the embedding of certain classes of functions of several variables

THE EMBEDDING OF CERTAIN CLASSES OF FUNCTIONS

OF SEVERAL VARIABLES

V. I. Kolyada UDC 517.512.6

w I n t r o d u c t i o n

Le t R k be a k - d i m e n s i o n a l Euc l idean space of points x = (x l . . . . . Xk) with rea l coo rd ina t e s , and let A k = [0, 27r; . . . ; 0, 2u] be a k - d i m e n s i o n a l pe r iod ic cube. Le t L(. k) = L~)(Ak) (1 -< p < or denote the space o f al l m e a s u r a b l e funct ions f(x) on Ak hav ing pe r iod 2~ in each of~the var ' iables x i (i = 1 . . . . . k) and such that

IlsE<;)----tl/ll~ = { f l/('<)l~dx)~"< ~ Ak

Furthermore, let

E . . . . . . , (])L(pk) = inf It f (x) - T ........ k (x)lls(p~)

be the comple te bes t a p p r o x i m a t i o n of f by m e a n s of t r i g o n o m e t r i c po lynomia l s of o r d e r not exceeding nj in the v a r i a b l e x i (i = 1 . . . . . k). In addi t ion , c o n s i d e r the pa r t i a l bes t app rox ima t ions o f

E~? ~ (/)L~) = ~ It/(xl . . . . . x,) - - T~ Ix, . . . . . x,-1, (x0, x,+l . . . . . ~ , ] [L~) '

w h e r e T = Tn[x I . . . . . Xi_l, (xi) , x i + i . . . . . Xk] is a t r i g o n o m e t r i c po lynomia l in x i of o r d e r _<n with c o - ef f ic ients in L(k- i ) . I t is well known (see, fo r example , [1], p. 44) that*

P m ~ E(~, ~ (I)L~) ~< E . . . . . . . . ~ (I)%~) (l < p < oo), (1.1)

k ~-~ E(i)~i' "~ oo). E ........ ~ (i)~(~)_ ~< c (k, p) ~ ~. ~ .~ ( / ) (l < p < (1.2)

If ~l(x), ~k(X) a r e cont inuous d e c r e a s i n g funct ions on [1, o~) such that lira ~i(x) = 0 (i = 1 . . . . .

k), then we denote by Ep ~l . . . . . Zk (1 ~ p < o~) the c l a s s of al l funct ions f(x) E Lp (k) fo r each of which

E (~)~,| (t)L(k) = 0 {~(n)} (i = l , . . ., k).

Le t us in t roduce one m o r e definit ion�9 If f(x) E L (k), then by the pa r t i a l modulus o f cont inui ty o f o r d e r

of f with r e s p e c t to x i in L~ ~'),''" we m e a n the funct ion (0 __% 5 1)

r 6) @)---- sup ]A~%t(x)ll.(~), Lp o<a<~ ~p

w h e r e

('~ x . . . . + ] h , . . . ,x,). (1.3)

Now cons ide r the k - d i m e n s i o n a l unit cube I k = [0, 1 ; . . . ; 0, 1], and let Lp(Ik) (1 ___ p < ~) denote the

s p a c e of a l l m e a s u r a b l e funct ions f(x) on I k such that

* Hence fo r th C (a , fi . . . . ) denotes a pos i t ive cons tan t depending on the p a r a m e t e r s a , f l . . .

Translated f r o m Sib i r sk i i M a t e m a t i c h e s k i i Zhurna l , Vol. 14, No�9 4, pp. 766-790, Ju ly -Augus t , 1973.

Original article submitted July 6, 1972.

�9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West t7th Street, New York, N. Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

530

If f(x) E Lp(Ik) , then by the modulus of cont inui ty of f in Lp(Ik), we mean the function (6 E i k)

i - -h l ~--h~ l ip

(I)P(];~)~(I)P(/;~I . . . . . ~A) : 0~h,<5~sup { ! . . . ! l](xi +hi . . . . ,x~+hQ-](x, . . . . . xk)[vdxi...dxr,} J0~ht~8/~

By the pa r t i a l modulus of cont inui ty of f with r e s p e c t to x i (i = 1 . . . . . k), we m e a n the funct ion (0 s 5 s 1)

i f f } ({) [ o)p ( / ; 5 ) = s u p . . . . . . IA,.h](x,,.. x~) dx,...dx~ O~h~b ~. 0 O

I t is e a s y to see that the following inequal i t ies a r e valid:

(~) max0~p (f ;&)~<op(f;8~, . . . ,6 ,) " ~ (~)

A n o n d e c r e a s i n g cont inuous funct ion w(6) on [0, 1] tha t sa t i s f i e s the condi t ions ~(0) = 0 and w(5 + v,) -< w(5) + w(~?) for 0 -< 5 -< 6 + V -< 1 is ca l led a modulus of cont inui ty. If wl(5) . . . . . Wk(5 ) a r e modul i of cont inui ty , then le t H~I . . . . . wk (1 -< p < r162 denote the c l a s s of al l functions f(x) E Lp(i k) fo r each of which

i) (f; co 5) : O{wi(5) } (i : I . . . . . k).

The m a i n p u r p o s e of this p a p e r is to find condi t ions on the functions ~i(x) . . . . . ~k(X) and the modut i of cont inui ty COl(6 ) . . . . . COk(~ ) that gua ran tee the embeddings (1 -< p < q < ~)

E ~ ...... ~ c L (~), (1.5) P

Hp c Lq (I~). (t .~)

In the p a r t c o n c e r n i n g the embedding (1.6), this p rob l em cons i s t s in extending the r e su l t s of Ul 'yanov (see [2], pp. 675-677) to funct ions of s e v e r a l va r i ab l e s . T h e s e ques t ions w e r e inVestigated e a r l i e r in a s e r i e s of p a p e r s by d i f f e ren t au thor s (see [3-6]). Indeed, T i m a n [3] es tab l i shed the fol lowing p ropos i t i on :

T H E O R E M A . L e t f ( x ) EL(p k) ( 1 - p - < 2 ) a n d p < q < oo. Then:

1) if the condi t ions

Z nk~/p_(k+1 ) //?(i) , . . . . (/)L(k) )~ < ~o (i : t . . . . . k ) , . p

or the equiva len t condi t ions

(~=~ . . . . . k)

a r e sa t i s f i ed , then f(x) E L(q k) ;

2) if fo r somec~ i > 0 ( i = 1 . . . . . k), o q + . . . + e k = 1 ,

~/~-~(E~!~ (/)~))~ < oo (~ = ~, , k)

Or

then

c~

~ - - T ' ~ = i . . . . . k ,

/ (x) 0 r. '~' ~ q .

(1.7)

(1 .s)

The suf f ic iency of condi t ions (1.7) fo r embedding (1.8) for p > k q / ( k + q) can be concluded f r o m T h e o - r e m B below with the help of an inequal i ty o f T i m a n (see [I], p. 363). This r e s u l t is conta ined in an a r t i c l e by Pandzh ik idze [5]. L e t us r e p h r a s e T h e o r e m A f r o m the point of view of the embedding of the c l a s s e s Ep~i . . . . . )~k.

531

T H E O R E M A ' . I f l - O, e l + �9 �9 �9 + e k = 1 )

y xqX"-Ut ? 'q ( x ) dx < t.

a r e sa t i s f i ed , then embedd ing (1.5) holds .

(~ = , l . . . . . k) (1 .9 )

(~ = ~ . . . . . k) ( 1 . 1 0 )

The next t h e o r e m was obta ined by Pandzh ik idze [4] and independent ly (although in a somewha t d i f fe ren t form) by T e m i r g a l i e v [6].

T H E O R E M B . L e t l - < p < q < ~ . If

2 r~kqLl,-(k+i)ia)iq(~/n ) < oo (i = i , . �9 �9 k), (1.11)

then embedding (1.6) holds .

