a n n a l e s s o c i e t a t i s m a t h e m a t i c a e p o l o n a e Series I: COMMENTATIONES MATHEMATICAE X V II (1974) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I : PBACE MATEMATYCZNE X Y II (1974)
Magdalen a Ja r o sz e w sk a (Poznan)
Some properties oî the spaces W (ap,Z)(Û)
1. Introduction. In this paper we investigate the space W^'^Æ) which consist, roughly speaking, of functions / defined in a product Q = Р ”=1£,- such that the generalized derivatives of orders less than or equal to s belong to the space M^P,^(Q) with mixed powers p = {px, . . . , p n), where A = (A1? ..., AJ (for the information about the definition of M^P,X){Q) see Section 2) and s > 0; if s is not an integer, the additional assumption (18) is to be imposed. The space W{P,X){Q) is a generalization of the space W^P,X)(Q) defined and investigated by Campanato [4] to the case of mixed powers. Campanato’s case is obtained if we put n = 1, and if additionaly A = 0 we get Sobolev’s space (see [1 0]).
The aim this paper is to investigate properties of functions belonging to 1Г^,Л)(£). In Section 2 there are introduced some notations and defini
tions and there is proved a lemma, applying the methods of Campanato.
Section 3 contains theorems concerning inclusions, continuity and Holder’s condition in case of integer s, in Section 4 we study the same properties as in Section 3 but in case non-integer s. The results contain those of Campanato [4] and [6]. In Section 5 completeness of the space
HŸ’A)(,Q) is proved. '
I am indebted to Prof. J. Musielak for his kind remarks in a course of preparating of this paper.
2. How we introduce the following notation. The index i = 1, ..., n everywhere, unless otherwise stated.
Let В be the set of real numbers, and let т { be positive integers.
We take two systems p = {px, . . . , p n), q = {qx, ..., qn) of non-negative integers such that > ... > p n, qx ^ ... ^ qn and we consider numbers A* satisfying the inequalities 0 < ?H < m{. In the following we shall apply vector notations, i. e. m = (ш1, . . . , ш й), x = (x1, . . . , x n) etc. We shall
n n n
write also A = a{, A = A*, N = m*. Let be open, bounded,
г= 1 i = l i = l
connected subsets of the real Euclidean space B mh By d(£ J we denote the diameter of Д-; and we write Q = Pf=l = P”= 1 Q{. Let I{x% а В т *
be the ball with centre at х\ей{ and radins > 0, and let 8t = I (х®, @{)п n û (J I (a?0, q) = P?=1I(a>“, qJ , 8 = I{x0, g ) n ü . We suppose that the boun
dary dQi of Q{ is of Lebesgue measure zero in R rn{and that û i satisfy condition (A) (see [8]). To simplify the notation, we shall write for example
/ If(x)\dx = j . .. j If{x)\dx1 ...d x n,
s sn Si
f \f(x)\pdx = ||/||* = / [ . . . ( /\f(x) \ ^dXi)P2lPldx2 . . .f JPn~4xn , etc.
bs sn Si
Let g(y) be a function defined and integrable on Q, and let V6\(®)= I
д(У) лл.
N - A ^
\x-y\
g{y)
\x-y\N - A d y i ... dy„
and a = (a1? ..., an), 0 < щ < m{.
In this paper we use the definition of L^P,X)(Q) and M^P,X)(Q) spaces originated from [8] and [9]. The definitions are slightly modified so that numbers At- related e. g. to M^P,;)(Q) should be substituted, in the definition and elsewhere, by numbers Х^п/р{. Again, if we refer to results of these papers, we accept the modification. Then e. g. condition Х{ > pnт {jp{
should be substituted by X{ > m{ etc.
First we prove the following lemma.
Lemma. I f g e M {l,X){Q), then
1° if 0 < b < m i - a t , then U°A(x)e Ж(1’а+А)(£);
2° if Xi = ~ , then и А(х)е for ju{ < m{.
