On imbedding theorems oî Orlicz-Sobolev space

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X X (1978)

H. ^{H u d z i k} (Poznan)

On imbedding theorems oî Orlicz-Sobolev space W kM(Q) into Gm{Q) for open, bounded, and starlike Q R n

A b s tr a c t . This paper is a generalization of Theorem 1, [7], p. 64, an imbedding theorem from W® into [7], p. 77, and of Theorem 1, [7], p. 91, to the case of the Orlicz-Sobolev space W kM (Q) (l ).

I am indebted to Professor J. Musielak for his kind remarks in course of pre­

paring of this paper.

0. Introduction. Let Q be an open set in R n, R + = <0, oo), ц — Le- П

besgue measure in R n, r(u) = \\u\\ = [ Щ 112 for и = (ulf un) e R n, i=1

J/' = the set of all natural numbers. We denote by F the real linear space of all complex-valued functions / defined and Lebesgue measurable in

Q (with equality almost everywhere).

A function M: R ^ >R will be called a 99-/unction if:

(1) M(v) = 0 if and only if v — 0,

(2) Щ а люх -f- a^Vo) ^ o-i ilf (vf) A a2 M (п%) for G R+ and cq, a2 e e 1 ai a2 — 1?

A ^-function M will be called К -function if:

M{v) ' M{v)

--->0 as , ---> 0 0 as v - ^ -0 0.

v v

A 99-function M is said to satisfy the {A.2>r )-condition if there are constants

^ > 2 and v0 > 0 such th a t the inequality

(*) M ( 2 v ) ^ x M { v )

is satisfied for every v > v0. If there exists a constant x > 2 such th at inequality (*) is satisfied for every v > 0, then we sa y 'th a t M satisfies the (A2 fconditimi.

i1) For definition of see p. 342.

For / 6 l}°° ( Q) we define the following functions /а И = a* x - y

f{y)dy; » > 0,

where y> e C°°{Q), гр(и)^ 0, ^{^(г^ )} = 0 for ||w|| > 1, гр(и) — y>(t) for \\u\\

= ||i||, j ip(u)du = 1. The fonctions / й are called a mollification or regu- larization of /. L M{Q) denotes, of course, the Orlicz space generated by a 99-function M, i.e.

L*M(Q) = { /e jP: 3 f M (л\f(x)\)dx < Ц .

' A >0 i >

Given a non-negative integer Tc and 99-function M, we denote w * , (Q) = {/ 6 L*„(Q): V 3 D af e L* M(Q)}

, |a|=^fc

and

W®(£>) = { / e X i(Û ): V 3 D “/ e ^ ( i 3 ) } , - n o ­

where a = (a1? ..., a j with аг-> 0 for i = l , 2 , . . . , n is a nmltiindex,

|a |= «! + ... + an and D “ = д]а] /дх*1 ... dx^n is the operator of generalized derivative (i.e. derivative in the sense of the theory of distributions) of order |a|. The sets W kM{Q) and W (ÿ { ü ) are real vector spaces (see [2]).

We define

QM(f) = J M(\f(x)\)dx, f e L*M{Q),

a

\\fÏÏLM(o) = iBffe > ° : QM(fle) < !}»

Ы Я = У е м ( О У ) , НЛ1И,*Ш, = i n f { e > 0 : ë i f ( № ) < l } , / 6 # S f (O)-

leK* }

11/Нрр(*)(0) = ll/llijf(û)+ ^ \\DafWlm(q)> /eWSÎ(û).

|aj = ft

The functionals and are convex modulars in L*M{Q) and TF|f(i2), the functionals ||*|| fc and |H L (*),0, are norms in WkM{Q) and W{${Q), respectively. The pairs <WkM(Q), ||- | | ^ (fl)> and <W(§{Q), 11*11и/щ(й)> are Banach spaces (see [2]). If p ( Q ) < 00, then L*M(Q) a L x(û) for every 99-function M (see [2], Lemma 1). Moreover, if the 93-function M satisfies the (/d2)-conditioh (or (zl2j„0)-condition if p ( Q ) < 00), then li/—/д\\ьм(п)~*®

as <5-^0 (see [3]).

For any continuous function / defined on Q, we define s u p p / = {x e Q: f(x) Ф 0}.

Imbedding of Orlicz-Sobolev space 343 We denote by C(Q) the space of all continuous functions on Û together with the paranorm

oo

\f\c(Q) = У ^ 2~

m= 1

Pmif) l +Pmi f ) ’

where p m(f) = sup \f(x)\ and x is a sequence of subsets of Q

— XtQm 00

such th a t Qm c= Û for m = 1, 2, ... ; Qx <= Q2 c ... and U Qm = Q.

, m—1

Cm(Q) denotes the space of all functions defined on Q such th a t Daf e C { Q ) for every |a| < m, with a paranorm

\f\cm(n) = max \Daf \ C(S}).

'a|<*

We denote by C°° ( Ü) the space of all infinitely differentiable functions on Ü, and Cf(Q) denotes the subspace of C°°(Q) consisting of functions such th a t supp / is compact in Q.

We write also

K{oc, r) = {y ^e R n: \\x-y\\ < r}

for every x e R n, r > 0 and

K{ x , r ) — {y e R n: ||a?-2/||<«)

for each x e R n, r > 0. Moreover, for every set Q a R n, we write (0.1) Qd — jo? e Q: d(x, Г(й)] > <5},

where Г( й) is the boundary of Ü and d {x, Г(й)) denotes the distance between x and Г(О). Moreover, we write

a! = a x! ... an\

for every multiindex a = ( a 1, . . . , a n) with a{ > 0, i = 1 , . . . , n ; a > /?

means th a t af > p{ for i = 1, .. . , n, ... for every a > ft.