Thus c e r t a i n suf f ic ien t condi t ions for embeddings (1.5) and (1.6) a r e a l r e a d y known. However the ques t ion of how p r e c i s e these condi t ions a r e has r e m a i n e d open, and fu r the r inves t iga t ion is r equ i r ed in connec t ion with this ques t ion . The p r e s e n t pape r is devoted to this inves t igat ion. It will be shown below that if the o r d e r s of s m a l l n e s s of the modul i of cont inui ty r ) . . . . . Wk(6) a r e equal , then condi t ion (1.11) is n e c e s s a r y for embedding (1.6) (where p ~ k q / ( k + q)). A s i m i l a r a s s e r t i o n (under an a s s u m p t i o n on ~i(x)) is e s t ab l i shed for condi t ions (1.9). In the r e m a i n i n g c a s e s T h e o r e m s A ' and B, gene ra l ly speaking , can be s t r eng thened . Indeed, in this a r t i c l e we obtain m o r e p r e c i s e condi t ions on the s y s t e m s of functions { l l (x) . . . . . ~k(X)} and {r . . . . . C0k(5)} which a r e not only suff ic ient , but (under c e r t a i n c o m p a r a t i v e l y genera l r e s t r i c t i o n s ) a l so n e c e s s a r y for embeddings (1.5) and (1.6).

We note that the p r e s e n t a r t i c l e uses the me thods of Ul 'yanov (see [2] and [7]) r e l a t ed to the c o n s i d e r - a t ions of e q u i m e a s u r a b l e funct ions .

The au tho r is deep ly g ra te fu l to l~. A. S torozhenko , under whose guidance the p r e s e n t p a p e r was c o m -

pl eted.

w A u x i l i a r y A s s e r t i o n s

LEMMA t . Le t f(x) ~ LpClk), and fo r na tu ra l n i (i = 1 . . . . . k) le t

(x )= tp . . . . . . . . k ( x ; f ) = h i . . . n ~ I f(t) dt--=a ........ k

fo r x E Q v l . . . . . Vk = { v l / n l -< xl < (vl + 1 ) / n l ; " " " ; v k / n k -< xk < (vk + 1 ) / n k } ' w h e r e v i = 0 . . . . . n i - 1

(i = 1, . . . . . k). Then l l f - r IIp -< 2k/pa~p(f; n{ 1 . . . . . nkl)-

This l e m m a extends U l ' y a n o v ' s L e m m a ([7], pp. 106-107) to the k - d i m e n s i o n a l c a s e and can be p roved

in p r e c i s e l y the s a m e way.

LEMMA 2. L e t f (x )ELp(Ik) , p E [ 1 , or Then :

1) if p = l , then for n a t u r a l n i ( i = l . . . . . k ; n l . . . n k - > 2 )

c~ (1; l /n l . . . . . l /nk) > 2 k+l (k -F t) EcIk ~:~s,~ [El=(n~...~k)_ ~ E IEl=(n~...nk)-,

2) i f p > 1 and the n i a r e na tura l n u m b e r s (n 1 . . . n k -> 2), then

(n,... E C I k

IEl~(n~...nk)-~

- - s u p i n f ~ [ f ( x ) [ d x ~ - - 2 - ( ~ l ~ + ~ ) D p ( n l , . . E~CZ k EC.Ez E

,) J

IE~f=2ln l ' "nk) - I IEI = ~-- led

t D (n 1 . . . . . nk; f); I II(x)]dx} ~ 2k+l(k-t- t~) E

11 (x) I ; ~

., n~; t).

532

L e m m a 2 is the k - d i m e n s i o n a l a n a l o g of U l ' y a n o v ' s L e m m a ([2], pp. 652-655) . The m e t h o d of p r o o f is b a s i c a l l y the s a m e .

P r o o f . I t i s e a s y to s e e tha t

T h e r e f o r e i t s u f f i c e s to c o n s i d e r the c a s e f(x) _> 0. L e t b 1 . . . . . by (P = n I . . . n k) be the n u m b e r s w r i t t e n in n o n i n c r e a s i n g o r d e r . If p = 1, then by v i r t u e of (1.4)

aPi,..., Pk

f --1 . ( i ) 1 n*--2 a*--I n.~--t

i=1 ~1=0 v,~=O vk:O

D(n nk; W), + / , " " s ~ ........ k-l'~k+l--a ....... / k ~ - - k

~1~0 Vlr v~ =0

and by u s i n g L e m m a 1 we ob ta in

D(nt , . . . ,n~; l )<~D(n~ . . . . . n k ; r sup [ I I ( x ) , ~ ( x ) l d x

IE/=i lV

~< ko, (tb; n,-* . . . . . na-') + 2~+'o,. (1; n, -~, . . . . n~-') ~ (k + t) 2~+'e, (]'; n [ * . . . . , n ~ " ) .

Bu t i f p > 1, then (see (1.4))

e~(r hi-',..,, no')/> ~ " (~; n,-' . . . . . n~-')

n~--2 "n~- t nk~t

. . . . . j-. ~ ( n , . . . n ~ ) - ' / * ( b , - - b ~ ) . vt:O ~=0 v~=O

L e t B 1 and B 2 be d i s j o i n t s e t s such tha t IBi[ = [B21 = (n 1 . . . n k ) - i , r = b~ f o r x ~B 1 and r = b~ f o r x 5B2. O b v i o u s l y ,

[ ~)(x)dx = b~.(n~.., n , ) - t s u p

E~Ih [E[=I/~ E

F u r t h e r m o r e , if E 1 = B i U t ~ , then

[E~] = 2(n~:.. n,) -~ and inf [~p(x)dx = b~(n~.., n~)-~; E~E~,[E[=JE~]/2

and since it follows from the definition of b i that ~(~) _1 (n . . . . m)~-ilPDv(n . . . . . , nk; W)

and by v i r t u e of the M i n k o w s k i and H o l d e r i n e q u a l i t i e s and L e m m a 1

Dp(n~ . . . . . nk; f )<~Dp(n~, . . . ,n~; W ) + 2 sup [ I f ( x ) - - W ( x ) i d x

IEl=i/~

~ ( ~ ; n J ~ . . . . . n Z l) ( m . . . m)~/~-' + 2 ( n , . . . n J ' /~ -W - ~ li~ ~ 2~/~+%~ (f; n j ~ , . . . . n [ ~ ) (n . . . . n~) ~/~-L

T h i s p r o v e s the l e m m a .

Def in i t i on . T h e nonnega t i ve func t ions f(x) and g(y) de f ined and m e a s u r a b l e on the cubes Ik and I m r e - s p e c t i v e l y a r e s a i d to be e q u i m e a s u r a b l e if f o r any r e a l n u m b e r t

rues {x: f(x) > t} = rues{y: g(y) > t}.

I t i s obv ious t ha t if ~(x) and g(y) a r e e q u i m e a s u r a b l e and fE Ll ( Ik) , then gE Ll ( Im) and ~ f ( x ) d x = ] g (y)dy .

I t i s a l s o e a s y to s e e tha t i f a nonnega t i ve funct ion F(u) is m o n o t o n e on [0, ~o], then , s i n c e f(x) and g(y) a r e e q u i m e a s u r a b l e , so a r e F(f(x)) and F(g(y)) .

* If n i = 1, t hen the c o r r e s p o n d i n g t e r m in th i s s u m is a s s u m e d to be z e r o .

533

LEMMA 3. F o r any nonnega t ive m e a s u r a b l e funct ion f(x) on lk, t h e r e ex i s t s a n o n i n c r e a s i n g funct ion go(x) - ~p(x; f) on the s e g m e n t [0, 1] e q u i m e a s u r a b l e with f (x ) .

P r o o f . Set

qo(x)= sup infvra i ] (x) ( 0 < x ~ i ) , q~(0)=limq0(x).

IEI=x

The l a t t e r l i m i t e x i s t s , s i nce go(x) is n o n i n e r e a s i n g on (0, 1]. L e t us show that for any t _> 0

mes{x: q)(x) > t} = mes{x: /(x) > t}.