In both cases 1° and 2° there holds the inequality (1) IIUA(^)||м(ь^)(й) ^ ох\\д\\М(1,ц{п),
where — аг- -f лг- if 0 < Аг- < т { — аг- and /л{ < mi if Х{ = — at ; 3° if —- at < Аг- < , then UA (х) is bounded in Q and
sup I V°A (x)I < cM \M(i,x)(a).
Xe£i
P ro o f. Let К be the norm of g in and let х°{е й ^ > 0, 2q{ < ^ = a{ Qt), where d{Qf) is diameter of Qt .
First we suppose that 0 < X{ < т { — щ. If xe I ( x 0, q), then
(2 ) V°a (x) \ < / \x-y\
g(y) , .
N-A “Ь I \x-y\4 = : * *
= Т В Д 4 - W ° A ( X ) .
Properties of the spaces 1\Т(^ ’Х) {Q) 361
We put
*Px ir d) I I g{y)\dy.
l-P*=1 Qi - F ni ^ X1 ( 4 l 1 (xi> ri)
Then cPx{rd) = 0 for 0 < r{ < q{ П
cpx {rd) < К f ] 4 < K fi for Qi < Vi < o l
i= 1
If we argue analogously to Campanato [4], we get ed
V!b(x)< J r j - N<p'x (rd)drd < |l + ] K ' ей
. ^ - w +л
<l‘ + ï ? ï b ! - n a,
where J . — IV + Л < 0 . Integrating the above inequality with respect to on 8t a successively for i = 1, ..., n, we obtain
( S ) / < 03(N, A, A)K-fJ g ? + 4
Now, we estimate W°A (x ). We have
(4) J W ^ d x = J' \g(y)\dy J dx
I{Xq,2q)^Q \x-y\N - A dx
J (Xq, 2 g)r\Q
Р^=1/(^,4ег) те
e4 ( J f , A,A)-K-f]eï< + Ч
Because |ж — y| (iV ^ < П1^г — Уг1 (m< inequalities (2), (3) and (4)
imply *=1
(5) J IU ^ (x )ld x ^ e 5- K - [ ] t > ? +%
and we get 1°.
Now, we take Лг- = mi — ai1 then ge M (l,A~e){Q) for every 0 < e{ < miy e = (fij, £n). Repeating the above argumentation we obtain 2°. Ine
quality (5) yields (1). Finally, let us consider the case т { — щ < Яг- < m{, Xx(rd) = J ... f \g{y)\dy = J |flr(y)|<fy,
I (xn’ rn)r'Dn I(x1,r1)r,nl P f= iHxi>ri)^ °i
where 0 < rt < Arguing as Campanato [4] we get easily П
X x i r a X K 'l J r p ^ K r i ,
i= l
and
| Е а д к f r i - N-x l(r d)drd < д:|е$-"г+л+ л ^ 7 - д г
thus ohtainig 3°.
3. Let 1 < Pi < oo, 0 < < m{, S > 0, integer.
Defin ition. We shall denote by W f ,X){û) the space of functions / such that / and its generalized derivatives in Sobolev sense (*) of orders less than or equal to s belong to the space M^P,X)(Q). Then
(6) where
N
\ K \ = Z h‘ > I P ] =
i= 1 dx^ dxkN
N is a norm in W{P,^(Q).
We assume that Q have the cone property.
Defin itio n. We say that an open set Q have the cone property, if for every xe Q exists cone of revolution with angle and height inde
pendent of x.
Let, at first, s = 1 and
Pn = Pn P n - ]
Рп(Щ -?ч)
Щ - k - P n l n
for 0 < < m i—p nln and q* = oo for Аг- = m{—p nln .