Further, we denote by

(+ ) N(v) = sup {uv— M(u)} for every v e R +

m>0

the W-function complementary to the JV-function M in the sense of Young. If M and N are complementary W-functions, then the following Holder inequality holds I

I f f(x)g(®)dx\^2\\f\\LM(a)\\g\\LN{Q), V (feL*M(Q), geL%(Q)).

Q

De f i n i t i o n. A set ü e R n is said to be starlihe with respect to a set c Ü if for every x e Qx, y e Q, 0 < t < 1, the point tx + (1 — t) у belongs to Q.

1. The imbedding theorem from into (Jm{ Q) . First, we shall prove three lemmas.

Lem m a 1 .1 . Let M and N be N -functions complementary each to the other in the sense of Young satisfying the (A 2)-condiUon (or (Лг ^ -c o n ­ dition if f i ( Q) < oo) and let f e L*M(Q). I f there exists a sequence of func­

tions fi e C°°(Q),l = 1 , 2 , . . . , such that

( i.i) II/—Alii ^я)->0 as Z-*oо and dlalf t

dx°l ... да>%* < A

for l — 1, 2, ..., where the constant A is independent of l, then the function f has the generalized derivative

dla'f

dx°l ... dx^n e L*M(Q).

P ro o f. By our assumptions, the Orlicz space L*M(Q) is reflexive and separable. Hence there exists a subsequence {Dafih}Z=i °f the sequence 2 weakly convergent as 7 t-> oo to a function coa e L*M(Q) satisfying the inequality \\o)a\\LM(Q) < A (2). Thus, for every g e L*N(Q),

/ g(x)(D°flh(x) — (oa(x))dx-> 0 as h-+oo.

Q

Integrating by parts, we get

f {Лл(® )И >(а) + ( - 1 ) |”Н1у М Л ”/ |4(®)}й® = » for every ip e Cf(Q). Hence, taking h~>oo, we obtain

/ {f (x)Daip(x)-\-(- 1)|Q| 1ip(x)ma(x)}dx = 0 Q

for every ip e C™(Q). Thus, the generalized derivative D af — o)a ^e L M ( Q ) exists, where ||ft)a||iM<û) < -d- The proof is completed.

Lemma 1.2. Let Q be an open and bounded set in R n and let B Q be a real positive number such that K (y, B 0) => Ü for every y e Q. Let f e L*M(Q) and let there exist real numbers h > 0 and À such that

(1.2) Глг((г(ж)) X)dx = f Ж(||ж|| X)dxx . . . d x n < oo,

r<h, IkclKft

where N is an N-function complementary in the sense of Young to the N-func- tion M. Moreover, let the functions M and N satisfy the (A2 VQ)-condition.

(2) A Banach space X is reflexive if and only if its closed unit ball К (0, 1)

= { x e X : {x} < 1} is weakly sequentially compact.

Imbedding of Orlicz-Sobolev space 345

Then the function

Uf (y)= f

I lx— j/IKK

where f x{x) = f(x) for x e Q and f x{x) = 0 outside Q, belongs to C(Q) and there holds for all y e Q the inequality

(1.3) I W K ^ I I / I Il^ ) ,

where К is an absolute {that is, independent of f ) positive constant.

P ro o f. Prom (1.2) it follows th a t

(1.4) ' Ï N{ r ~x)dx^> 0 as Л->оо.

r< h

Applying Holder’s inequality, we obtain

j f \\Х ~У''\ Afii®)dx^ < 2 \\fx\\LM(K(y,H)) ||II® — У\\ A||-LjV(fi:(s/,/i))

< 2 II/1|Lj^Q) \\Г А|1п^(Щ0,Л)) *

Since N satisfies ( J 2>^-condition, (1.4) implies Н^Иь^ .що.й))- ^ as ^->0.

Thus

(1.5) Uf,h(y) = f r~xf x(x)dx-+0 on Q as h->0.

r<A We have

Ufiy) = f II® — y r A/i(®) + Uf<h{y)

h<r*ZR

for all y e Q. The first integral on the right-hand side defines a con­

tinuous function for every h > 0 and, by (1.6), it tends to Uf uniformly on Ü. Thus, Uf is continuous on Q. Applying again Holder’s inequality with h = R 0, we get

\Uf(y)\

% \\fi\\bM{K(v,R0))

where К = 2 The proof is finished.

R e m a rk 1.1. If Я ^ 0, then condition (1.2) is satisfied by each A-function N.

Lemma 1.3. Let R 0 and Q be as in Lemma 1.2 and let 0 < Я < n, f 6 L*M{Q)y where M is a <p-function. Then Uf e L*M ( Q) and the following

inequality holds

№f\\bM(Q)

where K x is a positive constant independent of f e L*m{Q).

P ro o f. Since 0 < A < n, we have

K x = J r~xdx — J \\x\\~xd x < oo.

г< Щ ||аг!|<Д0

Applying the integral Jensen’s inequality and defining f x as in the proof of Lemma 1.2, we obtain

J M(\Uf (y)\)dy < j J II® —

a a ||a ;- î/IK R0

= / ж ( / -|Ж Д

< / { / 1!--~ - м ^ ш В с к ^ а у .

а ||х - |/ ||< й0 1

Putting х — у = и and applying the Tonelli theorem, we obtain JjtfflP ^ îO D d S K J { J i ^ - J f ( Z 1|/ 1(« + y )|)* (Jdy

a a ^iim|1<jB0 1

= J { | ж ( ж 1|Л(«+у)|)<гу}<г»

llM||<iî0 1 Я

« / - ^ { / л г ( г . 1 Л М 1 ) <Ч * *

Нм||<2г0 1 a

= J M ( K x\f(y)\)dy.

a

The last inequality may be rewritten as

Qui^f) ^ Qm(E-\$)? f e L M{Q).