Indeed , if q~(x) _< t fo r a l l xE [0, 1], then f(x) -- t a l m o s t e v e r y w h e r e on [k- Now le t 4 t = (x : r > t} ~ 0 a n d x 0 = s u p ~ t . T h e n x 0 > 0. F o r each n a t u r a l n > X o t w e h a v e go (x0-1 /n ) = t + (fn, w h e r e a n > 0, and we can find a s e t En ~ I k wi th m e a s u r e [Enl = x 0 - 1 / n such tha t f(x) > t f o r x E Ea. T h e r e f o r e

levi------rues {x: /(x) > t} >~xo = Ir

On the o t h e r hand , i f we had [Ft] > x 0, then , s i nce

F~ = (x : f (x) > / t + ~ } U ( U {x : t + f / ( n + l ) < / (x) < t + l/n}),

we could find a s e t E 0 = I k such tha t [E 01 = xl > x0 and inf v r a i f(x) > t. T h e r e f o r e we would have go(xi) > t, x~Eo

which would c o n t r a d i c t the def in i t ion of x 0, s i nce x l > x 0.

LEMMA 4. L e t the nonnega t ive funct ions f(x) E Ll(I k) and g{y) E Ll( Im) be e q u i m e a s u r a b l e . Then for any 0 < a -< l / 2 and i = 1 . . . . . [a -i]

sup inI i ](x) d x = sup inI i g(y) dy" E~CI k EcEt ~ EiCI m Ecgt~-

P r o o f . L e t go(x) = r f) be the funct ion c o n s t r u c t e d in L e m m a 3. L e t us f i r s t show tha t ict

sup inf I (p(x)dx= i ~(x)dx ( i = l . . . . . [a-l]). (2.2) E i c [ ~ ECEi ~ (i--.~ I g i l = i a IEl=a

I n d e e d , l e t E i ~ [0, 1], [Ei[ = i a . Set i

A~ = [ (~ - l ) ~, ~al, tt~ = E, \ U &"

T h e n go(x) _< ~( ia ) for x E R i , and so

i

But i f E i = f~ Aj , then j=!

i n f ! ep(x)dx<~ S q~(x)dx+((z-lE~OAil),p(i,~)~ i ep(x)dx.

IZl=r

inf ~r = g~Ei g (i--t)~ IE[=~

r (x) dx.

= {x:~(ia) < go(x) < (p((i-l)a)}, G~ ~ = {x:r < f(x) < r Obviously, Now le t A~

I (p(x)dx= I f(x)dx- (o) A~ O) G i

F u r t h e r m o r e s e t A (i) = (x : q~(x) = q~(ia)} N Ai , A~ 2) = {x : q~(x) = ~ ( ( i - 1 ) a ) } N Ai. and choose s e t s ' i

G, , )= lx : / (x )=(p ( i c r IG~('~I=IA~ l) I, (2) (~)

G~(Z)=ix: f (x)=~(( t - t )a)} IG~ t=lA~ I.

(Such a cho i ce i s p o s s i b t e s i nce f(x) and ~p(x) a r e e q u i m e a s u r a b l e . ) T h e n A i = A(~ U A! t)t U A! ~)t and. s e t t i ng

G i = G. (0) U G! l) U G!2), we have i 1 1

534

As above for ~o(x), we see that

~ (x)dx = S ~ (x)dx. A 1 o t

sup inf / f ( x ) d x = I f(x)dx.

l~:~l=ict IEl=a~

To comple t e the p r o o f we note tha t r f) = r g) e v e r y w h e r e excep t pe rhaps for a countab le se t of poin ts . Indeed , l e t ~9(x; f) be cont inuous at x 0 E (0, 1]. If we had cz(x0; g) < 9~(x0; f), then we could find an s > 0 such that r f) > ~v(x0; g) and m e s {x : ~v(x; f) > ~(xr g)} _> x 0 + ~ > rues {x : ~(x; g) > r 0' g)} for all x > x 0 + a, which is imposs ib l e . S imi l a r ly , we see that the inequal i ty r g) > ~(x0; f) is imposs ib l e .

LEMMA 5. L e t wl(5) . . . . . r ) be s t r i c t l y i n c r e a s i n g cont inuous funct ions on {0, 1] such that wi(0) = 0 (i = i ..... k). Then there exist a number 0 < ao~ -< 1 and nondeereasing continuous functions ~i(5),

.... Gk(6 ) on [0, i] that satisfy the conditions

O~a~(6) ~<t for 66 [O, t ] , a~(O) = 0 ( / = i . . . . ,k ) , (2.3)

~ , ( ~ ) . . . ~k(8) = 5 , ~ ~ [0, l ] , (2 .4)

~ i ( a i ( ~ ) ) = ~ = ~ ( ~ ( 6 ) ) fo~ ~ ~ [0, aol . (2 .5 )

P r o o f . Without loss of gene ra l i t y , we can a s s u m e that r ) _< . o . _< Wk(1). F o r xE [0, wi(1)] le t w*(x) co inc ide with the i n v e r s e funct ion of wi(6), and le t wi(x ) = 1 for xE [~i(1), ~k(1)] (i = 1 . . . . . k). Set

~'(x)=Hco,'(x), 0 < z < o ~ ( t ) .

The function ~]*(x) is s t r i c t l y i n c r e a s i n g and cont inuous on [0, r and its s e t of va lues is the s e g m e n t [0, 1]. Le t ~(x) denote the funct ion defined on [0, 1] tha t is the i nve r se of fl*(x). The se t of va lues of ~(x) is the s e g m e n t [0, r Set

a, (a) = o / ( ~ ( 8 ) ) , 0 ~ < 6 < 1, ~ = 1 . . . . . k.

Obvious ly , al l the ai(5) a r e n o n d e c r e a s i n g and eon thmous . Since 0 <- ~ ( x ) _< 1, we have 0 -< Gi(6) <- t . I t is a l so c l e a r tha t ai(0) = 0. F u r t h e r m o r e , c q ( 6 ) . . , ak(6) = ~ ( ~ ( 5 ) ) . . . r = ~*(~2(6)) = 6, i .e . , (2.4) a l so holds . F ina l ly , se t t ing aw = fl*(r we see that 0 -< ~(6) -< r for 0 -< 5 -< a~ , and. by v i r t ue of

�9 * the def ini t ion of the func tmns w i (6), fo r 6 E [0, aw]

~ ( ~ , ( ~ ) ) = ~ , ( ~ o ; ( ~ ( 6 ) ) ) = .q(~) , ~ = ~ . . . . . ~. (2 .~)

This p r o v e s Eae l e m m a .

R e m a r k 1. Since the funct ions c~i(6) a r e n o n d e c r e a s i n g , it follows f r o m (2.4) that the funct ions 5 / a i ( 6 ) 5 ~ [0, 1], i = 1 . . . . . k a r e a l so nondec reas ing . As is well known, this impl ies that the funct ions c~i(5) a r e s emiadd i t i ve . T h e r e f o r e , taking into accoun t the p r o p e r t i e s of ai(5) noted above , we can a s s e r t that these funct ions a r e modul i of cont inui ty .

R e m a r k 2. L e t ~wl(6) . . . . . Wk(6)} be a s y s t e m of s t r i c t l y i n c r e a s i n g modul i of cont inui ty . Then the funct ion fl(6) c o n s t r u c t e d in L e m m a 5 will be ca l led the m e a n modulus of cont inui ty for this s y s t e m . It is e a s y to s ee that ~2(6) is a l so a modulus of cont inui ty . Indeed, fl(6) is cont inuous and s t r i c t l y i n c r e a s i n g on [0, 1] and ~2(0) = 0. In addi t ion, if wi0(1) = m a x o f ( l ) , then ~(5) = ~i0(Gi0(6)), 6 ~ [0, 1]. But k~e funct ions

wio(6 ) and ai0(6 ) a r e semiadditive; t h e r e f o r e so is ~(5).

