On problem o f density o f C ~(i2) in generalized Orlicz-Sobolev space W k M{Q) for every open set Q <= R n

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1977) ROCZNIKI POLSKIEGO TOW ARZYSTW A MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE X X (1977)

H. H tjdzik (Poznan)

On problem o f density o f C ~(i2) in generalized Orlicz-Sobolev space W k M{Q) for every open set Q <= R n

Let Û be an open set in R n, ц = Lebesgue measure in R n, R + = ( 0, oo), W + — the set of all non-negative rational numbers and let Ж be a real­

valued function defined in Q x R + and satisfying the following conditions :

1 ° M ( u , v ) is an Ж-function(J) of the variable v for almost every

U e û ,

2° 'M(u, v) is a continuous function of the variable и for each v e R +) 3° there is a constant c > 0 such that M(u, 1) > c for almost every

U e Q .

We denote by F ± the set of all complex-valued functions / defined and Lebesgue measurable in Q. F 0 will denote the set of functions feF^

such that f(x) — 0 almost everywhere and F will be defined as F — F x jF0.

According to the assumptions, the function M {x , |/(ж)|) is Lebesgue meas­

urable for each f e F 1 and M[x, \/г(х)\) = M(x, ]/ ² (ж)|) almost every­

where, if f x- f ^ F Q.

Given a non-negative integer h, we denote

TtrkM(Q) = { f e F : V 3D°feL*M(Q)},

|a|<&

where a = (ax, ..., an), a{ are non-negative integers, |a| = ax + ... + a n, D a = d|a|/d#ai ... dx^n is the generalized derivative operator (i.e., deri­

vative in the sense of the theory of distributions) of order |a| and L*M{Q) is the generalized Orlicz space (see [ 8 ]). Let

N ( îi , v ) = sup{ a v ~ M ( u , o')} = sup {av — M (u, a)}, ue Q,

cr>0 OeW

i.e., N(u, v) is the Ж-function of the variable v complementary to Ж-func-

(Ь See [6], p. 12-75.

5 Roczniki PTM — Prace Matematyczne X X

tion Ж (u, v) in the sense of Young with respect to the variable v. Further, let us denote

Q(f) = j M (x , \f(x)\)dx, feL*M(Q), о

Qa(f) = f M( x , \Baf(x)\)dx , /с W k M{Q), \a\ < Tc,

a

e ( f ) = 2<>ЛЛ,

\a\^k

W f h ^ = Ы |« > 0: г ( Ц < 1 j , feZ*M(D), Ш1м(о) = sup \\jf(oo)g(x)dx\\, ftL*M(Q) {2),

11 / ¹¹ ^ , = in ff > 0: < i } , A W ir(fi), П i2e = {#€,£?: inf |[ж — 2/11 > « } , where ||& || = ( V a^)1/2,

V*r(Q) * ' é l •

and Г { й ) is the boundary of Ü.

The Holder’s inequality (see [ 6 ], p. 98)

(1) J j f{®)y{x)dx I ^ ^{\\Л\ьм(о)} Н^/Иь^дя)» f €IjM(Q),geLN(Q)1 a

and the inequality (see [ 6 ], p. 97)

(2) И/Иьд^Я) ^ Wf\\lLM{Q) ^ 2 ||/|1х,м(Д), f e L M(Q) hold.

First, we shall, prove some lemmas.

L

1 . Let Q <= R n , Qx c= 6e open sets and let f ( x , y) be a com­

plex-valued function defined and Lebesgue measurable in Q x Qx and /(•, y) eL*M(Q) for almost every y^Qx. Then the following inequality(3) holds:

(3) II / / ( • . y ) fy 111 (B| < / ! ! / ( • , y n lu f^ d y.

Six Qx

P r o o f. Let J(x) = J f ( x , y ) d y , g €L*N(Q) and |lsr||ijv(0) < 1, where N {u, v) is the complementary function to the Y -f unction M (u, v) in the sense of Young with respect to the variable v. Then

J (x)g(x)dx I ^ I ( / l/(®» y)\dy}dx

n ox

^ / ll/(* ? y)\\LM(Q)\\9\\LN(Q)dy < J ll/(* , y)\\LM(Q)dy .

Q Q

(2) See [6], p. 98.