Hence we obtain inequality (1.6) and Uf e L*M(Q). Thus, the lemma is proved.

De f i n i t i o n o f t h e s p a c e L kM{Q). In the space W kM(Q) we define an equivalence relation It: f i t g о Y D af = D ag and we define L ku W

|a | = k

= WkM(Q)jB. The equivalence class [ /] e L kM{Q) of the function / e W kM(Q) is defined as follows:

[ / ] = {g e ТП,(Й): fBg).

Under addition and scalar multiplication:

[ /] + M = [/+ f f], «Lf ] = M ; « 6 * 1.

Imbedding of Or Пег-Sobolev space 347 L kM{Q) is a real vector space. I t is easily seen th a t L kM{Q) = W}^ { Q) l 8 k, where 8k denotes the real vector space of all algebraic polynomials of order 1 (the simple proof is omitted). Hence it follows th a t each function f e W kM{Q) may be written as

(1.7) / = / * + / . , where f x e { f \ e L kM{Q), f z e 8 k{Q).

Indeed, taking any function f x e [ /] e L kM(Q), we have / —f x e S k . I t suffices to assume

f~fi —

The following functionals will be of importance:

J

= - / ж [ ( ^ 4 г | 1 ) 0 / И | 2 ) ] * ■ >

fc-i

N=&

1/2- fc-1

J !(P ) = ] tor =

7 = 0 |а| = г 7 = 0 |а| = 7

J .< [ / ] ) = 2 е и ( П ’Л , f e [ /] e Lm(Û),

f al = A:

ll/IU ,a = m f{ e > 0 : J ^ f h X U , / e [ / ] s X b ( f l ) , ьм<и>

‘Н Л Ц ^, = £ \ m \ \ LMm, f e L kM(Q),

|a| = fc

2II/IIL» lût = inf {e > 0: J 3(//e) < 1}, / e i j , (û ), ll/ll£ = inf {« > 0: J,(P/<>) « 1}, P 6 S*, l l / l l ^ , = ll/llijH(fi, + ll/li£5f(0), f e W km ( Q ) ,

2 № ° Л Ь м т , / e w b ( û ) .

M |a|<fc

Obviously, functionals J x, J 3, and J 2 are convex modulars on L kM{û) and 8k, respectively. For example, we shall show th a t the functional J x is a convex modular on L kM(Q). Since for all f e [ /] the number J x(f) is the same, we can write J x(f) in place of J x([/]). We have:

(1) if / g 8k = [0], then for every |a| = Tc, D af = 0. Hence J x{f) = 0.

^ (/) = 0, then B af(oo) = 0 almost everywhere on Q for every \a\ = Tc.

Hence it follows th a t f e 8 k(Q);

(2) the fact th a t J x{f) = J x(f) is obvious;

8 — Boczniki PTm' Prace Mat. XX.»

(3) convexity: let f x e Ш , / 2 e [/„], ax, az ^ 0, cq + a o ^ l ; then applying the Schwarz’s inequality for sums, we get

+ «2/ 2 )

( “ U

4 — *k

^ ( « i / i + aa/2)

dxh ... дщк

Г )

г „г I v l |Y \ / v l а‘Л(*) 2\1,!1^

J -¥L“-U к

+“2 2 J***

2 L

г1 гк \ ' н--лк 1 к '

Q i x...i k

^ «гЛ(/1) + «2^ ( Л ) ,

where %e {1, 2 , . . . , n} for l = 1 , . . . ,7c. Thus, the functionals ||• ||J£,

^ ( A ) ’ ^ ^LkM{QŸ ^ ^ ( Я ) ’ '1° V ^o ), Xl I j. are norms in the w*^Q)

spaces 8k, L kM ( Q) and W kM(Q), respectively. Moreover, the following inequalities hold

(1^.8⁾ \\Daf\\LM(Q) ^ ^ la l ~

(1-9) \\Л)т к < С к>пт ж \ m \ \ bM(Q), where

Gk>n =

|a| = fc

"M(0) |a| = fc

Inequality (1.8) follows from the inequality

oM( B af ) < J x(f) for all / e [ / ] e 4 ( i 3 ) and |a| = 7c.

Now, we shall prove (1.9). Writing y ( f ) = Ck>n max \\Baf\\LM(Q) and N=*

applying Schwarz’s inequality and Jensen’s inequality for sums, we obtain

J M [ ( 2 1^ И /у12Г ] < ^ < / [ 2 ^ 1-»7 (*)/г|]<*®

fl |a| = fc |a| = fc

fl '|a| = A: К,п * y j _ l l См ( ^

0]. n J \ a|= * К,П A

— 2 ^ =1-

ск K,n |al = /cf i cc ! Hence inequality (1.9) follows immediately.

\Daf(x)\

dx

dx

Imbedding of Orlicz—Sobolev space 349 Definition. A linear operator 77: WkM(Q ) (Q) is called a projec­

tion operator if 772 = 77, that is, fl(ITf) = I l f for every / e WkM{Q).

If for every / e Sk(Q) there is a function g e such th a t Пд = /, that is, the operator 77 maps 17^(7?) on Sk(Q) с 1Г^(7?), then 77 = 7 on Sk{Q), th a t is, 77/ — f for every f e Sk. Obviously, Sk(D) is a vector subspace of WkM{Q) for any bounded set Ü c; R n.