R e m a r k 3. If the funct ions wl(6) . . . . . Wk(6 ) a r e i n c r e a s i n g in the b r o a d s ense , then an appl ica t ion of L e m m a 5 to the funct ions (1 + 6)wi(6) y ie lds the ex i s t ence of funct ions ~1(5) . . . . ; ak(6 ) that s a t i s fy c o n - d i t ions (2.3) and (2.4) such that

r I o~(a~(5) ) = 0 ( t ) (0 < 5 < i, 1 ~< ~ < k, ~ < ] ~< k).

LEMMA 6. (Ul 'yanov [2], p . 660). L e t the n u m b e r s q > 1 and a E ( l - q , 1) be given~ Then if r is a modulus of cont inui ty and

E n-~o~.(t/n) .= ~,

535

we can find numbers Bn such that:

1) B~.O,B,~o)(t /n);

n

2) Z Bm=O{no(i/n)}; r a ~ l

3) ~ 2~(~-~)(BP - B w ' ) q = r n ~ t

w Embedding of the Classes E~I ..... Zk

In this section we study the question of necessary and sufficient conditions for the embedding (1.5).

We shall first prove an auxiliary theorem, which is of some interest in itself.

THEOREM i. Let the nondecreasing sequences of natural numbers ~(I) ~ ~ as n ~o (i = 1 ..... k) n

be such that the sequence v n = yn "). . . ~n t~)~" satisfies the conditions

Vo = l; t ~ r ~ v , + , / v ~ ~ s (n=0 , ~,.. .). (3.1)

le t 1 -< p < q < ~ and f(x)~Lo(k). If Fu r the rmore ,

~/p-~ q ( 3 . 2 )

n ~ 0

then f(x) ~ L~ ) and

tllll~ ~ C(p, q, k, r, s) ( l l l lh + D'~').

Proof . Let ~o(x) be a nonincreasing function on [0, 1] equimeasurable with If(2nx) I (x EIk), and let

T~ (I) ~ (k)(x) be the trigonometric polynomial of order (~(n I) ..... ~(n k)) that realizes the best approxi- n ,... , n (k~

mation of f(x) in the metric of L~ ". Then (see Lemma 4) for n = 0, 1 .... and ~(x) =- I~(t)dt 0

Ec~',~I~I--~* ~'clkf.~.l=.,7~ 1

_ : r ( , ~(~) (2~x) I dx + sup f [ T(,) ~(~) (2~,01dx_ A~" + A~ ) . . . . . . . Ec~.k, 1~l=~Ta ~ ~ . . . . . . . . �9

Using Holder ' s inequality, we find that

(2n) k/p f ( x ) - T (1) ~(k) (x) rdx */p �9 ,~ . . . . . . (2n)k/p Ej~) ...... (~) (])5(k) .

For the es t imate of A~ ) we use an inequality of Nikol 'skii ([8], p. 256) for t r igonometr ic polynomials:

I k , ,1/p

I IT . . . . . . . . kllc -~- m a x [ T . . . . . . . . ~ (x) ] < 2k ( l I 1 nl ) tl T . . . . . . . . ,~ IlL(p '~). x E A k =

For n = 1 , 2 . . . . we obtain (see (3.1))

. . . . , 1 , " " , m - ~ l , " " , m - } - i

n - - 1 n - - 1

< 2%XI{]] T~ ..... 1[l* + 2 v'/ '~*[I T~a) ~) • T ( ~ ) ,(k) IIv} ~ - * v~,,+,( l lT ~) ...... ~,--flip , " r e + l , ' " , m + l ,

m ~ 0 ? R ~ O

" ' " m ~ 0 v m , - " , ~ m L p j

Taking into account the es t imates found for A(n l) and A~ ), we obtain (see also (3.1) and (3.2))

ti 1 ~1/q

0 0

536

{ i - y l q/p_l q jl/q <(8-- ~.)~/qs~-l/q(2~) ~/q (2~)-~illt]~ T v ~ , Er ..... ~<~;(l)L(~) (2~) -~/"

oo

+ 2 , + 1 , 1 , [ E , V ~ I ( E I p 'qII/q Dl/q ,~i /q) , ~:~ EJn j,)(l)~<~)) | } ~ Cl(ll/!h + +

where C I is some constant, and

A ~ v~, ~ ~/~ q ":m E4> ' , ~ ) (I1~> .

We estimate A by using Holder's inequality and condition (3.17 :

n ~ l m ~ 0 m : O n ~ l m=O

C (q, r) E qm-'/, q -", s ~ , ~-~ ~:q/~-1 E = ~:n--1 Ev(1) , jk> (1) (k) V~ -~ C' (q, r, j / ~ ~ j~) jk) if)L ( ) �9

n : l n - - l ' """ n--1 Lp ~ n , , " ' , n ph m ~ n ~ 0

Tak ing into accoun t the e s t i m a t e for [[fl[q obta ined above, we see that the t h e o r e m is val id.

Note tha t by v i r t u e of inequal i ty (1.2) the fol lowing t h e o r e m a l so holds :

THEOREM 1 ' . L e t the sequence u(s ~) (i = 1 . . . . . k) sa t i s fy the eoMi t ions of T h e o r e m 1, and let f(x) (~L(k), w h e r e 1 < p < q < ~ . If

P oe

D~ ~E~/~-~ [E(~l) (]).(~)l~< ~ (~ = i . . . . . k), (3.35

then f(x) E Lp (k) and

k

i=t

Let 2i(x) ..... ~k(X) be continuous, strictly decreasing functions on [I, ~) such that lira ~i(x) = 0

(i = 1 ..... k). An application of Lemma 5 to the functions c0i(6 ) = ~i(6 -i) (0 < 5 -< I) yields the existence of a system of continuous increasing functions pi(x) ..... Pk(X) on [i, ~) that satisfy the conditions

v J x ) ~ as x->-~, v~(x) >1t for x ~ i ( i = t . . . . . k), (3.4)

~ ( x ) . . . ~ , ( x ) = x , x~ [t , o~), (3 .5)

k,(v~(x)) = . . . : ~ ( v ~ ( x ) ) = A ( x ) for x / > a ~ t . (3.6)

The function A(x) decreases strictly to zero as x ~ ~, is continuous on [I, ~), and is the inverse of the function

k

A.(t)=H~,,'(t), o<<.t<-< max ) , , ( i ) , i = t

w h e r e ~/(t) co inc ides with the i nve r se funct ion of hi(x) on [0, Zi(1)], and l*( t ) = 1 for tE [~i(1), max ~j(1)].

The funct ion A(x) is in s o m e s e n s e a m e a n for the funct ions ~l(x) . . . . . 2k(X).

THEOREM 2. L e t 1 < p < q < ~ . If

S xq/~-2Aq(x)d x < o% (3.75 1

537

then

EX,,...,~. ~ L (*). P q

P r o o f . Set v ~ i ) : l a n d ~( i )= [ v i ( 2 2 n ) ] + l f o r n = l , 2 . . . . ( i = l . . . . . k).

2 2n ~. V,~i)... V: h)= vn ~- (Vl (2 2~) -- 1). . . (Vk(2 z") + l) ~< 2 ~ -- O (2 ~=) < 2 ~+~

T h e r e f o r e 2 _< V n + l / " n -< 8 f o r n _> n 0.

L e t f(x) 6 E ~ i . . . . . ~k. Then for i = 1 . . . . . k (see (3.6) and (3.7)) P

[E(:!, ~ (/)L:)] q = 0 2~"(q/P-~'k~*(v,(22")) = n = 0 n = 0

and by v i r t ue of T h e o r e m 1' f(x) E L k.

THEOREM 3. L e t l _ p < q < o~ If fo r s o m e a > 0

x~Ei(x)f as xto~ ( g = l . . . . . k), (3.9)

then condi t ion (3.7) is n e c e s s a r y for embedding (3.8).

