 2/tL

c[t w h i c h p r o d u c e s a f i n i t e c o n t r i b u t i o n in D 2. COROLLARY.

3[p q~ 2--p)

i t M/'~- Idt, i

T h e l e m m a is p r o v e d .

In L e m m a 4 l e t p = l / t - < 1.

Then

!' f,' (t, u).~'r-~d. ~ ' c i )-'"(~Ou--~a'ub

(44)

In f a c t . f o r g = 1 / t the s e c o n d t e r m in (40) d o e s not e x c e e d ckq(t), which is m a j o r i z e d b y the r i g h t - h a n d s i d e of (44) s i n c e X(t) a d m i t s an e s t i m a t e of p o w e r type. R e m a r k . T h e f u n c t i o n p(t, u) a d m i t s an e s t i m a t e of p o w e r type w i t h r e s p e c t to t. f r o m the e x i s t e n c e of s u c h a n e s t i m a t e f o r 4,(t).

w i%

Proof of Theorem

I.

PROOFS

OF

THE

THEOREMS

Suppose that condition (12) is not satisfied. A ( ~ ) ~ t (I)q ( ] r ~ )

This follows easily

Then clearly

uM(c~_i)_ld~ = ~ .

(45)

H e r e and h e r e a f t e r we l e t M = N - m > 1. Since n > ( M + l ) / 2 , t h e r e e x i s t s e, 0 < e < 2q, s u c h t h a t n > M ( 1 / q 1 / 2 ) + 1 / 2 + s / q . W i t h o u t l o s s of g e n e r a l i t y we c a n a s s u m e tilat a l o n g with (45) r e l a t i o n (31) i s a l s o s a t i s f i e d f o r the i n d i c a t e d e. In f a c t , i f A ( e , 4,) = 0o then we c a n find fl > 0 s u c h t h a t f o r e l ( u ) = 4,(u)u-fl we have A(4,l) = ,o, A(e, 4'1) < r T h e e x i s t e n c e of s u c h a n u m b e r fl >0 f o l l o w s e a s i l y f r o m the e x i s t e n c e of Y0 > 0 s u c h t h a t A(79 (I,) = ~ f o r y < Y0; A(7, 4,) < ~ f o r y > 70. L a t e r a r g u m e n t s then s h o w t h a t f o r the c l a s s L ~ I ~ N ) t h e r e is no, t r a c e on R m in the s e n s e of the d e f i n i t i o n s in w M o r e o v e r , the t r a c e d o e s not e x i s t in the l a r g e r c l a s s L~(RN). T h u s we h a v e (45) and (31). By L e m m a 2 w e o b t a i n (32), and by L e m m a 1 we know that (45) i m p l i e s that r~[t (0, ')!lt.q('.~) : : ~o. We fix t o > 0 and c h o o s e a s e q u e n c e {Wk(~)]~~176 w k ~ C~r

(46)

, s u c h t h a t ( 1 / p + 1 / q = 1)

io~;ii.t,(B~r)-~O (k---~-~ ) ; "h(t)~ I' (o~(;)T(t,~)d~+o~

(47)

(#-+oo).

(48)

T h e l a t t e r l i m i t is u n i f o r m with r e s p e c t to t ~ [0, to]. T h e p o s s i b i l i t y of m a k i n g s u c h a c h o i c e f o l l o w s f r o m (46), a r e s o n a n c e t h e o r e m , a n d the r e p r e s e n t a t i o n a k ( t ) = a k ( 0 ) + 5k(t) , w h e r e 6k(t) --* 0 f~< ~ ~ ) u n i f o r m l y with r e s p e c t to t ~ [0, to] b y v i r t u e of (32). T h e a i m of o u r s u b s e q u e n t a r g u m e n t s w i l l be to c o n s t r u c t a s e q u e n c e of s m o o t h f u n c t i o n s fk(x) ~ A [see (4)], x ~ R N, s u c h that ill, m

f,i) ( R N

--~0

(k-~r

9

(Fh(x'),

cr(x'))-~-oo(k--,-c~),

(49)