Theorem 1. I f Q, p, q satisfy previous assumptions and fe W[P,X)(Q), then
1° if 0 < 1, < Щ —Рп/п, then fe M {Q,fl)(Q) for every 1 < qx < q* and for every
Qi Qi Ял
Pi — Aj + аг- < m i--- mi -\--- h Аг- H---
Pn n Pn
and there holds the inequality
(x) For the definition of generalized derivative see [2], p. 225.
Properties of the spaces W ^ ,X) (Q) 363
2° if т { — р п/п < ^ < m{, then f is equal to a bounded and continuous function almost everywhere in Q, for which we have
sup|/(aOI<c7||/|| M .
xe(i w 1 I " *
P roof. Let fe W^’A)(D). Applying (2.2) of Campanato [4] to the product space of W-dimension, we obtain
<7) \ m \ « ce(û) f %
where
N
g(y) = \f(y)\ + ^
i = l
Let, at first, 0 < Яг < Щ —рп1п> Vn ^ <h < 2* • Let a{ be fixed numbers satisfying the condition
(8) 0 < щ < m< —— m{+ — + k(gi~Vn)IVn-
Vn n
Let e be the positive numbers such that
df(y)
дУг
N - ~ - N + qxJr{ A -e ){ q 1 - p n)lpn > A . Vn
We write the equality
< 9 >
дл Pnl9l {y)
y ^ N - A ) ! a i - P n)lQi P r X X \x — y\N~1AA- e)(Qi-Pn)^lPn-(^-A)lQi dy.
We apply the Holder’s inequality for the product of three functions, to the right-hand side of equality (9) and we get
й ^ \x-y\" * A IJ Ic c - y [
x dy
l N - 1 - ( Л - e)(Sl - p n)lqlP n - ( N - A ) q 1]p n
\x-y\
IIP*
The assumptions made for e and A are followed by the inequality [N —1 — (Л — e)(g'1— р п)1дгр п — (N — A )lq l \Vn < N and hence the third integral at the right-hand side of inequality (1 0) is finite (see example
8 b), XYIII.5, [6]). Let us denote it by c9. Now, we estimate U (x)
— J —~— d y . If fe W[P,X)(Q), then gPne М*-1’^ (Q), where MSl,X){ü)
о I® У i
denotes the space M (P,^(Q) for which p t = 1.
Applying Lemma 3° and Theorem 2 [9], we obtain
(1 1) 1/ И Г < ва„||/“||ж(1.Ч«1) < ^ <,и11/|Цг,,л)(0) • Prom (7), (9), (10), (11) we get
'jPn
\ x - y l1 (12) l/(*)l* < « . ,11/11^ / ]x f j lir_A üy.
Integrating the above inequality with respect to x{ on successively»
for i = 1, ..., n, we obtain
m ^ â x < с1г||/||^р)(о) j ax j f n(y} -Â ay .
\x-y\N A
g P n ( y )
/dx J --- d y .Г У \if )
S Q Iй7 УI
If Х{ < ш{~ р п/п and p n < y i< g * , then Х{< т {~щ and for every а satisfying (8) we have
(13)
p n n
/*> / -J s y w - i * » < ел я \ г^ Л{а)- [ j ф г * < « .« / c > v , - / 7 a?i+“ •
S Si ' г = 1 г = 1
Prom (1 2) and (13) we obtain iiffi
( l 4) ll/ll^(ei,a+A)(a) — И/®1 Hm(1.- + *)(£) ^ С12’С1зИ/11^(р,А)(д) ~ GU WfW^iP, X)(Q) * Simultaneously \\Л\м(д,а+х)(0) < cxs\\f\\M(qi,a+x)(0), hence
fil
where = щ -f Xt < m{ ----т {-\--- \-Xt —Ух . Ух , , Ух
P n П P n
We proved 1° for pn ^ уг <у*. If Pi = y{, then
WfWjtf(P>ft)(Si) ^ °6 l l / i U , V ) a n d
where
Properties of the spaces W f ’^ (Q) 365
If p. = p n1 then fe М {рп’и)(й ) for [ii < X t+ p jn . At last we study the case when qx < p n\ then < p {. Applying Theorem 2 [9] we obtain (16) ll/lljtfte.AOtfl) ^ C16 И/ИмО’п’/*)^) ^ с17 М\\м(Р>!*)((3)
^ ^18 11/11 Xy(Pn>^)(Q) ^ ^19 *
where = ,mi —q1mi lpnJr q1pi lpn. Inequality (16) yields (15), because (16) is also true for < А{+ р п1п. Hence we deduce that fe M{q,li)(Q) for ^ < mi - q 1mi lpn+q1ln + q 1ki lpn.