P r o o f . Le t

i xq/P-~Aq(x) dx = ~. (3.10) 1

L e t us p rove the ex i s t ence of f(x) 6 E~i . . . . . Ak belonging to L (k).

Tak ing into a c c o u n t the fact that Zl(x)x~ and x / v l ( x ) a r e i n c r e a s i n g (see (3.5)) and equal i ty (3.6), we s ee that fo r x _> a~ the funct ion A(x)x ~ is a l so i nc r ea s ing . T h e r e f o r e

A(x) ~< 2~A(2x) (x1> a~). (3.11)

Le t us choose a na in ra l n u m b e r ~ so that the condi t ions

n~ >~a~, max vi (n , )< 1/~6 (3.12)

a r e sa t i s f i ed , and le t us c o n s t r u c t a sequence of na tu ra l n u m b e r s {nm} by se t t ing n m + i (m = 1, 2 . . . . ) equal to the s m a l l e s t na tu ra l n u m b e r N such that Z (N) < 2-(~A(nm). Then

A(nm+;) < 2-~h(n~), (3.13)

A ( n ~ + , - t ) /> 2-=A(n~) (m = 1, 2 . . . . ). (3.14)

Since (see (3.14) and (3.11))

aa n m + t ~

n l m = i nra m = t m = ~

we have (see (3.10))

q / P - - I q n,~ A (nm)= ~.

m = i

F u r t h e r m o r e , the fol lowing inequal i t ies hold (m = 1, 2 . . . . ):

~ ( n ~ ) < ~ ( n ~ + , ) (i = 1 . . . . . k) .

Indeed, o t h e r w i s e we would have (see (3.6), (3.12), and (3.9))

A(nm+,) = L(~(n~+i)) > %~(2v~(nm)) >~ 2-~k~(v~(n~)) = 2-~A(n~),

which would c o n t r a d i c t (3.13). F r o m (3.16) and (3.12) fol lows the inequal i ty

- ' ( r~) < '18. max vi

(3.s)

Then (see (3.5) and (3.4))

(n >/no).

(3.15)

(3.16)

(3.17)

538

We i n t r o d u c e the fo l lowing n o t a t i o n (m = 1, 2 . . . . ):

L - t (0 - t a , , = 4 max v, (nj), r~ = ~ , (n~) ( i = ~ , . . . , k),

By v i r t u e of (3.17), Qm c Ak (m ~ 1, 2 . . . . ) and Qm i n Qm2 = 0 fo r m 1 ~ m 2. s e t

]<x): tH,a t - -~ t Xir~>am for x~Qra(m-~t, . . . . )

0 at the other points of A~,

Note that here we use the idea of the construction performed by Ul'yanov in his consideration of similar questions ([2], p. 676).

First of all let us show that f(x) E L(n k) but f(x) ~ Ln(k). Using LiouviHe's formula (see [9], p. 397), we find that (see (3.5))

/q(x)dx = H ~ q r (k /2 ) r . . . . r= ( i - = C(k, .

T h e r e f o r e ( see (3,13) and (3.15))

t[/l]~= C(k,p) 2 Ho,~n~ ' = C(k,p) A~(n=)<

1711:= C(k, q) ~ ~/"-'-~"

L e t us now e s t i m a t e the b e s t a p p r o x i m a t i o n s of f(x). F o r th i s p u r p o s e we f i r s t c o n s i d e r i t s p a r t i a l m o d u l i of s m o o t h n e s s of o r d e r v = 2 [a ] + 3, and then we u s e J a c k s o n ' s i n e q u a l i t y ( s ee [1], p. 288):

~.<<.C(k,.~) ~ ' ~ ( I ; " ' (~' ,~, )%. (3.19) E vl,...,~ ~ t '$" p

i=1

L e t

and b y v i r t u e of (3.20)

1 / ~(n,+~) <<.vh<l/v~(n,) (s >~ 1). (3.20)

= a m - r . T h e n f o r m = l , 2 . . . .

- r ~ o) ~ o) ( 3 . 2 1 ) O) ~ O~ _~4 max r~ ~) .. >t~r~. v m - - t t m+ t ~'m-b t �9

b(" + vh b ( r e s p e c t i v e l y xj > g ) fo r s o m e j , o1" if b~)+ ~ < x i < b ~ ) f o r a~c l e a s t one i , t hen x.~ Qn (n = 1, 2 . . . . ) and f(x) = 0. T a k i n g th is r e m a r k and (3.22} into

a c c o u n t , we o b t a i n

~ 2n

. . . . . . h f(x~ . . . . . ~k) dx~...dxk 0 0

-(~)

0 0 0 1

g(1) ~k)

0 0

2n 2a

539

Since the s equence A(nm) is l a c u n a r y (see (3.13)), by v i r t ue of (3.18), (2.20), and (3.9)

~l)l b-<~ 1

0 0 m~s-~-2 Qm m=s -}-1

L e t us now e s t i m a t e AN(h). L e t Q~) denote the o r thogona l p r o j e c t i o n of Qm onto the hype rp lane x 1 = 0, and

l e t x} m) = xt(m)(x2 . . . . . Xk) and x(m) = ~ m ) ( x 2 . . . . . Xk) be the roo t s of the equat ion

r~ ) ~ - " " ~- k r~ ) = I (xlm)~ ~Im)), (3.23)

w h e r e the point (0, x~, . . . , x k) E Q~) is f ixed. The equal i ty A (1)u, ht.(x ) A = 0 holds in the fol lowing c a s e s : �9 ~ ( , ) -

1) (x l . . . . . Xk) ( Q and (x. 1 + uh . . . . . Xk) ( Q m ; indeed, in this c a se u , h [(x) is a f ini te d i f f e r e n c e of o r d e r 2[c~] + 3 of a po lynomia l of d e g r e e 2[c~] + 2;

- ~ ( 1 ) 2) (0, X~,.. . ,Xk) t ~ , but [xl, x t + v h l , , n t •

3) (0, x2 . . . . . xk)(~O~) ( m = i , 2 . . . . ).

But if (0, x 2 . . . . . Xk) E Q(m 1) and ~ m ) _ vh -< xl -< xl(m), then for m -< s (see (3.23) and (3.20)

IAX!hJ(xl . . . . . xk) l'~Hmi~=o(;)ll--{(xl-[-]h--am)2r~) J r - . . . + (xk-~-a")~}l[a]§ = H m i ~ o ( V ) l j ~ r~) )

v - - 1

r~ln ) = H~ (r~)) ~-~ I(x[ ~) - - xt - - ]h) ( j ~- x~ -}- ]h - - 2a,~)]-{- 1=0

v - - I v - - I 1 - - v v - - I

< C1Hm (r~))~-*h - V (2r~) ~- 2vh) -Y" % C~It,~ (r~)) ~ h

Obviously, the resulting estimate holds in the case (0, x 2 ..... x k) e q(m I), ~m) _ .h _< x i -< x~ m) (m -< s).

Thus (see (3.20) and (3.5)) s

= �9 . . "'" ~ , h ] ( x ~ . . . . . z~) I" d x ~ . . , d z k }

_ _ v - - I s

: 0 (h p -E-) h I Q~)I tI,~ p (r~)) p ~ = 0 (h ~ --7-) h r ~ l . . , r~)nmA~ (nr~) (r~)) ~ ~

~ - - i s v - - I

= O ( h P ~ - ) 2 )~l~(vl(n~))(vl(n~))P -E- .

But fl = [ ( u - 1 ) / 2 ] - o ~ = [c~] + 1-o~ > 0. T h e r e f o r e (see (3.9), (3.16}, and (3.20))

~--i s v--i v - - i

A" (h) = 0 (h ~ -~) ~i p (%'1 (Yts))(u (ns))Pa ~ (u (nm))P~ : 0 (h" -{-1 ) ~ (v~ (n~)) (h (n~))" --V

= 0 (h v -C) )hp \ vh ] = O {k~ (h-~)}.