For any projection operator, we have

/ / / / //*/• V / e ^M < û).

where /7* = 7 —77 is also a projection operator mapping WkM{Q) into L kM{Q). Indeed, (Я*)2 - ( 1 - П ) о { 1 - П ) = 7 - Я о 7 - Я о 7 + Я 2 = 7 - Я - Я* and if / ^g TF^(£), t h e n / —Я * / - 77/ e SA, whence Я * / ^g [/] ^gL kM{Q).

The spaces L kM{Q) and {II*f: / ^g WkM{Ü)} are isomorphic. The operator Я* is an isomorphism between these spaces.

Lemma 1.4. I f Q is open, bounded and starlike with respect to a ball K ( 0 , H ) c= Q and M, N are N -functions complementary in the sense of Young and satisfy the ( A2>&0)-condition, then for any function f e WkM{Q), к — 1, 2, ..., the following Sobolev equality holds almost everywhere on O:

(1.10)

/ И = J ? %a j Ca{y)f{y)dyY j ^ w a{æ, y) Baf (y)dy,

| a | = * - l Щ0, Н) S i ^{| a |} = fc

where the functions 'Qa are continuous on Q, vanishing on Q \ K ( 0, H) for

\a\ < к — 1 and the functions wa are continuous on JjxTÏ for |a| = к with the exception of (x, y) with x = у and such that

3 V у _ К ( ж^ ) 1 < Б -

B > 0 I a | = fc (x,y)eSJx Si

Х ф у

P ro o f. If / ^gCk{Q), then (1.10) holds for each x e Q (see [7], p. 62, (7.13) or [1], p. 35, (11)). For the sake of simplicity let us denote the two terms on the right-hand side of (1.10) by (TIf){x) and (Я */)(x), res­

pectively. Я / is well defined by inclusion L*M(Q) <=. L\00(Q) (see [2], Lemma 1) and the function Я*/ by Lemmas 1.2 and 1.3. The maps Я :

(О) э f^->IIf e Sk and Я* : WkM ( Û) э /н>Я*/ e I kM ( Q) are projection op­

erators. Let us write

F{x) - (Щ)(х) + {ll*f)(x), f g WkM{Q), x e Q . There is a sequence {/^“ i c= C°°(Q) (see [4]) such th a t

<1Д1) i ! / - Л Ц , (0)- °

There holds the equality

U-12) /,(®) = (Л/,) (æ)+ (#*/,) (æ), w e Q , 1 = 1 ,2 ,...

Now, we shall show th a t

(1.13) № —fl\\bM(Q)~^Q aS I-+OO.

By (1.11) and by inequality ||/||il(fl) < K\\f\\bM{Q) with an absolute con­

stant K > 0 (see [2], Lemma 1) it follows th a t II/—/IIl^ - ^ O as l-^oo.

Hence, writing

max sup |af| = K x, max sup \Са(У)\ = 772,

|a |s Ç f c - l ж е д Л $ \ a \ * Z k - l у е Щ О . В )

лус have

\ (Щ)(х)-(Щ, )(х)\ < ^ к гЕ ‘ { ШУ) - МУ) \ * У

|a |< :fc — 1 Q

= ^ 2( у l) ii/ - / ,ii£i(D)^ o as г-*со.

|a|<fc-l

Since the last estimate is uniform with respect to x e Q, the sequence {77/} tends to 77/ uniformly on Q. Hence

\\n f—nfl\\LM(Û)->® as l-*yoo.

Moreover, we have

(1.14) (П*Л(х)-(П*/, )(х) = £ У ) ( & т - 1 Г М У ) ) * У -

|a| ==/<; Q

Hence it follows th a t ЦЯ*/—77*/||iM(a)->0 as l-^oo. In order to prove this we consider two cases.

(1) Let condition (1.2) be satisfied with Я — n — k. Since the func­

tions wa are bounded on Ü x D \ {{y, y)}, applying Lemma 1.2, we obtain

|(Л*я (®) - ( л*/,) и I « к w t -D7<n LMm ■

Hence it follows th a t 77*/ tends to 77*/ uniformly on Û. Thus

\\n*f~ft*fi\\LM(Q)-+0 as Z->oo and

(1.16) ||F — /Ньж(л) ^ \\Щ~Щг\\ьм(0)~\- \\n*f—Я*/||х,м<я)->0 as l->oo.

So, we proved (1.13). From (1.11) and (1.13) it follows th a t ЦТ1 — /\\ьм (о) = ® i.e. F{x) = f{x) almost everywhere on Ü.

(2) If condition (1.2) is not satisfied with Я — n — k, then we have n — k > 0. But in this case Я = n — k satisfies the inequality 0 < Я < n.

Applying Lemma 1.3 in (1.14), we obtain

11Л*/-л*/,1Ьлг(0) < я. У-11Л7-Я7,11хж(я,->о

|a | = fc

as 1->oq.

Imbedding of Orlicz-Sobolev space 361 This implies condition (1.15) and further (1.10) for almost every x e Q.

Thus the lemma is proved.

Now, we define in the space W kM{Q) the norm

(1.16)

ml

M(S3) \m \g k+ \\f\\Tk .

L M(0)

Th e o r e m 1.1. Let Q be an open, bounded set in R n, starlike with respect to a ball K( 0, H) <= Q, and let Ж, N be N -functions complementary in the sense of the Young, satisfying the (A2f#0)-condition. Moreover, let condition (1.2) be satisfied with 1 = n — k. Then the inclusion W kM{Q) c C{Q) holds and

\f\c(a) <K\ \ f \ \ 1wk (3)>

w M(Q)

where К is an absolute positive constant, i.e. the imbedding operation from

<WkM{Q), I NI * ,m> into <C{Q),\-\C{aj> w continuous.

w m\ q’

P ro o f. Since I l f e 8k , so in order to prove of inclusion WkM(Q) c C(Q), according to (1.10) it suffices to show th a t 77*/ e C(Q). This follows from Lemma 1.2. Now, we shall prove the continuity of the imbedding operator.