Let now Яг- > mi —pnjn. We choose a{ such that mi — Ai < at < p n[n and аг- satisfy condition (8). It is easily seen that лг- > т г- — . From (7), (10) and Lemma 3° it follows that the function / is bounded in Q and there holds the inequality
(17) sup\f(cc)\ < c7||/|| (РгЛ) .
X*Q W 1 ( }
Inequality (17) yields the continuity of the function / in Ü. Let {fx} be the sequence of functions fe Cl {Q) and/г -> /in the norm ||*|| (p>A) ; then (17) gives a uniform estimation with respect to l. Hence {/} is uni
formly convergent in Q to a continuous function, which is equal to / almost everywhere in Ü.
It is possible to generalize Theorem 1 for the space Wj/’ ^(O) for s > 0, integer. It is sufficient to apply Theorem 1, successively, to the derivatives of order s — 1, s — 2, ..., 1, 0.
„ . , * Pnimi~ h )
Before it, we introduce the notations : qx = --- ---— for Щ — h ~ sPn!n о < Яг- < т { — 8рп1п and q* = oo for Яг- = т { — 8р п/п.
Theorem 2. I f Q has the cone property and f e ТГ^,А)(Н), then 1° if 0 < li < m{ — spn In, then fe M^q,fi)(Q) for qx < q* and /q < mi
+ qx qx qx
s--- L — and there holds the inequality
П P n P n
2° if ш{ — Ьрп/п < Яг- < т {, h — integer, 1 < h < s, then the derivatives of order s — h are bounded in Q and
| * | = S - f t а ’
P roof. If feW [p’x)(Q), then /е1У<^’я)(£) (see (9) [9]) and the suitable inequality for the norm holds. We deduce, applying Theorem 1
to fe W(s n,X){Q) and 0 < A{ < mi — s f nln that fe W^-l \&) for
рЛщ —К) P i <
Щ - k - P n l n and At < mi + P\
Pn Pr
p\ . 3 p\
m i 4" '
P*
and we get the inequality analogous to inequality (15).
Subsiquently, we apply Theorem 1 to fe \@) and so on to f t w f* , n (û).
Hence, under suitable assumptions, from the transitivity, taking into account (9) from [9] we obtain the thesis for 1°. Number 2° of the thesis simply follows from 2° of Theorem 1.
N ote 1. We can formulate Theorem 2 in a different way: we can assume that f e W(f ,X){Q) and DsfeM ^v,X){Q), then, applying Theorem 1, successively to derivatives of order s — 1, ..., 1, 0, we deduce that fe W ? ^ { Q ).
Theorem 3. I f Q has the cone property, Q is a convex set and f e A) ( D), s > 0, integer, m{ — {s — l ) p nln < A{ < then f is equal almost everywhere in Q to a function satisfying for every x, y e Q the condition
N
\ f(x )-f(y)\ < c22£ \Xi-yt
P ro o f. Let Ai > m i — (s — h)pnjn, h — integer, l < k < s —1. We apply Theorem 2 to derivatives of the function / of order h = 1, ..., s — 1 and we deduce that derivatives of function / of order < h, among them the function /, satisfy the Lipschitz condition in Q.
4. We shall study here the space for s > 0, non-integer.
It is impossible to apply previous argumentation, because (7) is not true in this case.