Thus )L(p k) = Obvious ly ,_ for any 1 -- i -< k

(o(~ ~) (f; 5)L(} ) : 0 {)~i (5-~)}. (3.24)

H e n c e , us ing inequ~Ii ty (3.19), we deduce tha t

and (see (i.I)) E(i)oo(f)Lp (x)n, = O{~i(n)} (i = 1 ..... k), i.e., f(x) ~E~l ..... Zk This proves the theorem.

R e m a r k . It is e a sy to see that T h e o r e m s 2 and 3 r e m a i n val id if the funct ion A(x) in condi t ion (3.7) is r e p l a c e d by one of the funct ions Zi(#i(x)) (i = 1 . . . . . k), w h e r e the #i(x) a r e cont inuous n o n d e c r e a s i n g funct ions on [1, ~) tha t s a t i s fy condi t ions (3.4) and (3.5) and such that

540

~ , ( g , ( z ) ) / ~ j ( ~ j ( x ) ) = O ( l ) ( x ~ [ l , ~ ) , l < i , ] - < < k )

(see R e m a r k 3 to L e m m a 5).

To conc lude this s e c t i on we s ta te s o m e c o r o l l a r i e s to the t h e o r e m s that we have p roved .

COROLLARY 1. L e t 1 0 be r e a l n u m b e r s . The embedding

xa, ,,x ak ~ L(~ k)

holds if and only i f

a = ( l / ~, + . . . + l / (z~): ~ > t / p - l l q.

Indeed, in this c a s e , a s is e a s y to see , A(x) = x ft.

L e t ~(x) be a cont inuous funct ion on [1, ~) such that Z(x) t 0 as x ~ 0.

the c l a s s of al l funct ions f(x)E L(p k) fo r each of which E n . . . . . n(f)Ln( k ) = O{ ~(n)}.

(1.2)) =

{3.26)

(3.27)

Le t E ~ , k (1 _ 1 (see

C O R O L L A R Y 2. L e t l _ < p < q <

then

Indeed, if f(x) E E # , k , then, se t t ing v (1) = n

we see tha t the embedd ing f(x) E L~)n. fol lows. a l e n t ;

. If

1

(3.28)

E ~ " (~) (3.2 9) p,h ~ L q .

o o . = v (k) = 2 n in T h e o r e m 1, f r o m the condi t ion n

Z 2~(q/,-,)%q (2 .) < ~ (3.30)

n = i

But, a s is ea sy to s ee , condi t ions (3.30) and (3.28) a r e e q u i v -

C O R O L L A R Y 3. L e t 1 _- p < q < ~ . I f fo r s o m e a > 0 the funct ion Z(x)x ~ is i n c r e a s i n g , then c o n - d i t ion (3.28) is n e c e s s a r y fo r embedd ing (3.29).

Indeed, wi thout Io s s of gene ra l i t y , we can a s s u m e thgt Z (x) is s t r i c t l y d e c r e a s i n g . Set t ing Zi(x) = Z(x), i = 1 . . . . . k, we see tha t in this e a s e A(x) = Z(x l /k ) , and the fu l f i l lment of condi t ion (3.28) impl ies

the equal i ty f x q /p -2Aq(x )dx = ~ . Using T h e o r e m 3, we s ee that a funct ion f(x) 6 L(p k), f ( x )~L~ ~) ex i s t s 1

such tha t (see (3.25)) En . . . . . n(t)Lp(k) - O{Z(n)}.

R e m a r k . By v i r t u e of C o r o l l a r y 2 and inequal i ty (1.2) fo r 1 < p < q < ~o, embedding (1.5) fol lows f r o m condi t ion (1.9). H o w e v e r , in the genera l c a s e , if the ;~i(x) have d i f f e ren t o r d e r s of s m a l l n e s s , (1.9) is not n e c e s s a r y for (1.5). Le t , for example Ai(x) = x a i , ~i > 0 (i = 1 . . . . . k), and suppose that not aU the a i a r e equal . T h e n condi t ion (1.9) is equivalent to the inequal i ty 1 / k min cq > 1 / p - I / q . If 1 / p > 1 / k

�9 min(~ t, t h e n w e can find q > p such tha t this inequal i ty is not sa t i s f i ed but (3.27) and, t h e r e f o r e , embedding (3.26) hold.

We fu r t he r note that condi t ion (1.10) a l so is not n e c e s s a r y fo r embedding (1.5). This fac t , howeve r , is a l r e a d y ev idenced by c o m p a r i n g (1.10) and (1.9). If , for example ,

k

~ i ( x ) = ( x l n x ) ~' ( i = l . . . . . k), p > l , k > 2 , q = 2 p ,

then it is e a s y to see tha t fo r any ~i > 0 (o~ i + . . . + ffk = 1) a t l e a s t one of the i n t eg ra l s (1.10) d i v e r g e s (for c~ i = 1 / k (i = 1 . . . . . k) al l these i n t eg ra l s d ive rge ) ; a t the s a m e t ime condi t ion (1.9) is sa t i s f ied .

w E m b e d d i n g o f t h e C l a s s e s H ~ i . . . . . C0k

F i r s t of al l we shal l expla in the gene ra l na tu r e of the condi t ions on the pa r t i a l modul i of cont inui ty of f(x) E Lp(Ik) tha t gua ran t ee its be longing to Lq(Ik).

541

THEOREM 4. L e t f(x) E Lp(Ik) , p E [1, oo), and le t the cont inuous n o n d e c r e a s i n g funct ions at(x) . . . . .

( t = i ..... k), (4.1)

ak(X) on [0, 1] sa t i s fy condi t ions (2.3) and (2.4). I f p < q < oo and t

u, =- ~ ~-~'~[ ~; ~) (1; ~,(~))pd~< o

then f(x) (~ Lq(Ik) and

k

(4.2)

P r o o f . By v i r t u e of (2.1), wi thout l o s s of gene ra l i t y we can a s s u m e that f(x) _> 0. Le t v -> 0 be an i n t ege r . F o r each i = 1 . . . . . k - 1 choose an in teger s(i) _> 0 so that the fol lowing inequal i t ies a r e sa t i s f i ed :

2 -(~(~) +~) < a~ (2-') ~ 2 - ~!o (~ = i . . . . . k -- i). (4.3)

�9 (1) , _ s . k - L . ~ ~J 2-* ~ al (2-~) . . . a , - i (2-*) ~ 2 _ _ _ * "" *

k--1

Set s~)= ~ - y ' , s(~0. since 2-~ = ~ ( 2 - ~ ) . . . ~ k ( 2 - ~ ) ~ ~k(2 -~) i = l

By virtue of (2.3) and (2.4)

T h e r e f o r e ~ _>s + . . . + s .

I S ( i ) + �9 + s(k-1)) have �9 2 -~' P " " , w e

s (k ) / 2- ~ % ct k (2-'i. (4.4)

Thus by v i r tue of the p r o p e r t i e s of a m o d u l u s of cont inui ty (see (4.3) and (4.4))

k k k (k) . s(0 r 2-~ ~),... ,2-~ ) ~< 2r 2- * ) < 2{o~)(f; 2cq(2-~))~< 2 2~o;)(];ai(2-~)). (4.5)

Now le t q)(x) = ~o(x; f) be the funct ion defined by L e m m a 3. If p = 1, then by v i r tue of L e m m a s 2 and 4 and (4.5), for 2 - ( n + l ) < x < 2 -n (n = 1 , 2 . . . . )

2 - n

O(x)~fg(t)dt~< I g(t)dt= sup i/(x)dx~< inf ~l(x)dx-F(k § -~),...,2-~)) 0 0 E C I k E "Eclte

I N l = 2 - n [ E [ = ~ - n

i k

% i r (t) dt ~- (k -F 1) 2 k+2 2 ~~ (/; ai (2-~)). 1 - - 2 - n i = l

Since the functions o~i(x) (i = 1 . . . . . k) a r e modul i of cont inui ty (see R e m a r k 1 to L e m m a 5), o~i(2 -n)

_< c~i(2x ) _< 2c~i(x ) fo r x _> 2 - ( n + l ) . F u r t h e r m o r e , i

(p (t) dt ~< 2-"q) ('/2) ~< 4xllfll~. l - - 2 - n

Thus k

(I) (x) <~ 4xilfll, + (k + t) 2 h+~ 2 ~176 (f; ai (x)),

and we obta in

llfllq={ i (Pq(x )dx} '/q <<-{'i l , },/q r x -~ dx + ~ - r A) o o t ~ t o

We have thus p r o v e d the va l id i ty of T h e o r e m 4 for p = 1 and d e t e r m i n e d the value of the cons t an t in (4.2) ;

n a m e l y , C(1, q, k) - (k + 1)2 k+3.