915

where Fk(X') = fk(x', 0), x' ~ R m ,

and ~p(x') is a function in C~~(Rm). To do this, we consider

~.,,(x)=o)(x')(o,,(x") (x'~R,,,, x"~th,, z=(x', x")~Zt~), w h e r e the wk w e r e c h o s e n e a r l i e r and w(x') ~ S(Rm) will be c h o s e n l a t e r . is f~(.r)=~(Vi+tEJ-)v~([)

(x)~A;

T h e d e s i r e d s e q u e n c e of functions

( k = t , 2 . . . . ).

(50)

Here

IIhIILcp(RN) = IlVktI~(RN) = Ik01]~p(R,.) II0~b.p(R~) -+ 0 (k ~ ~), so that the f i r s t r e l a t i o n in (49) is s a t i s f i e d .

L e t ~ = (~', ~"), w h e r e ~' ~ R m, ~ " ~ R M.

Then

F r ([') = (2~) - u / z /~ ~N (~,, [,,) d[".

(51)

We s u b s t i t u t e the e x p r e s s i o n f o r ~N in (50) and use the identity (f, (~) = (f, ~0), w h e r e ~ ~ S ~(RM), ~0 ~ S(RM), and by (30) and (48) we obtain (~') -- (2~)--M/20) m (~') a~ ([~'(). If the function ~b(~') ~ Z, w h e r e Z is the s p a c e of F o u r i e r t r a n s f o r m s of functions in C O (Rm), then Rm

We c h o o s e w (x') ~ S(R m) so that ~m(~,) >_ 0, ~m(~,) ~ C~ (R m) and supp ~ m = {~, ~ R m : I ~' I --- to}. T h e n we choose r ~ Z so that ~b(~') _> 0 (~' ~ R m , I ~' I ~<- t 0) and c = (2n) -M/~ y 7o"~ (~') ~ (~') d~' > O. Rm

We u s e the notation~o(x ~) = ~m(x') ~ C o ( R m ) .

Then

(F~, r = ( ? ~ , * ) ~-~ c min a~ (t), which a c c o r d i n g to (48) i m p l i e s the s e c o n d r e l a t i o n in (49).

T h e t h e o r e m is p r o v e d .

2 ~ P r o o f of T h e o r e m 2. 1. Suppose that condition (12) is s a t i s f i e d and f(x) ~ A (see [4]), F(x') = f(x', 0) (without l o s s of g e n e r a l i t y we c a n a s s u m e that a = 0). T h e r e e x i s t s a function v(x) ~ S(R N) such that 7~(~).~_O(]~l-}-l~12)v~(E)

( ~ = ( ~ ' , ~")~R.~-).

(52)

Here

II/liL~@~ ) = [IvIIL~(RN). We will a l w a y s let M = N - m >- 1. in S' (R M) it follows that

F r o m (51), (52), (30), and the g e n e r a l i z e d P a r s e v a l identity f o r functions

w h e r e vm(~ ', ~) is the F o u r i e r t r a n s f o r m of the function v(x', ~) with r e s p e c t to x' ~ R m. By v i r t u e of (6), a f t e r m o v i n g the F o u r i e r t r a n s f o r m a t i o n f r o m ~' to x' in R m and i n t e g r a t i n g with r e s p e c t to ~ R M, we obtain Q~ (F, x') = (2a)--M/2 f r~(~')~(l~_'I, ~)~(~-', ~)I,~ (x')d~ RM