De fin itio n. We shall denote by Wlp,X)(Q), s > 0, non-integer,
1 < pi < oo, 0 < Ai < the space of functions / such that fe
and there is a positive constant M (f), such that] there holds the inequality
(18)
j j |^_2^|(3-1Э])+ЯЧ/РИ /
n
г= 1
tor every
8' = P U & i, # = 1 (у%, Qi)n Qf
Properties of the spaces (Q) 367
Then
(19) l l / l l ^ v , = { 11/11%. V ) + П
sup П QrhPjPi f dy
i— i S'
uce^e?
is a norm in W(P,X){Q).
If Xx = . . . = Xn = 0, then W ^,X)(Q) = WP{Q), it is 'the Sobolev space with mixed norm. If Xx = ... = An = 0, p x = ... = p n = p, mx
= ... = mn = m, Qx = . . . = Dn = Ü, then W{P,X)(Q) = WP(Q) it is the Sobolev space.
If l x = ... = l n = A, Pl = . .. = p n = p ,m x = . . . = mn = m, Ql
= ... — Qn = Û, then we obtain the space studied by Oampanato in [4] and [5].
Theorem 4. I f condition (A) is satisfied, feW^/’^iQ ), 0 < s < 1, 1 < p < oo, 0 < < m i t Ai + spi > mif then fe Hp,li)(Q) for = sp< +Af and f satisfy the Holder's condition almost everywhere in Q:
n X Ш
(20) \f(oc)-f{y)\ < c23llfllw(p,*)m)f ] l^i-ViV S Щ = ^+- г — ,
8 ^ f = i P i
for every x, y e Û.
P roo f. Let x 0 e Ü, x 0 be fixed ; then we have for almost all y e S (2 1) inf f \f(x)-c\pd x ^ f \ f (x )- f (y )jpdx.
c<Rbs bs
Integrating the above inequality with respect to y{ on successively, for i = 1, . . . , n, and applying condition (A), we obtain
+ |fc| = [s]2 ;
\Dkf ( x ) - D kf(y)\ \* ylvn bs ' П \хч - у {\8 W+miton
\p у
I dx\
(22) inf
CeR
J \f{oo) — c\pdx <
П
J~J h 1 Qï 4 f dy J \f{x)-f{y)\pdx s bs
П
2 pn + mi Q»pn
bS Inequalities (2 1) and (2 2) yields (23)
1/0») ~f(y) 1
П
П \Щ-Уг\8+Щ/Рп i=1
(23) inf f \f(x)-c\pd x ^ c 2i\\f\\
bs w\(p>*\ЩП 1+^iPnlPi
p
I dx.
and f e H p,^(Q ) for = s p i -[-Xi . Inequality (20) follows from (23) and Theorem 2 [8].
We introduce additional notations in the proof of the next theorem :
°< =pU O ,.
Theorem 5. I f Д have the cone property, fe W [p,^(Q), 0 < s < 1, 1 < P i < oo, -f spi < nii, then /е М^р,^ (й ) with /u{ < ?H + sp{ and
^ ^25 \\f\\-py(p, A)^ "
P ro o f. Let x 0e Q. From the cone property for Д it follows:
there cones Тг*(а?°) с Д , for a?0, with vertices in x\ and having angles at and heights Щ independent of x\. Let a{ he a real number such that (24) 1 < cq < 1 -f spijrrii.
Let Q{ < min {H{, 1}, — fixed. For almost all points xe 8 we have
\f(x)fi < 2*|/(y)|»i + ^ \ f ( x ) - f ( y ) \ Pl.