(3) This irtequality is a generalization of Minkowski inequality (see [2], p. 148).

Density in generalised Orlics-Sobolev space 67

Hence and by definition of we obtain (3).

L emma 2. Let f t W kM(Q). I f for every compact set 8 in R n,

' (4) — 0 (ek)i as e~> 0 ,

then the function

(5) Ф{х) =

[ ⁰ ,

X€Ü, X4Ü belongs to W k M{Rn).

P r o o f. It is known (see [1], p. 55-66) that there exist generalized derivatives D a<P for \a\ < h and

D a 0 {x) D af ( x) , x e Q ,

0 , x i Q . Hence, Lemma 2 follows immediately.

L emma 3. Letfe W\I (Q). I f there exists a sequence of functions q>se Cf(Q) such that

lim||/—<pe|| k = 0,

S—►oo

then Фе Wk M(Rn).

P r o o f. It can be assumed that cpse C f {Rn) for s = 1, 2, 3 , . . . Thus, w r , i e W k M(Rn), and

< \\<p,-nwki(0) + l l / - ? ,‘ llIF «jf (OX 0 > a s s >

Thus, the sequence {ys}fLi is a Cauchy sequence with respect to the norm

||*|| k . But the space Wk M(Rn) is complete (see [3]), so there exists

w M ( R n )

a function F e Wk M{Rn) such that

lim\\F—<pt\\ k = 0.

s-+oo

Hence, in particular, it follows that

( 6 ) lim ||_F — <р 8\\ьм(цп) = 0 .

s->oo

Since (ps{x) = 0 for xeR n\ Q , so from the last condition it follows that

= o. Thus, F{x) = 0 for almost every x e R n\Q. From ( 6 ) it follows that

P 1 - ÎW

(Q) ^ llJ^ ~ ? )s llijM(Rn) + I I /— aS S - + 0 0 .

Hence F(x) = f{x) for almost every xeQ, and so F (x) = Ф(х) for almost

every x e R n and Фе Wk M{Rn). Thus, the proof of Lemma 3 is complete.

Now, we denote:

f s{x) = e

h [

Л '

x - y

f (y)dy, feL{oc№)

/;<*) = 8 - f y> ( ~ - \ f ( y ) d y , f e Z ^ O ) (4),

a ' '

where ipeC°°(Rn), supply = А (0 ,1 ) = {xeRn: ||a?||<l}, tp(y) = y(a?) for IM ¹ = ||y||, y { x ) ^ 0, J y > { x ) d x = 1 . Obviously, f e, fle C ^ R 71). If £ is

R n

bounded, then f eeC™(Q), f eeC™{Rn). Further, let us introduce the following condition:

4° there is a constant к > ² such that inequality

{Af) M(u, 2v) < xM{u, v)

is satisfied for almost every u e û and each vc R+ .

Let p(u) be a continuous function in R n satisfying the following conditions : there exist positive constants сг, c2 such that p (u) > c1 for almost t every ueR*1, p ( u ^ Au^ff) < c2p(u^)p(u^) for ||w(1)|| <

Let us put

M ( u , v ) = p(w)Jf ¹ (v),

where М г{и) is a real-valued A -f unction satisfying condition (zl2) for every 0 (see [7], p. 35)(6). As well known(7), such a function M(u, v) satis­

fies the following condition

5° /Ж (# , \fe(x)\)dx < C JM [x, \f(x)\)dx, 0 < е < 1 , A c Q, is a

л A

measurable set, where C is an absolute constant. For such M(u, v) we shall prove the following

T

1 . I f f e Wk M{Q) and the condition (7) ll/lln^nv^) — 0 ( ek) holds, as s->0, then there exists a sequence of functions (pse Cf (Q) such that

(8) II/— 0>eIL* as s->oo.

(4) If feL % [(Q ), then feL\oc(Q) (see [3], Lemma 1).

(5) See [4], Examples (22), (23), (24).

(6) In the case ц (Q) < oo it suffices to assume that condition (A2) is satisfied for large v.

(7) See [4], Example 1.

Density in generalized Orlicz-Sobolev space 69

P r o o f. We shall consider two cases.