First, we shall show th a t there is a constant K x > 0 such th at ( к m \

---- — I > 1 for every x e Q, where K x is independent of / g W kM{Q).

КДЛ(®)1/

Let us write Cla — l/al for |a| = l, A = max sup |af|, СЬгП = —

|a |< fc - l xeQ ’ |a| = l Ct !

and aa = f £a(y)f{y)dy• We shall determine the constant K x. We have що,н)

«7,1-

Г Кг 2 Cla\aa\ 1 *-i -, ...--- :--- У А ж

1 = 0

> кМ

к— 1 а 2 2

3 = 0 m = q

к~1 K l 2 °a K \ I

j1 = 0Lj к

k x 2 c«K l _____ N -i______

к— 1

A@i,n 2 2 4 \at

3 = 0 \f}\=q

k-1

i-o kAci n 2 2 G$\ap\

3 = 0 |/î| = q

> kM

( ___ Ь ___)

\k max с1гП'А}

for every x e Q. Since Ch>n < GhtU as lx < l2, we have max Cl>n = Ck_ hn.

(3) This means that there exists a function f xeC(Q) such that f (x) = f x (x) for almost every x e Q and the inequality holds with f x insted of /.

^Choose any e > 0 and let К г = 1cACk_1>n31 1 We have

\ m m I/ L U - e / J ^{k — e > 1.}

Hence, by definition of IT*f, Lemma 1.2, and inequality (1.8), we obtain

\(п*л(х)\ < jt4 j r wdjwl^ ü) < к * ( £ i) m Lk

I a| = ft ^{|a| = fc lm}S°)

for all x e Q, with an absolute constant _ЙГ4 > 0. Hence and from (1.16) we get

\f(x)\ < \(lTf)(x)\+\(Il*f){x)\ < -ff5i|/||W^(O) for all x e Q, where K 5 > 0 is an absolute constant.

Let us choose a sequence of real positive numbers,

as n->oo, and let us define a sequence of sets {Qd } by formula (0.1).

Then

'Pm(f) = SUP |/( * ) |< KsWfW][wk W = 1 ,2 , ...

xeaôm Further, we have

OO OO

m~ 1 +-^5

< K s \\f\f k5UJUW^i(Q)

oo

œ

The proof is finished.

Theorem 1.2. I f the assumptions of Theorem 1.1 are satisfied and if

(1.17) 3 3 / oo,

h > 0 m e /u {0 }

Йеге holds the inclusion WhM(Q) c Cm(Q) and

\f\cm(Q) < ^ 1!Я1^(Л)

with an absolute constant K 6 > 0, i.e. the imbedding operator from ( W kM{Q), ||-|l fc,J> into {Cm{Ü), \-\c^(Q)> ^ continuous.

"MS >

P ro o f. First, w'e shall prove the inclusion WkM(Q) cz Cm(Q). I t suffices dm(II*f)

to prove continuity of --- on Q. where the derivative is under-

F J da$ ... дя%*

stood in the usual sense. We have (see [7], p. 79-80)

Imbedding of Orliez-Sobolev space 353 (f

д-xi1 ... dxpnn \ fjn—kWai...an(®, У) y j i - k + m K ; :i%(x, y),

where wa{x, y) = wa an{x, y) are the functions from (1.10) and the functions У) = w i(x >y) are continuous on D x D with the exception of point (x , y) with x = y for |a| = 1c, \(}\ < 1c—1 and such th a t

3 V V Ko», 2/)i <

# ! > 0 \a\ = k |/9 |< fc-l

Thus, we have

(1.18)

r

6

J дхЬ ... дхРп Q la| = fc 1 n

I 1 w«<*. , У)x P f ( y ) — ; p d <*y» Л

■ ■ ;¥nn

< _f ,» -*+ » . У)I e*/(ÿ)

|a|-* dyî1 д у у Ay.

By Lemma 1.2 and condition (1.17), the integral (1.18) is uniformly con­

vergent; hence it defines a continuous function. Furthermore, the integral (1.18) is also uniformly convergent for m = 0. Hence, by the theorem on differentiation of integrals with respect to a parameter, there exists the usual derivative

dmm * f )

— -— J-^— eC(Q).

d x ... dx°n1 П By (1.18) and Theorem 1.1,

\Daf \C(0) < ,11/11^^)

for all a such th a t [a| = m, with an absolute positive constant K m. Since condition (1.17) holds also for every s, 0 < s < m, so arguing as above, we get

и > 7 1 с « Ч ^ K s 1 1 / l l L f c

™ M'u>

for |a| = s, s = 0,1, ..., m —1, with an absolute constant K s > 0. Taking into account the two last inequalities, we obtain

l/|o » (û, < Æ ||/ l k * (o), feW fcS ?), with an absolute positive constant K.

B e m a rk 1.2. Since for 1c = n + m condition (1.17) holds for any -^-function N , so if all other assumptions of Theorem 1.2 are satisfied, then there holds the inclusion W n+m{Q) <= Gm(Q).

In some cases a stronger inclusion may be obtained.