H e r e iZs(~ ') = (1)KS is the c h a r a c t e r i s t i c function of the s e t K s (s = 0, 1 . . . . ). Changing the o r d e r of the F o u r i e r t r a n s f o r m a t i o n and i n t e g r a t i o n is justified b y FubiniVs t h e o r e m s i n c e the function has c o m p a c t s u p p o r t with r e s p e c t to ~' (because of (1)Ks), and with r e s p e c t to ~ the i n t e g r a n d is i u t e g r a b l e at z e r o (for ,I~(I ~'], ~) this f o l lows f r o m (12) and L e m m a 1, in which A < ~o) and d e c r e a s e s r a p i d l y at infinity s i n c e v(x) ~ S(RN). By u s i n g the g e n e r a l i z e d Minkowski inequality (the n o r m of the i n t e g r a l does not e x c e e d the i n t e g r a l of the norm) and then the r e l a t i o n .~ ( ~ ' ) = ~,(~') ~ ( ~ ' )

916

and Eq. (6): Qsm(v(., ~))(~') =gs(~')~'m(~ ', ~), we obtain fm RM

The aim of this stage in the proof is to establish the est.~mate

[IO, (F)',lLr,(n,0 ~

(v (., g), X')JILp(RZI)d~,

c ( p (2 ~, ]r ffM

where oCt, u) is defined by (37) and the Lp-norm

(53)

is taken with respect to x' ~ Rm,

F o r this it s u f f i c e s to e s t a b l i s h the following e s t i m a t e on the m u l t i p l i e r n o r m (1 < p < ~ ) :

I9 (I "i, ;) .~t~(.)I:G (R,,,) ~ co (2 ~, I;1) (s = o, 1, . . . ) .

(54)

F i r s t we c o n s i d e r the function

q,'~(t, ~)=,~r(t, ~),-~(t)

( t > 0 ; s = 0 , t . . . . ).

H e r e v0(t) = l f o r t ~ ( 0 , 2q-~); v0(t) = 0 f o r t >- 4]-~"~ 0 - < % ( 0 - < l f o r the r e m a i n i n g t > 0 . For s =1, 2.... we have us(t) = v ( t 2 -s) (t >0), w h e r e v(t) = 1 f o r t ~ [ 1 / 4 , 2q-~]; v(t) = 0 f o r t ~ ( 0 , 1/8] a n d t - > 4 ~ , 0 v(t) - I f o r the r e m a i n i n g t > 0. We a s s u m e that %(0 and v(t) have continuous d e r i v a t i v e s up to o r d e r k 0, w h e r e k 0 is the s m a l l e s t i n t e g e r f o r which m / 2 < k 0. We will show that I ' L (1~'1, ~)IM;,(R,~, )

-

F r o m the definition of ,I,s it follows that 'I*(IVI, O > ( Y ) - - ' ~ , ( I Y I ,

O~',(V),

s o that the l e f t - h a n d side of (54) equals

w h e r e e depends only on p and m. We t u r n to the p r o o f of (55).

Therefore,

(55) i m p l i e s (53) and (54).

F o r s = 0, 1 . . . .

we have

l~%(t)t
O
(1>0,

and t h e r e f o r e by the L e i b n i z f o r m u l a (0 -< k -< k 0) we have -< c ~ ' I o' w (*..a)[ 9~"-k) 8th

~

~ =0

8t l

-

.

This inequality should be e o n s i d e r e d f o r t ~ [0, 44-m] if s = 0, and f o r t ~ [2s-3, 2 s + Q - ~ ] if s = 1, 2 . . . . . since only f o r these t is the left side not equal to z e r o . By v i r t u e of L e m m a 3 [see (38)] and the c o n n e c t i o n between t and s it follows that

'~

/

u2"~(t,;)~cp(t, ig[)

(t>0;

s=0,1 .... ;

0
Since ,o(t, I~1) -> c0(1, !gl) f o r 0 < t <- 4g-~,, and sinee we have an e s t i m a t e of p o w e r type f o r oct, 1721)(see the r e m a r k a f t e r L e m m a 3), we c a n r e p l a c e p(t, I g I) by p(2 s, [g I). F r o m these a r g u m e n t s it follows that f o r an a r b i t r a r y m - t u p l e o e - > 0, oe= (a t . . . . . a m ) ; 0-< [el =oq + . . . + a m - < k 0 w e h a v e