1 ° Ü is an open and bounded set in R n. Then f ecC™(Q) for s > ⁰ . We show that the functions f B satisfy condition ( ⁸ ), as e->0. From (7) and the inequality II/II z ^ a ) < C \\f\\LM^0) (see [3]? Lemma ¹ ) it follows- that Фе Wk M{Rn) (see Lemma ² ). Furthermore, f e{x) — f'e(x) for xe Q3t,

_ / x — y\

Indeed, if х е й Ъ е and y e Q \ Q 2e, then \\x — y|| > e. Hence ^1---1 = 0

îot y e Q\Q2e, X€ÛZeând

f e{x) = £'

/ * ( ■ x - y

f(V)dy =/,(®).

Thus

||Ф—Фе| | k — ||Ф—Фе|| к п 9 J &S £~^0 ( 8) ,

11 elV^f(Rn) ellPF^f(Rn) ’ п

and

ll/-/e llTFy 0 3e) = aS £^ ° *

Next, we can write

II/ / e l l ^ (13) < II/ f^w^Oto) + + ^^w k M(a\n3t) * We have ||/|| k ->0 as e->0, by virtue of Lebesgue bounded convergence theorem, and by condition 4°. Now, we prove that

ll/ell к ->0 as e-^ ⁰ . Since the function M ( u , v ) satisfies condition 4°

and the following inequality holds

(9) m ax\\Daf\\LM(Q) < ll/l! fc ... < cn}km^\\Daf\\L {Q) (9),

|a|<fc

|a|<fc

where спЛ = ^ ¹ , it suffices to show that f M (x , \Daf s(x)\)dx-^0,

|a]<A; я\дз в

as £-> ⁰ , for |a| ^ ~fc* Since D Jg — 0 for cc € i Q ^ j oc | ^ so W ⁶ liâ)V ⁶ f M(xj \Daf s(x)\)dx = j M ( x , \Daf e(x)\)dx for |a|<fc, e > 0 .

Clearly, we have

Л°Л(*) = -péüT J ^<pi~\f(x^z)dz.

x —s*Qo, 1*1 <«

(8) See [4], Theorem 5 for Q = JRn.

(9) See [3], Proposition 2.

Now, let us put

ca = f \DaW{z)\dz for |a|</c(10).

i * k i

Then e~n j \Da4/{zle)\dz = ca. Applying the integral Jensen’ s inequality, we obtain

M (x , \D°f.(x)\) < M x ’ 4 г f |ДТ (г/е)1 • c „ l / ( œ - * ) l < i * ' e и «. ⁶ c-

„ r jB « r ( ⁷ e)| Jr( cx \f(x-z)\\

^ J ?c. МГ —F —

1*1 <« “

dz.

Hence, by virtue of Tonelli theorem, we have

( 10 )

J M (x , \Daf e{x)\)dx^ j j J

Лб\ЯЗв °е\й3б N<e

/ { J

а з \ а 3 е

'l * l < «

\Baf ( x - z ) \

--- П--- М \х ’ Вс„ \ в

• W M >/(— M J

е"о„ М\х,

,!«1 dz)dx

1*1 <«

where na is a natural number such that ca < 2n°, the constant к is defined by condition 4°. Let us assume

Ji(s) _{S£\ û3 e} J ^{M (x,} _' l/(® -* )l \

e,a| ) dx for |#| < e < ¹ ,

and let us estimate «^(г). Let м = x — z; then \du\ = \dz\. If \\z{\ < e, then

( £ в\ £ ?3е) — 2 c= Q \Q U. Hence for ||«||<e<-l, we have (11) Jx{z)

where p( ¹ ) = sup p(z).

11 ^* 11= 1

( 10) I t c a n b e a s s u m e d t h a t

Density in generalised Orlics-Sobolev space 71

By (10) and (11), for 0 < s < £, we have

f

ü e \ Q 3 e . й \ " 4 е ' 8

as e->0, |a| < Jc. Thus, the proof in case 1 ° is complete.

2° Q is an arbitrary open set in R n. Let gs{x) — f e{x)xAx )i where XÀX) = *(«»), %eC°°{Rn) and %{x) = 1 for \\x\\ < 1, /(a?) = 0 for ||æ|| > 2, 0 < %(x) < 1 . Obviously, ge€C™(Q) and supp ge cz J}en { x e R n: ||a?|| < 2/e}.