Example 1.1. Let M(v) = vpfp, where 1 < p < oo, v ^ O . Then N (v) — vp'/p’, where l j p + l l p ’ = 1. The functions M and N satisfy the (/Ja)-condition. Let Ü be an open, bounded set in R n, starlike with respect a ball K { 0 , H ) a Q. We shall show th a t m = к — [n[p] — 1 (where [a]

denotes the integer p art of a) satisfies condition (1.17) for k > nIp. Since к > [nip] +1, we have 0 < к — [nfp] —1 < к —nIp. Hence k — ( k— [nip] —

—1 ) — n > k — ( k —n/p) — n = n l p — n = —n ip ’. There exists an s > 0 such th a t к — ( k — [nlp]—l) — n > —п / р' +е/ р' = — (n — s)/p’. Hence

дг^-(*~[п/*]-1)-») < jy •>/»') y V 0 < r < 1, and

J

j r-n+*dæ<

r < i ^ r < 1

Thus, the inclusion W k(Q) a Ck~ 1 (Q) holds for k >n l p . So, we obtained a known inclusion proved by S. L. Sobolev in [7], p. 78.

In the following example we shall show th a t Theorem 1.2 is a gen­

eralization of this inclusion.

Example 1.2. Let Q be as in Theorem 1.1 and let N(u) = wp,ln (l + u), и > 0 ,1 < p ’ < oo , k > nip', where Ц р + Ц р ' = 1. Moreover, let M be a complementary function to the Y-function N in the sense of Young.

Then the inclusion WkM(Q) a Ck~lnlp]~1 (D) holds and the imbedding operator is continuous.

P ro o f. As is known (see Example 8.2 and Theorem 4.3 in [5]), the functions M and N satisfy (d 2j„0)-condition. First, we shall show th a t for every e > 0 there exists ue > 0 such th a t

(1.19)

U>Ug

Щ ».) up,+e.

Let us denote by q the derivative of N . We have q (u) = p ’uv' 1ln ( l- f ^ ) + -\-up'j(l-\-u). Hence uq(u) = p 'u p’\n.(l + u) -}-ир'+11(1 +u), and further

щ( и) и , 1

—_ —_ — *)' _|______________ <; ж' j _________

N (u) ( l + 'it)ln (l + /it) "" ln(l-t-г^), Since 1 /In ( 1 -}- w) —*• 0 as u-> oo, we have

Vw > 0.

V 3 Ve>0 Me>0 p'

uq(u)

N(u) < p ' + e.

Hence we conclude th a t

« V 3 V>0 и >0 м>м. J еиf

ф ) N(t)

U

dt < (p' + e

и

dt

T *

Imbedding of Orlicz-Sobolev space 355

Thus

V 3e>0 ue w>we -*■’ \™e)

Hence we obtain inequality (1.19). Thus, if Jc<m + n, then for every e > 0 there is an re > 0 such th a t

X T ( J c - m - n \ - N ( r e ) (k - m - n ) ( p ' + B)

Х V ' ^{^} r ( k - m - n ) ( p ’ +e) ’ ' e

for all 0 < r < re. Hence

(1.20) ‘ j N ( r k - m~ n)) d x ^ x s f r (k~ m- n)(p' +e)dæ

r< re r< rg

for Jc<m-\-n. Let us pu t m = Jc — [n/p] — 1. From Example 1.1, we conclude th a t there is an ег > 0 such th a t Jc — (Jc — [n/p]— 1) — n >

—{n — Sjffp'. Hence

+ e) >

For s2 = exp'l(n — el) we have (n — s^j p' = мКр' + ez), whence

—({n — ex) l p' ) { p' - \ - e) >—n for 0 < £ < £ 2.

Thus,

С -ЖТ/ l) — »v -,

I N( r ' vp* ' ) d x < oo for re> 0 . r^rs

By Theorem 1.2, we obtain our inclusion.

2. The space W ^ ( ü ) and continuity of the imbedding operator from W $ (£ ) into W {S { Q ) . I t is obvious th a t WkM(Q) <= W ÿ ( ü ) for every non-negative integer Jc and th a t for Jc = 0 or 1, W {$ ( Q ) = WkM(Q) for every open set Q c R n. Moreover, if Jc = 0 or 1, then 41 * IUfc)(fl) = Ч# Чж№)(13) and if Jc is an integer greater than 1, then we have ^ M

(ад) V / ^ w î f ( û ) -

Since the norms 1|| • II j, and II • II * are equivalent, so for each non-

WkM(Q) "wkM(Q) 41 ’

negative integer Jc and for any open set Q c R n, the imbedding operator from <WkM(Q), into <Wÿ( Q) J - I l ^ (û)> is continuous.

We shall prove th a t in case of Ü such as in Theorem 1.1, W$ ( Q )

= WkM{Q) and the norms ||-|| к WkM(Q) and ||-|| ш 11 V jf’lfi) are equivalent.H

Lemma 2.1. I f f e W${Q), where Q is an open, bounded set in R n, starlike with respect to a ball K (0, H) <= Q, M and N are N -functions complementary in the sense of Young, satisfying the (A2>v^)-condition, then for every m, 0 < m < k, there exists the generalized derivative I)af e L*M(Q)

for |a| = m.

P ro o f. There exists a sequence of functions f t e C°°{Q), l = 1, 2, . . . , such th a t for a — (0, ..., 0) and \a\ = к

(2.2) \\Daf ~ D afi\\LM(Q)~>^ as ?->oo

and

(2.3) \\Hafi\\bM(o) ^ \\Daf\\LM{Q) (see [4], Theorem 1).

The functions f t satisfy equality (1.10) for each x e Q. For any positive integer к and for every m such th a t 0 < m < k, the following two cases may occur:

(i) 0 < n — к -f m < n, (ii) n — k Jr m ^ 0.