In the c a s e p > 1 we shal l s t a r t with the inequal i ty (n = 1, 2 . . . . ) 2-n 2-(n-t)

~p( t )d t= 2-~(p(2-~)+ [ ( p ( t ) - - ~ ( 2 - ' ) i d t ~ 2- 'q ) (2 - ' )+ q ) ( t ) d t - ~( t )d t .

3 2'-.,.,,I,,. { i t : lqd=}"' (k+ 1)2.§

(4.6)

542

By vir tue of Le mmas 2 and 4 (see also (2.2) and (4.5)), for 2-(n+l) s x < 2 -n (n = 1, 2 . . . . )

~ - n 2 - ( n - l )

i 9(t)dt-- i 9(t)dt= sup ~EI(X)dx - sup inf I](x)dx<2~m§ 2 - ~ )) 0 l - n E C I k E ~ C I k E C E ~ ~ " " " ~

IEI = 3 "~n }E~l=~_(n_l ) I E I = 2 _ n E

< 2 t c /P+4 (2X)l-1/P 2 03~) it; (Zi (X)).

Furthermore, for any m _> i

'/~m ~/~m

( 2 ~ ' ) - 9 ( ' -~- )~2m [!9(x)dx-,!m r <~2m[

Thus (see (4.5) and (4.7))

(p(2-~) = ~ [9 (2-') -- 9 (2-('-'))1 + qg(V~)'~< 211ftI~ + ~ 2 ~ [ S " ~ 2 v ~ 2 0

lira 2Ira

0 l /m

(4.7)

2-@-2)

~-0-I)

-..<211In, +2k/P+~ Z 2 " + ' 2 ~ ( V - ~ ) c % ( f ; . . . . . . 2-~)) ~< 2l]]t[~ + 2k/,+4 ~..jZ2~/pw~)(]; ai(2-?))

h + l ~5 k n - -1 2 - @ - 1 ) k4-1 ~ 1

~ 2 ] I ] t ' x + 2 v Z Z I t-(l+lZ')c~ 2i]flli+2=-~:+~Y--J s t-(l+lm~~ i=l v=l 2- -v i = l 2 - ( n - l )

Returning to inequality {4.6), for 2-(n+l) _< x < 2 -n we find that (see (4.7))

" '+' g i - g w , + 2 p x~-~,'p (I; ct~ (x)). r IT(t) dt<2x[ 21]lllx + 2 -~-+5 t-(l+l/P'a)(P'iliaiit)dt ] k-x+~ 0 i~l x i=l

Using this estimate, as above for p = I, we obtain

~1'~ '~ I k--1 5 " '

0 i=l 0

~+i k i i

+ 6 V ~ ( ~ " z ~q l i /q +27 t-(l+ih)0)~ )(1;"` (t))dt) dx} , X,,.~ ld td i ~ l 0 x

and to conclude the proof we apply Hardy ' s inequality ([10], p. 294):

d~< q~ x - ~ [ ~ ' (f; ~,(x))l~d~. 9 0

In conclusion, we note that the constant in (4.2) does not exceed ( 2 ~ / P + 2q)2k+t/2 +5 for p > i and (k + 1) �9 2 k+~ for p = 1.

Remark. Obviously, condition (4.1) will be mos t p rec i se if the sys tem of functions at(5) . . . . . Gk(6) is such that

~ (/; ~,(6))/~o~' (I; ~ ( 6 ) ) = o ( ~ ) ( l < ~ , ] ~ < ~ )

(see Remark 3 to Lemma 5).

Le t {wl(5) . . . . . wk(5)} be a sys tem of s t r ic t ly increas ing rnoduli of continuity, and let ~(5) be the mean modulus of continuity for this sys tem (see Remark 2 to Lemma 5). The function ~(5) plays an im- por tan t ro le in questions of the embedding of the c lasses H~I . . . . . ~k. This follows f rom the following two fundamental theorems :

THEOREM 5. L e t l _<p < q < ~ . If

i x-q/pf~q(x)dx < 0% (4.8) 0

then

543

The va l id i ty of this a s s e r t i o n fol lows i m m e d i a t e l y f r o m L e m m a 5 (see a l so (2.6)) and T h e o r e m 4. Note tha t condi t ion (4.8) r e l a t i ve to f~(6) has p r e c i s e l y the s a m e f o r m as the condi t ion

i x-q/pc~(x) dx < o% f~

which is n e c e s s a r y and suf f ic ien t for the embedding H~(0 , 1) ~ Lq(0, 1) (UPyanov [2], pp. 675"677).

THEOREM 6. C0k(5 ) be such that

~-~ ~o~(l/m) = O(no)~(l/n) )

Then condi t ion (4.8) is n e c e s s a r y for embedding (4.9).

Le t 1 _<_ p < q < *~, and let the s t r i c t l y i n c r e a s i n g modul i of cont inui ty r . . . . .

(~ = t . . . . . k ) (4.10)

P r o o f . Le t

x-q/ ,~q(x)dx= xq/p-~q( t / x ) d x = ~. o 1

By v i r t ue of a t h e o r e m of Stechkin ([11], p . 78), for each i = 1 . . . . . k t he re ex i s t s a convex modulus of

cont inui ty ~i(5) such tha t

(o~(6)<%(6) <2co~(5), ~ [ 0 , l ] , i = i . . . . . k.

T h e r e f o r e we can a s s u m e that the funct ions 5 /w i (6 ) a r e n o n d e c r e a s i n g as 5 t Set ~i(x) = cJi(1/x) (i = 1,

. . . . k). Then , as is ea sy to s ee , x~i(x) ~ as x ~ ~ , A(x) = ~ ( 1 / x ) , and i xq /P-2Aq(x)dx = ~ ' Using T h e o - 1

r e i n 3 , we see that funct ions g(x) ELp (kS exis t such that g(x)~ L(q k5 and E( i)n,~ (g)Lp (k) = O(r176 By v i r tue

of T i m a n ' s inequal i ty ([1], p. 363) and condi t ion (4.105,

n

Thus the fuact io~ f(x) = g(2ux) (x ~ Ik) be longs to H~l . . . . . wk but f~ Lq(Ik). p

COROLLARY. L e t l _ i l p - t t q, (4.115

then

Lip (oh . . . . . ak; p)_~ H~a: ..... ~ak ~ Lq (Ik). (4.12)

If , in addi t ion, all c~ i < 1, then condi t ion (4.115 is n e c e s s a r y for embedding (4.12).

Indeed, in the c a s e under c ons i de r a t i on , as is e a s y to s ee , ~(x) = x a . If c~i < 1, then (4.10) is s a t i s -

f ied.

R e m a r k . Condi t ion (4.10) is sa t i s f ied by a wide c l a s s of modul i of cont inui ty that i n c r e a s e as 5 ~ 0, gene ra l l y speak ing , no f a s t e r than 5 ~ for 0 < c~ < 1. It is l ike ly that T h e o r e m 6 is val id even without r e - s t r i c t i o n (4.105. Some r e s u l t s a long these l ines a r e conta ined in T h e o r e m 8, which we p rove below. In

a def in i te s ense this t h e o r e m c o m p l e m e n t s T h e o r e m 6.