We shall prove that the sequence ge satisfies condition ( 8 ), as s->0.

We have

( ^{1 2} ) ITg,{x) = B “/ . (*)*.(*) + £ \a\^k.

Д+У=а

УФ0

Similarly as in case ¹ °, we have

II/

^ II

. ~Ь ll/ILfc , о ч n > ~f" ll^elLfc , о ч 0 . •

W M W WmS °

3

) ^ ( “ 4 “ Зе)

It is known that ||ye| | k -^ 0 , as e - > 0 (see case 1°). In order to prove t^M( 0\Q3s)

that ||grj| k -^ ⁰ , as e ^{- > 0} it suffices to show that each component

W M (a \ a 3e)

of the sum ( ^{1 2} ) tends to zero with respect to ^ ж(о\.о ³ £)_convergence.

We^shall apply also case 1°. Let |a| < h. Since for every xeQ, mAx)Xs(x)\<\I>afe(x)\,

so by case ¹ °, ||.Da/ e*e|| * .-*<>, as e^O. Now, let p + y = а, у Ф ⁰ , then D v[xÀx )] = sMx(sx).

Denoting

{ 1 21

1 /eP2/s = I xeRn: — < Ця?Ц < — J,

we have D v[Xe(x )] = ® for &€Pn\ ¹ /eP 2/E. Moreover, if the following inequality holds:

(13) |D7.(*)-Dv[z.M ]l « v ^{1' 1} |-07.(®)1х1/Л,.(* ),

where cy = sup \D 7%{x)\.

Hence, by virtue of the proof of case 1° and condition 4°, for У Ф /5 + у — a, we have

/ M {x , \Dpf s(x)Dv[xe( ^ ) l \ ) ^ ^ 0, as e -» 0 .

°\ Q3e Thus

\\I>pf eI)vXs\\LM(ü\a3e)-^0, as ^{£ ^ 0} for \0 + y\ = |a| ^ lc , у ф ⁰ . Now, we prove that

(14)

(û .“ ^ 0 , as £ ^ 0 . wМ^а ^з«)

Since f'e(x) = f e(x) for x e Q ie, so

\\f—ge\\ k =\\ф~ф х \ \ . < ||Ф — ф'ув|| k , ,

lJ yeV y o 3£) вЛ eAB"wk M(Rn)’

where Ф is defined by (5). It is known, that Фе W k M{Rn) and (16) ЦФ— ^ellTFfc^(RW)-^0) as £-> ⁰ .

By virtue of (9) it suffices to show that

\\Da0 — D a{ 0 e Ze)llz,M(RW)_>0, as £->0, ^Jot I < h.

Further, we have

(16) j M ( æ , \ 0 ( x ) — 0'e(x)xe{oo)\)dx= j M (x , \Ф (x) — Ф'е(х)\) dx +

Rtl IMKl/e

+ j M ( x , \ 0 { x ) - 0 'e(x)xe{%)\)dx+ J M (x , \ 0 {x)\)dx.

1 /е<||ж|К ² /е ¹ И ¹ > ² /е By (16),

J M {x , \ 0 {x) — 0 'e(x)\)dx ^ J M {x , \ 0 (x) — 0 's(x)\)dx->O, as £-> ⁰ .

1 !ж||< 1 /е Rn

We have also

(17) J M (x , \ 0 (x)\)dx->Q, as £-^ 0 .

M > 2

Indeed, by virtue of condition 4°, the function M(x, \0(x)\) belongs to L ^ R 71). Furthermore, 0{x)%Rns^((i)2js){x), where K(0,2je) = {xeRn: ||a?||<

< 2/fi}, is convergent to zero for almost every x e R n, as £->0. By contin­

uity of the function M (u , v) with respect to v, M(x, \Ф(х)хцп\щ0>21е){х)\) -^0, as £— >0 for almost every x t R n. Moreover, M(x, |Ф(ж)хйп\щ 012 /е)(ж)|}

< M (x , \0{x)\) for every x e R n. Thus, by virtue of the Lebesgue bounded-convergence theorem we obtain (17). Now, we consider the integral

/17/) J M ( x , I 0 ( x ) - 0 'e(x)xs(x)l)dx, as £ ^ ⁰ .