I.:,],~,] J~'T~(}~-'I,~.)t~c9(2s, ,~.)" In this e s t i m a t e , as b e f o r e , the l e t t e r e r e p r e s e n t s a c o n s t a n t which in e a c h e s t i m a t e does not depend on s, ~, t = I~'i. By a known t h e o r e m on m u I t i p l i e r s the above inequality i m p l i e s (55) f o r k 0 > m / 2 (see [3, Chap. IV]}, T h e e s t i m a t e (53) is p r o v e d . 2.

F r o m (53) we obtain the following e s t i m a t e on the n o r m (7)

[FtI~c(B,+B~)

(O=p),

(56)

917

where

~

B~) = ~ ~ ( 2 ' ) - p

=

j" s=0

(57)

P(2~,l~])llQ~(v(',~),x'tIL~(.~)d~,

(58)

,0(2

9

1~1>2--s

In (58) we use once a g a i n the inequality (1 < p < ~)

[[O~ (v (., ~), x'llLp(Rm) < C flu (', ~)[ILp(R,~),

(59)

w h e r e c does not depend on s, ~, o r the function v. T h e s e c o n d s t a g e in the p r o o f is d e v o t e d to the e s t i m a t i o n of B p f o r 1 < p < co. Since p(t, u) and $(t) a d m i t e s t i m a t e s of p o w e r type with r e s p e c t to t [see (15), (16), P r o p o s i t i o n i in w and the r e m a r k a f t e r L e m m a 3] we s e e that 0o

(60) l~I>:It

We s t a t e an inequality which follows f r o m (37) f o r u > t -i , t --- 1:

p (t, u) <~ c 9 (t) u -M -4- r (t) U - ( M - I ) / 2 - j + l~--(M--t)/2--JX (t)i/qt M(t/r

.f

i/u

12(M--I)/2--Jdu

= C{p: (t, u) + p~ (t, u) + Pa (t, U)},

w h e r e the function X(t) is defined in (39), and we a l s o m e n t i o n the inequalities (1 < p < ~o)

~( t) >~cx ( t ) uq>~clk ( t),

(61)

which follows f r o m (15) and (16). It s u f f i c e s to p r o v e f i n i t e n e s s of the quantities which a r e obtained f r o m (60) b y s u b s t i t u t i n g pi(t, I ~ [) (i = 1, 2, 3) f o r p(t, I [ I)~ When we s u b s t i t u t e P* (t, I ~ I) and change the i n t e g r a l with r e s p e c t to ~ into s p h e r i c a l c o o r d i n a t e s , ~ = ru, and use the r i g h t - h a n d inequality in (61), we obtain a quantity which does not e x c e e d ~

P

c ! t- ~ - i

lit

r-: dr S llv(', ru)I]Lp(,,n) dtt

dt.

lul=t

If we extend the i n t e g r a l with r e s p e c t to t onto (0, ~) and change t to l / t , we c a n apply the H a r d y inequality (21), obtaining an u p p e r bound in t e r m s of

c!rM--t(lu~=ll]v(',rl~)lfLp(Rm)dU

dr~cl[vilLp(~N)"

T h e l a t t e r inequality is obtained by a p p l y i n g H~Jlder's inequality to the i n t e g r a l with r e s p e c t to u and u s i n g F u b i n i ' s t h e o r e m f o r the i n t e g r a l of v ( x ' , ~) o v e r R N = R m X R M. A n a l o g o u s l y , by u s i n g the l e f t - h a n d inequality in (61) we can e s t i m a t e the c o n t r i b u t i o n in (60) m a d e by the t e r m c o n t a i n i n g P3(t, I ~ I) (for j > M(1/q - 1/2) + 1/2). T o the r i g h t we a g a i n obtain ci]v iILp(RN). In (60) we now substitute P2(t, I~1) f o r p(t, I~l) and use the identity