THEOREM 7. L e t 1 - p < q < ~r and let w(5) be a modulus of cont inui ty . If i

~ x-~q:P+k-qo q (x) dx < o% (4.13) 0

t h e n

H,., --== Hv (A) ~ Lq (A). (4.14)

Indeed, we c a n a s s u m e that ~(6) is s t r i c t l y i nc rea s ing . But then for wi(6) = w(6) ([ = 1 . . . . . k) ~(8) = w(51/k), and condition (4.13) is equivalent to condition (4.%).

544

THEOREM 8. Letl_<p < q < ~, p ~kq/(k+q), and letw(6) beamodulus of continuity. Then con-

dition (4.13) is necessary for embedding (4.14).

P roof. Let

. ~ x_~ql~+~_,o)q(x~d x _~ x~ql~-(~+i)(o~( ~ /x )dx ~ ~o. (4.i5) 5 i

We consider two cases separately.

i) p > kq/(k + q). The method of proof in this case almost coincides with the method of proof for

Theorem 3, and so we shall set forth explicitly only the most essential aspects.

Let ~ = k + 1 - k q / p .

Set

Then a ~ ( 1 - q , 1). We useLemma6 (see (4.15)), and for m= 1,2 .... set

a =3.2-(~+~), r =2-(~+~), H, = 2 ~ m / ~ , ( B ~ - B ~ , ) ,

( 0 at other points of f~.

for x(~Qr~, m = ~ , 2 . . . . .

In a way completely analogous to Theorem 3, we find that

~ /*(x)dx = C(v, k )H .Jr~ (~ ~ 1)

and f(x) ~ Lp(Ik), but f(x) ~Lq(Ik). Let us now est imate w(~)(f; 5). Let P

2 - ( ' + ~ h < 2 -('+~) ( s > l ) .

Then

A(h)------ S . . . IM,'~ f ( z , . . . . M l ~ d z . . . . a x e = ~.. IA~,~f(x~ . . . . x , ) i ~ a z , . . . d x ~ o o o o o

(4 .16)

(4.17)

~-h i ( 0 X + ~ . . . [A,.~ ]( ~ . . . . . x ~ ) l p d x ~ . . . d x h ~ A ' ( h ) - k A ' ( h ) �9 2-(s§ 0 2-(~+J)

By virtue of (4.16), (4.17), and condition 1) from Lemma 6,

9 0 m ~ s Qm m ~ s

Let us now est imate A"(h). If (x 1 . . . . . x k) E Qm and (~ + h, x~. . . . . . x k) E Qm for m -< s, then

15~ ,~? (x , , . . . , x~ ) [= I(x, W h - a , ~ ) ~ - ( x ~ - a ~ ) ~ l ~ H - - - L ~ h ( I x ~ - k h - a , ~ i § ~ 2 h H . j r ' ' (4.18) Fm Fm 2

Let q(m 1) denote the orthogonal project ion of q m onto the hyperplane x i = O, and let x~ m) and x~ m) be the roots of the equation

(x~ = ~)~ + . . . + (x~ - a~)~ = r L _x[ ~) ~ ~'~).

wherexk) E ~(1)the.(mP~ s )(0'and ~ . . . . . x~ m) -hX k)<_ xlE Q(m 1)_< x~__ re)is fixed.(or ~AS)-h in the<_ xlPr~176 x~ m))'~ Theoremthen 3 . . . . . we note that if (0, ~ ,

( t ) - / h~,h l~x . . . . . . xk) = 0 (h) H J r m ; (4.19)

for all other points x different from those considered above, we have A~')f(x) = O. Thus (see (4.18), (4.19), and (4.17))

A " ( h ) ~ O ( h ~) IQ~) lhg~ / r~ -p+(2h ) p IQ,~lH~Vr~-P=O(hp) r:-~ (rm + h ) H ~ P r . z - p = O ( h p) 2~p(B2~-B2~.Op.

545

Using Jensen's inequality (see [I0], p. 43), conditions i) and 2) from Lemma 6, and inequality (4.17), we obtain

2 ~

A ' ( h ) = O ( h p)

Taking the resu l t ing e s t ima te s into account , we conclude that f(x) E HpWk .

2) L e t p < [kq/(k + q)]. Then for r = k / p - [ ( k + q)/q] > 0 and k(x) = x - 0 + ~ )

5 xkq/~-(~+')~ q (x) dx = oo.

Using Corollary 3 inw 3, we see that a function g(x) E L(p k) exis ts such that g(x)~ L(q k) and E n . . . . . n(g)Lp~Q

= O ( 1 / n l + e ) . By v i r tue of T i m a n ' s inequality ([1], p. 363), r 1/~fn)Lp(k) = O(1/n) (i = 1 . . . . . k).

Thus the function f(x) = g(2~x) (x E I k) s a t i s f i e s the conditions

/(x) ~ L~(I~), ](x) (~ L~(h), ~p") (f; 6 ) = O(~). (i = t . . . . . k).

But since (see (4.15)) w(6) ~ 0, by v i r tue of the known p r o p e r t i e s of a modulus of continuity, w(5) -> C5 (C > 0); t he re fo re fEH~, k. This p r o v e s the theo rem.

Remark . The suff ic iency of (1.11) for embedding (1.6) follows f rom T h e o r e m 7. If the o rde r s of s m a l l n e s s of w1(5) . . . . . r ) a r e di f ferent , then, genera l ly speaking, condition (1.11) is not n e c e s s a r y for (1.6). We see that this r e m a r k is val id by cons ider ing , for example , the c lass Lip (~t . . . . . ~k; P) (this is comple te ly analogous to the r e m a r k a t the end of w

LITERATURE C I T E D

I. M.F. Timan, Theory of Approximation of Functions Of a Real Variable [in Russian], Fizmatgiz,

Moscow (1960). 2. P.L. Ul'yanov, "The embedding of certain function classes H~," Izv. Akad. Nauk SSSR, Ser. Matem.,

32, No. 3, 649-686 (1968). 3. M. F. Timan, "Some embedding theorems for Lp-classes of functions," Dokl. Akad. Nauk SSSR, 19___33,

No. 6, 1251-1254 (1970). 4. L.K. Pandzhikidze, "Embedding theorems for functions of several variables," Soobshch. Akad. Nauk,

Gruz. SSSR, 60, No. i, 29-31 (1970). 5. L.K. Pandzhik---idze, "Embedding theorems and the connection between the best approximations in dif-

ferent metrics for functions of several variables," Soobshch. Akad. Nauk Gruz. SSSR, 6--i, No. I, 25-

28 (1971). 6~ N.T. Temirgaliev, "Certain embedding theorems for classes H- -- of functions of several variables,"

Izv. Akad. Nauk Kaz. SSSR, Set. Fiz.-Matem. No. 5, 90-92 (1970). 7. P.L. Ul'yanov, "Embedding theorems and relations between best approximations (moduli of con-

tinuity) in different metrics," Matem. Sb., 81, No. i, 104-131 (1970). 8. S.M. Nikol'skii, ,,Inequalities for entire fun-~tions of finite degree and their application to the theory

of differentiable functions of several variables," Tr. Matem. In-ta Akad Nauk SSSR, 38, 244-278

(1951). 9. G.M. Fikhtengol'ts, "A Course in Differential and Integral Calculus, Vol. 3 [in Russian], Fizmatgiz,

Moscow (1963). i0~ G. Hardy, J. Littlewood, and G. Polya, Inequalities, Cambridge University Press, New York (1934). ii. A.V. Efimov, "Linear methods of approximation of continuous periodic functions," Matem. Sb., 54,

No. I, 51-90 (1961).

546

the embedding of certain classes of functions of several variables

Documents