' l/e<l!a;!l< ² /e

Density in generalized Orlics-Sobolev space 7£

We have, |Ф(гс) — Ф'(ж)^е(ж)| < |Ф(а?)| + |Ф'(ж)|, for x c R n. Hence, by convexity of M (u, v) with respect to the variable v and by condition 5°r for ⁰ < s < ¹ , we have ⁴

j М(х,\\Ф{х) — Ф'е{х)%е{х)\)йх^ J M ( x , \ 0 (x)\ + \0 's{x)\)dx

l/e<||£C|K2/e ||£C||>l/e

J M { x , \ < P ( x ) \ ) d x + \ J M { x , |Ф' ( x ) \ ) d x

ll*!l> l/e !!язЦ>1/в

< -| (l+ C ) j M { x , \Ф(х)\)с1х->0, as s-^0.

ii * ii > i/«

Thus, integral (17') tends to zero, as e->0. By (12) for |a| < Tc, we have

=\\ в ° ф - 1>° ф : х, - £ - , )

Р+У=а УФО

< w ф - о ° ф : 2 ^ г \\(П11Ф ',-^ Ф )П -'хЛ ьм1нп)+

P+Y=a уф о

+ Gpy\\DP&DVXe\\LM(Rn)-

Р+У=>а уф О

Similarly as (17'), we have also ||Н“ Ф — ^ “ Ф^Н-Ьд^в»)- *® as s->0. Let /9 + у = а, у ф 0 . Then, by virtue of (13) and (15), we obtain

IK-Z/ Ф — ТФФ’,)V>x.\\L a m < e'” 1 CyIIBOФ - (Df Ф)'.\\С м т ^ 0, as e->0. Moreover, by (13), we obtain

т ^ Ф ^ х Л ь ^ < CyS»' w ®lliM(l(<P ² , ^ 0 , as e-M),

l^ + yl = N Ф 9.

Hence

||Ha{Ф —g ^{e) 0} , as ^{£ - > 0} and for every \a\^7c.

Thus, by virtue of (9), it follows that \\f— gB\\wk ^{- > 0} as e-> ⁰ , and the proof of case ² ° is complete. ^ ³⁸

Now, let us assume that M(u, v) = M(v), where M{v) is an W-func- tion satisfying condition (zl2) for every v > ⁰ . Thus, there exists (n ) a real­

valued and convex function К defined on R + such that: K(u) = 0 if and only if и = 0, К (1) = 1, К is an increasing, continuous function for и > ⁰ and

(Zl2) M ( u ‘ v) < K{u)M(v) for every u , v ^ 0.

(n ) К (и) = sup

« > 0

M (u -v )

--- for every и > 0, see [7], p. 57-58.

M (v)

Then conditions l°-5 ° hold. Let K ~ l denote the inverse function to K.

For such function M(v) we shall prove that if for f e Wk M{Q) there exists a sequence of functions cpseC™(Q) such that condition ( ⁸ ) holds, then condition (7) is satisfied. First, we shall prove some lemmas.

г dt

L

3 . I f the function M is such that J - — — —— < oo, f e C (Q),

о )

.8 c= Ü <= Вп, у (8) > ⁰ , Ü is bounded and starlike with respect to 8, then the following inequality holds

<18)

fx{Q)N~l

[x{8)N ~l

\\fl\lMm + 0 (n, M)d^

i = l

àf_

дщ

|i I LM(Q)

■where d — sup \\x' — x"\\,JY complementary to the N -function M in the

x ' , x " e ! i

sense of Young.

P r o o f. Since the set Ü is starlike with respect to 8, so for X€8,y e Q , t €<^0, 1>, we have x Jr t{y — x ) e û . First, we shall prove that

<19) Щ х + Ч У - Щ \ ь м(й),у < { ^ " ¹ ( n } “ ¹ ll/lliw(û).

Let us put x-\-t{y — x) = z. Then j M [ \ f ( x + t(y

Si

-ж))|] d y = t ~ n j M

Q

Thus, we have

\f(x + t{y-x))\ \ m i

!l / ll LM(£i) dz.

•Consequently, inequality (19) holds. From (19) it follows that

i

/ ^{dX; df_} (x + t{y — x)) dt <

% ( fi)

dt K ~ l {tn)

df

дх 4 LMiQ)

It is known that the following equality holds for fe C 1 (Q) (see [ ¹ ], p. 52):

f(y) = / ( ® ) + J -— -(x + t i y - x ^ i y i - x j d t .