Let (i) hold. Then, applying Lemma 1.3 and arguing as in the proof of Theorem 1.2, we have

(2.4) dm(n*f,)

f 1

dxp ... dx°n _{lm ( ° )} ^J r n - k + m

О *

v i«i=fc

wi(x, y) ^dkfi(y)

dy? • • • dyann L M ( Q )

я ^ \№afi\\bM(Q)i

|al = fc

where the constant is as in Lemma 1.3 and the constant B x is such th a t

V V V

(x, y)eQxSi l a l = ^ l ^ l = w i < f c

х ф у

Taking into account (2.3) and (2.4), we get

<2.5) 11-г>"(я7,)|| < XlB, y \\D°f\\LMia), у № = я»< ft.

|a| = fc

Let (ii) hold. Then, applying Lemma 2.2 and inequality (1.2), we obtain I ^(i7*/,)(® )| = ^ уП—к+т ■ E wi{s>, y)Daf l(y)dy\

О |a| = fc 1

K B X у 11-0711^,, V W = m < t , x e Q , where the constant К is as in Lemma 1.1. Let К 1 = m a x (l, M(l)p(Q)]- Then, by the last inequality and by convexity of M, we obtain

Imbedding of Orlicz-Sobolev space 357

(2^.6⁾ QM Г______i L k b xk i

JD^JTfr)

Z № af\\LM(Q)1 = Г л гГ ____

J J i K B . K 1

|a| = fc Z \\Daf\\LM (Q)

]

|a| = fc

K 1

Г „ г/1, л ^

M(l)n(Q)

J Jlf(l)d a? < --- —i--- < 1 ,

и K l

\ / \ft\ = m < к.

Taking into account both, cases (i) and (ii), we deduce th a t for every positive integer к there is an absolute constant К > 0 such th at

(2.7) ll-Df’(tf*/i)|lijf(0) < K V HDVIlijrfoi,

V \?\

|a|-fc Now, we shall prove the inequality

(2.8) « x l ||/||tjlf(0|, V li»l = m < k - If \fi\ — m < &, then we have

i f w m x ) = ^ f (»)«*»•

|e|<fc-i lltfIKH Let

where Ct

A = max |D^(a?a)|, В = max sup |£a(y)|,

|/3|=m,0<m<fc laKfe— 1 H y l K H

|a|<fc— 1

' k , n —

Z

|a|<fc— 1

Jensen’s inequality for sums and for integrals, we have

< f M {

i >

a|<fc— 1 Hl/IKBT

J

I <h

h Z, f x(ABC^ f

/ д(дг(о д > ) { f x {ABC!>‘.ni‘ {K(o>H))\f(y)\<iy\d3’

k >n |a|<fc-l Q

< i : Z

<

k >n |a|<fc-l ^Si 1*{Q)

!Л(К(0,Н)) ^j M( ABCt _nf>(K(0, Я » \f(y)\)dy « eM(ABCKnp(Q)f ).

Hence inequality (2.8) follows with constant K x — АВСкп/л(й). By (2.7) and (2.8), we obtain

(4) For /u{Q) < oo the inclusion L*M (Q) c LX{Q) and inequality WfÏÏLM(Si) <

< K II/ILl^û)) hold. We apply here (2.3) to the norm П-Иь^й).

(2.9) W S J k r tm = \\о“(Щ1+п*Ш\ьМ{щ

^ Къ[М\\ьм{а) +

|а|=*

for |/?| = т < к with an absolute constant K 2 > 0. Thus, the assumptions of Lemma 1.1 are satisfied. Hence the generalized derivatives-DpfeL*M{Q) for every |/?| — m < k exists and the proof-is finished.

Th e o r e m 2.1. I f Q is an open, bounded subset of R n, starlike with respect to a ball K { 0 , H ) a Q, M and N are N -functions complementary in the sense of Young and satisfy the (Л2 &0)-condition, then for every non­

negative integer Te, W {ÿ ( Q ) = WkM(Q) and the norms || • | | (A) , IHL* are equivalent, i.e. the imbedding operator from (W kM{Q), || • || k > into

wmW

<fW(${ Q) , j|• and the inverse operator are continuous.

P ro o f. From Lemma 2.1 follows the inclusion W {$( Q) c W kM{Q).

Hence W($ ( Q ) = WkM{Q). By continuity of the norm || • \\lm(Q)i passing to the limit as Z->oo, we obtain inequality (2.9) with function / in place of f . Summing both sides of (2.9) with respect to |/?| < k, we obtain

^WkM{Q) ^ -^3 11/11^7^(0)

with an absolute constant K z > 0. Hence, by (2.1) and by equivalence of the norms 11| • || k " "wkM(Q)and I k , we get equivalency of the norms

WM ^

SI WkM(Q)k and and the proof is finished.

Vjg(û)

3. Equivalence of norms || • || kWM{0) imbedding Theorem 1.2 for the norm

and || • II1 k . We proved the

wm(°)

|| ‘II1 k • Now, we shall prove WM(°)

this theorem in the case of norm I k wkM(a) . For this purpose it suffices * * и are equivalent. First, we

wkM(o) H 7

I1 k and

W kM ( Q )

and to show th a t the norms II • Ц1 k wkM(o) and

shall show th a t the norm || • \\°wk is equivalent to norms || •

||*|| ,k) . Hence, by Theorem 2.1, equivalence of norms ||*

a7 (о)

ll-ll^fc (fl) will follow. By inequality (1.9), we have

II/l b (0) < Ck,n l Wf\\Lk where Ck>n =

lmW It\- k a!

]wkM{.o)

We shall prove the inverse inequality with an absolute positive constant.

Imbedding of Orlioz-Sobolev space 359

The following inequality is obvious:

WD’f h ^ o ) < V l l ^ o , for a11 l“ l = / 6 [ /] 6 L kM(Q ).