y Iiv(.,~)[[Lp(R,~)[~l-(M-l)/2-i ]~l>/t/t

~t(M--:)/Z--r \~/l~l

"

d~=cYl ~(M-t}/2-~ /

0

r(M--l)/2--J S ]lv('ur)llLP(R,sdudr d~t. \i/~t

lul=t

,

By v i r t u e of the r i g h t - h a n d inequality in (61), a f t e r applying the Hardy inequality twice (first (20) for the i n t e g r a l with r e s p e c t to t and then (21) for the i n t e g r a l with r e s p e c t to g), we again obtain an e s t i m a t e in t e r m s

of clIvlILp(RN). Thus

B~ <~c [IV]I[p(RN)=cliff| P

Lv (RN~, (] < p <

3. We t u r n to an e s t i m a t e of B i in (57). we i n t r o d u c e the notation

We will use the Lii:~lewood- P a l e y r e l a t i o n (22).

T~={k:A~=_K~} 918

oo).

(s----~O,t . . . . ),

(62) To this end

We have the r e l a t i o n s

w h e r e K s is defined in (6) and Ak is defined in (22). oo

U Ts = A;

t)

2)

r~ R T ~ = ~

(63)

(s:#:l).

In this notation it follows f r o m (6) and (22) that

liO+(v(., ~), x')l]Lv(~,~)~c (~1% (v(., ~), X']~)+/Z]Lp(+,,, ).

(64)

This s t a g e in the p r o o f is d e v o t e d to a c o n s i d e r a t i o n of the c a s e 2 _< p < ~o. In (57) we apply the Hglder inequality f o r i n t e g r a l s with indices p and q and we use (44) and (15). T h e n

~), Xr'd!P

I!O~

I d;.

(65)

We e n l a r g e the d o m a i n of i n t e g r a t i o n to I g I -< 1 and r e v e r s e the o r d e r of i n t e g r a t i o n and s u m m a t i o n . We u s e the notation

A~ (z') = k~T.~ E i6~ (v (., ~), z')l~.

(66)

T h e n (to be c l a r i f i e d below) we see f r o m (65) that

I~i
Brn

/

\s=O

I'~l~
T h e f i r s t inequality follows f r o m (64), and the s e c o n d f r o m the inequality

"~' A~/2<~ \~o A,] ,% We a l s o u s e d (63), and the l a s t inequality follows f r o m (22). s t a n d s the quantity ctllViiLpfflN). T h u s (2 --< p < ~ ) ClqLp(RU)

4.

(p >~ 2). By v i r t u e of Fubini~s t h e o r e m , on the r i g h t side

(67)

,,Jl~~.tnu) " ~.p

In this final s t a g e of the p r o o f we p r o d u c e an e s t i m a t e on B 1 f o r 1 < p < 2. h ( t , z) :=

I

[

We use the notation

p ~ (t, [u])dul uvq (t, z:>O; u ~ R ~ ) .

(68)

t

Then 0 ~ h~,q(t,z), CMp q (t, z) Z M--~ .... O-'

w h e r e c M is the a r e a of the unit s p h e r e in R M, so that

[ d' (t, I~i)h-" (t, l.t) d~ - 1,.~' (t, t-'). I~d~l/t

If we m u l t i p l y and divide the i n t e g r a n d in (57) by h(2 s, I~l) and apply the H61der inequality f o r i n t e g r a l s with i n d i c e s p and q, f r o m the above equation we obtain (to be explained f u r t h e r ) cr

BI'<~-.~ t2(2~)-''h(2''2-~)';'''~

f h(2',[q) iQ~(v(":)iY.~,(R,-~)d::=

~

~'(2~'[;I)iiQ~(v(" ~)L.4R-') -

In this equation we i n t e r c h a n g e d s u m m a t i o n with r e s p e c t to s and i n t e g r a t i o n with r e s p e c t to ~ (which is e a s y to justify) and u s e d the notation ~t(t,z) =-,(t)-~gz(t,t-l)"~/qh (t, z) ~' ( t . z >0). Now v~e use (64) and take the i n t e g r a l with r e s p e c t to x ' ~ R m outside the s u m m a t i o n o v e r s. of (66) and (69) we obtain

(69) In the notation

9]9

As (x') ]~;l~i

Rm

]~[<:i

2s~I/l~[

2s~l/t~l

dx ~.