Hence

n 1

\f(y)\< № ) I+ Æ^ M-^ - ( я + % - я ) )

i = 1 Ô * %

dt for Xe8, y e Q ,

Density in generalised Orlics-Sobolev space 75

and thus applying inequalities (2) and (3), we obtain П 1 II df

m i MW,y < \\sm'bM{a),y+ d v / M - ( œ + « ( y - ® ) ) i=l ⁰

dt

II/ (ж) \\LM(a),y + 2 d IJ*

L]tf{p),y

1

— . ^\ y l j ^ 4. K -'it*1) Z j WdXi

'0 4 ' ' i = l 11 1

From [ ⁶ ], equality (9.11), we have

\\f(X )WlL M {0),V

=

where N ~l is the inverse function to N (v), N (v) is the complementary function to the Y -f unction M(v) in the sense of Young. Thus,

H

г = 1

Taking the norm || • ? we have

[ ^ ] < [ ^ ] M W

П

II

II1

+ C { n , M ) d V U - ^p{S)N-

é l 1 II dœ* ' L o » ) ] ’ i.e., we obtain inequality (18).

By inequality (2), we get

'B(û)-jrlRWl ^{* ii df}

< 2 0 ) l l / l l i M (a ) < 2 --- 1 1 / 1 1 ^{+ 2} a in, M)d У - L

L emma 4. I f f e C k{Q) and Ü is as in Lemma 3, then the followin inequality holds:

L M (Q)

<21) 11/11 ^LMm < C3 {(1 + d)*-1 C ( Q , S , M ) II/II b ^ V s , + ^d* у ll-D°/lltjl(0)} ,

|a|

where the constant C3 depends o n n , Je, M only, and

[//(D ) C( Q, S, M) =

2p ( Q ) N - ' \ - ^ \

p(S)N-

[м$)]

P r o o f. From (20) it follows that for к = 1 inequality ( ^{2 1} ) holds.

Let ( 2 1 ) be true for к — 1 . But Daf e C l {Q) for |a| = к — 1 , so by virtue of (18), we have

11/1Ьм(П)< e,{(l + d)*-2C(û, 8 , M)\\f\\wk_2iS) +

+ d k- 1 У \ c ( Q , S , M ) W f l \ L (s) + 2 C ( n ,M ) d V l l f w ) 11

i Æ . L ы » дх<

< 2c ³ m a x (l, C(n, M)) ^{{ ( 1} + d f - 1 C ( Q , 8 , M ) 1 1 / 1 1 ^ + ^ £ И ■

\a\=k

C o r o l l a r y 1 . I f f e W k M{Q), Q is bounded and starlike with respect to S and

there exists a sequence of functions <pseC™(Q) satisfying ( 8 ), then f satisfies ( 2 1 ).

C o r o l l a r y 2 . I f f e Wk M{Q) , Q is bounded and starlike with respect to S, then f satisfies ( ^{2 1} ).

P r o o f . This follows from the density of G°°{Q) in W^( Q) (see [5]) and from Lemma 4.

C o r o l l a r y 3 . I f f e W k M(Q), where Q is as in Lemma 3 and f (x) = 0

for xeS, then

(22) !l/lliM(o) < L ' W l w

|a|=A;

} dt

L e m m a 5. Let Ш be such that --- < oo and let Q a R n, Q ф R n

0J K ~ l {tn)

be an open set. Then for every f eC ^( Q) the following inequality holds : 2 3

( 2 3 ) ll/lln M( û \ f le) ^ c4 fifc ^ ll-D 7 lU _ M( a \ o 2e) >

|a|=*

where the constant c4 depends on n, к, M only.

n-times

P r o o f . Let В = { . . ., —2, —1 , 0 , 1 , 2 , . . . } , Bn = 5 х Б х ... х Б , and

G ! G

Further, let — l = [—

’ 2 \ 2

= \xeRn: X - - 1

2

/l(®) = /(®),

0 ,

Х ей , x i ü . e

In • • • i j for l = (h, • • 1 ^n) and

—L Then e )