Hence

од) 111/1ьШ1<с;,п2ц/|и , /6[/]sib(û), where C'k,n = 2 1. Further, we have

|a| = fc

J A f ) = У fjf(|D7(®)|)*)< V Гж ([ У^-|Л',/(*)|21 )<to

|a| = * Ô |сф=* fl = -* '

>/([^ I r i ^ / w i 2] )*»

л 'L|/?l=* r • J /

< f Ж ( [ ^ | ^ С*.»Л 7(*)1!Г ) * > = J ACk,„f).

a \i-m = k P ’ J /

Hence and by (3.1), we obtain

(3.2) 2|l/I U lo, < 0 ; , n | | / | | <; , f e L kM( Q) , - (3*3) 1 lljTILfc ^ ^fc.nll/ILfc / g-^m(^)- Now, we shall prove the inequality

(3.1) ||/||Ьм(0), f e W U Q )

with an absolute positive constant K. Let К г — h A B /Ж-1(1), where A = max , Б = max Gl>n,G lin = V l .

|a|<fc— 1 & • 1 |a| = i

Furthermore, let a a == J Ca{y)f(y)dy. Then, applying Schwarz’s inequality r^H

and convexity of M, we obtain

Щ \ _ y J 7 у » 1 ГГ

max |a„| / L \ £ j a! | IcAB тах |л „ / ,

|a|<jfe <-U [a‘~l |a|<fe M ^ i 1)

kAGi>n т а х |л а| /

|a|<fc

Thus

(3.5) \\Щ\\¥к < ^ 1 maX К !-

|a |< fc -l

Applying Holder’s inequality, we obtain

K K / lta(æ)||/H |dr< J |/1(a?)|d®<2B1||xollijv(o)ll/llij£(o)

r<H

— ^2 ll/llz^fip \ / / e • Taking into account the last inequality and (3.5), we get the inequality

(3.6) _{V l ^} XWkM(D)

for a l l / e WkM(Q) with an absolute constant K 2 > 0. Now we shall prove the inequality inverse to (3.6). We consider two cases.

(i) J N( r k~n)dx < oo for some h > 0. Then г<Я

i/ и к ж м ; ^

for an absolute constant К > 0, all x e Q and f ^e TYkM{Q). Analogously as in the proof of (2 6), we obtain the inequality

(37) ii/n£M,a )< x * н л ц (а), V f e w U Q ) with an absolute constant К й > 0.

(ii) f N(iik~n)dx = oo for any h > 0. Then 0 < h < n. Taking into

r < A

account (3.4) and applying Lemma 1.3 to function Я*/, we obtain again (3.7). Hence, we get

(3.8) m °wk < * 0 ш и ,™ + 11 /И

M г'м(°) ^ ^ ° + 1)

for any f e W kM{Q). From (3.6) and (3.8) there follows the equivalence of norms II • II1k and || • ||° k . From the equivalence of norms || • || fc

wMW WmW LKM(£i)

and ^Hl k 11 "ькы(щ follows th a t of norms |HI° k 11 V y e ) ’ , || • Ц1 л V y t y , and || * ||V ^ fl) and further, by Theorem 2.1, equivalence of J J [Ml1wkM(Q) k and ||-|| k v y o ) . We proved the following

Th e o r e m 3.1. I f the assumptions of Theorem 2.1 are satisfied, then the norms ||-|| k and ||• Ц1 k are equivalent in the space WkM{Q).

E e m a r k 3.1. Since all norms defined in the space W kM{Q) are equi­

valent pair, W kM{Q) provided with anyone of them is complete (because it is complete for || • \\wm(q) 5 see [2]).

Imbedding of Orlicz-Sobolev space 361

4. Compactness of the imbedding operator from W kM{Q) into Gm(Q).

In this section we shall prove th a t if the assumptions of Theorem 1.2 are satisfied, then the imbedding operator from (W kM{Q), ||• || k > into

wm^

<Gm(0 ), I -|а»г(й)> is compact. First, we shall prove the following

L e m m a 4.1. Let Q be an open, bounded set in R n, let M and N be N -functions complementary in the sense of Young, satisfying (A2 ^-c o n d i­

tion and let

(4.1) . 3 3 / N (r*)dx< oo.

Л>0 ЛеК1

Moreover, let B ( x ,y ) be a continuous function on П x Q \{ ( y , y)} satisfying the condition

0-2) 3 V _|Я(*, У)|«

■A>0 ( x , y ) e P x Q X ф у

Then the integral operator

(4.3) Uf = f \\x -y \\xB {x, у )f(y)dy

Q

as a map from <.L*M{Q), ||• ||£ (Д)> into <G(Q), Hc(a)> is compact.

P ro o f. I t was proved in Lemma 1.2 th a t the operator Uf is a map from L*M(Q) into G(Q). We have only to prove th a t Uf is compact. Let d(Q) = sup \\x — y\\ and let R > d(Q). Then, for every x e Q, K (x, B) => Ü.

x, yeQ

Let us suppose th a t a set of functions F c L*M(Q) is given such th a t

(4-4) Il/Над»)

for all / e F, where К is a positive constant. We have to prove th a t the set {Uf}feF is bounded in G(Q) and:

(4.5) for every e > 0 there exists a ô > 0 such that i f f e F, x, x-j-Ax e Q and \\Ax\\< ô, then \Uf {xJr Ax) — Uf (x )\< e .

Applying Holder’s inequality and taking into account (4.2), we obtain (4.6) \Uf {x) \ < A f \\x -y \\x\f{y)\dy < 2 A \\f\\LM(Q)\\\\x-y\\% MK(XiR)).

Q

By (4.1), we have ||||^ - y ||A||bjv(ff(a;,fl)) = ||№1|л||ь м в д >) < oo. Further, by (4.1) and (4 jf#o), we get

||llyllA||zw(*:«>.d>)~>^ <5->0.