(70)

z s=

T h i s inequality is obtained by applying the H~itder inequality for a s u m over s with indices 2/(2 - p) and 2/p (i < p < 2). If we substitute into (69) the e x p r e s s i o n s for h(t, z) and ~b(t) [see (68) and (16)], and use {44), we a r r i v e at the e s t i m a t e [for the notation s e e (39)]

~t(t, z)21(2-~)~c)~(t)qx(t)-l-:/q(2-P)l(t,

z) 2/qC2-p).

The function on the right-hand side of this inequality admits an e s t i m a t e of power type with r e s p e c t to t (each f a c t o r has this p r o p e r t y ) . T h e r e f o r e by P r o p o s i t i o n 2 the f i r s t s u m in the b r a c e s on the right-hand side of (70) is m a j o r i z e d u n i f o r m l y with r e s p e c t to ~, I~1-< 1, by the quantity D in (41). By L e m m a 4 this quantity is finite. By v i r t u e of the notation in (66), p r o p e r t y (63), and the e s t i m a t e (22) we obtain f r o m (70) the inequality

from which, as before, we obtain (67), this time for 1 < p < 2. The estimates (67) for 1 < p < co and (62) together with (56) complete the proof of T h e o r e m 2. In conclusion, the author expresses his deep thanks to V. A. If'in and P. I. Lizorkin for their attention

to this work. LITERATURE

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

920

CITED

S . L . Sobolev, Some Applications of Functional Analysis in Mathematical P h y s i c s [in Russian], Izd. Leningrad. Un-ta, Leningrad (1950). S . M . Nikol'skii, Approximation of Functions of Several V a r i a b l e s and Imbedding T h e o r e m s [in Russian], F i z m a t g i z , Moscow (1969). I . M . Stein, Singular I n t e g r a l s and Differentiability P r o p e r t i e s of Functions, P r i n c e t o n University P r e s s , P r i n c e t o n , New J e r s e y (1970). L . R . Volevich and B. P. Paneyakh, "Some s p a c e s of g e n e r a l i z e d functions and imbedding t h e o r e m s , " Usp. Mat. Nauk, 20, No. 1, 3-74 (1965). L. H i i r m a n d e r , L i n e a r P a r t i a l Differential O p e r a t o r s , Academic P r e s s , New York (1963). M . D . R a m a z a n o v , "On a c l a s s of function s p a c e s , " D i f f e r e n t s . Uravnen., 2, No. 1, 65-82 (1966). M . L . G o l ' d m a n , "On integral r e p r e s e n t a t i o n s and F o u r i e r s e r i e s of differentiable functions of s e v e r a l v a r i a b l e s , " Candidate's D i s s e r t a t i o n , Moscow (1972). S. Bochner, L e c t u r e s on F o u r i e r I n t e g r a l s [Russian translation], F i z m a t g i z , Moscow (1962). G . N . Watson, T h e o r y of B e s s e l Functions, C a m b r i d g e Univ. P r e s s . M . L . Gol'dman, " I s o m o r p h i s m of g e n e r a l i z e d H6lder c l a s s e s , " Differents. Uravnen., 7, No. 8, 14491458 (1971). A . B . Gulisashvili, "An interpolation t h e o r e m of weak type and b e h a v i o r of the F o u r i e r t r a n s f o r m of a function which has given Lebesgue s e t s , " Dokl. Akad. Nauk SSSR, 218, No. 6, 1268-1271 (1974).