2 j U #

UBn И) = R n. Let

’•i)

those balls K ^ l , ^ that £?\Ц, п . ёг |— Z, Ф 0 (if Q is bounded,

then there exists a finite number of such balls, if Q is not bounded, then

Density in generalised Orlics-Sobolev space 77

there exists a countable set of such balls). Let us denote these balls by L{

and their centers by ж(г). Further, let be a point from Г ( й ) such that d(x{i\ Г{ й)) = ||a?(i) — y(i)\\. Let us denote by the balls with centers y{%) and radius 2e. Since L { c QiJ so

Wfl\LM(0\Qe) ^ \\fl\\LM(Li) < \\fl\\LM{Qi)‘

i i

*

We have f(x) = 0 in Q — Qh for sufficiently small Ji. Hence there exist smaller balls in the balls in which f (x) = 0. By inequality (22), we obtain

\ \ f l \ \ L M {Q i) < Сз ( 2 е ) * £ \ № af l \ \ L M (Q i) •

|a|=fc

Thus

ll/lliM (0\£>e) ^ _|a|=fc

Since every point xeQ{ belongs to at most 16w balls of the family {Qj}jLi (see [1], p. 54-55), so

%OrsQi{x) < 16nXu(^^Qt.)(^) » x e R n.

% i

Hence

11/11ьж (Д \«е) ^ C4£fe1 6 n ^ ll-Da/ l l i ilf(i3\02s)j

|a|=A

i.e., inequality (23) holds.

C

4 . For every function f e W k M{Q) such that there exists a sequence of functions (pseCf (Q) satisfying ( ⁸ ), inequality (23) holds.

P r o o f . If ( 8 ) holds, then

№ ~ (P s \\ l m (0)-+ 0 and ^ \\Daf — В а(р 8\\Ьм(а\оя)->0 1 as e-+ o o .

|a| = k

Hence, we obtain

|a|=& |a|=&

Since ( ^{2 2} ) is satisfied by the functions (p 8, as e->0, so, by (24), it is satisfied by /, too.

T

2. I f f e W k M(Q), where Q is an arbitrary open set in Rn, M(v) satisfies condition (zl2), J ^{dt dt < oo} and if there exists a se-

о к ~ Ч П

quence of functions <pseC™(Q) such that ( ⁸ ) holds, then condition (7) is satis­

fied.

P ro o f . If Q = R n, then (7) is satisfied immediately. Let Q Ф Rn and choose <5 > 0. From ( ⁸ ) it follows that there exists s ⁰ such that

Thus, the proof of Theorem 2- is complete.

The autor is much indebted to Professor J. Musielak for his kind remarks in a course of preparing of this paper.

[1] В. И. Б у р е н к о в , О приближений функции из пространства Wp(Q) фини­

тными функциями для произвольного открытого множества, Труды Стек­

лова, C X X X I , Москва 1974, р. 51-63.

[2] H . G-. H a r d y , J. Е. L i t t le w o o d and G. P o l y a , Inequalities, Cambridge, Uni­

versity Press, 1934.

[3] H. H u d z ik , A generalization of Sobolev space (I), Functiones et Approxim ate II»

Poznan 1975, p. 67-73.

[4] — On generalized Orlicz-Sobolev space, ibidem 4, Poznan 1976, p. 37-51.

[5] — On density ofO °°(Q ) in Orlicz-Sobolev space W \l {Q) for every open set Q <= R n, ibidem 5, Poznan 1977, p. 113-128.

[6] M. А. К р а с н о с е л ь с к и й , Я. Б. Р у т и ц к и й , Выпуклые функции и прост­

ранства Орлича, Государственное Издательство Физико-Математической Литературы, Москва 1958.

[7] Н. M u s ie la k , Inequalities of Bernstein and Privalov type, Functiones et Appro­

x im a te I, Poznan 1974, p. 5 5-66.

[8] — and W . O r lic z , On modular spaces, Studia Math. 18 (1958), p. 49-65.

Let s0 be such that

1!9%И£м(Я\Де) — 0 f°r 0 < £ ^ £0.

Then, for 0 < £ < £0, by Corollary 4, we have

l l / l l i M( a \ o e) ^ ll/~ '9 % Н ь д/( о \ о г) + 119%11£м ( о \ л е)

References