Integrating the above inequalities with respect to on Ог-, succes
sively, foi* i = 1, ..., n, rising, successively, to suitable powers and applying condition (A) we get
n Pn
(26) J \f(x)fdx < c№f j е Г ъ т< f I №\*dy +
0 i—1 b8
+ b f l # ' ï a» ï ( i=î S b0 x f ] \Xi — yi\s+milp 'т ~ Ш '*
n
i—1
n . Pn Pn
\\s\\Pn 1 T
= < Ы \Л \м {0)[ [ e t *• *•
f=l Hence, for every gt < min {Щ{, 1}, we have
П J 1f ( x ) f d x< li/Ç(»,j) . П
bS s % =1
Applying the above inequality in (25) we get
dx
. f . Pn PnJ
mm |(aï. _ 1) _ - m i +Ai — , Здп +Л* — J
« “г—1 If Pn
“i—1 If (û/_т)--- --- ----}---
П ai Pi ai P i .1
о f=i
+ c / 7 ef » / # / ( № ) ~/W11’ v’
S ь» П \%i-yi\s+mi!Pr f=i
" . ( лг Pro “ г — 1 Pn .
f j £ Ш^ П + — щ т » 8Р^ щ \
< c29II/II^,a)(û)
Properties of the spaces W (f ,X)(Q) 369
and hence, for q{ < min 1} we get
. f Hxi Pn . ai— 1 Pri sp n + l
m i n { — --- — +
2 P i a? Pi m{,-
Pn
J \f{x)\pd x ^ c29\\ff^M [ J q JL\
bs 8 i= 1
Repeating this argumentation Z-times, we get foi* 0 < 1}
_ n , f xi Pn ai— 1 Pn sPn xi Pn \
j ш х г а х к с м ^ - П е Г ч ^ ' vsn.
bs s i =1
We observe that I- j ^i Pn . Щ—1 P
lim ■--- 1---j m\ = m.
J Pi ' and -, Ai --- h spn < — m{,1 I Pn
Pi Pi
l->oo \ (%i Pi Pi
lienee for sufficiently small q{ and a{ satisfying (24) we obtain , P n sP n + xi- , 77
J \f(x)\pdx < c25 и\йр,л) П Qi a*
bs 8 ( гЛ
and the thesis follows.
We shall study, at last, the case, when s > 1 non-integer. We put 8 = [8] + 0, 0 < 0 < 1. If fe then, from the definition, deriv
atives of the function / of order [s] belong to W^P,X\Q) it means D[s]fe W(g ,X)(Q). If 0 < ?H < т { — dpi, then, from Theorem 5, it follows:
Я ^ е Ж ^“>(Я),л .<Я» + #й .
Simultaneously applying Note 1 (because fe WPS^(Q)) we deduce that fe W^\X){Q) for < ?H-f and there holds the inequality
<36> [SJ s
Applying Theorem 2, we get fe M ^ ( Q ) for qx < q*, Я.1 <Zi
Vi < + [e] - ---nti + t o---
n Pn Pn
and
(27) I l l ' l lm(Q’v\q)^ ^ 2 7 '
From (26) and (27) it follows
\\f\\M(q,v[4a)< c,28 I
for qx <
if Яг- < mi — &pi — [8']pn[n
w{P’ X\q)
Vi < m{ + (s -iJ') — + — (Ài — mi ) + êq1 —
n Pn Pr
We introduce the notations:
* _ P n $ P i )
qi Щ - № «/» - h ~ $Pi
if h < mi — W p J n — ûpij q* = oo if = m i- [ s ] p nl n - ê p {.
We shall formulate the next theorem, based on the above argumenta
tion and the already proved theorems.
Theorem 6. I f Q, have the cone property, /e W^,Xi(Q), s > 1, non
integer, 1 < pt< oo, 0 < A* < m i — & р { — [ s ] p jn , then f e M ^a,v)( ü ) for h < I* and
^ h , r -, h , « h . , h
v{ <m{--- + [ * ] ---b vpt---b h—
P n ‘ ^ P n
and the inequality holds
^ ^28 \\f llpp(2?,’
5. We shall prove completeness of the space TF^’A)(£), but first we must prove completeness of the space M ^ P,X){ Û ).
Theorem 7. The space l I p,x*{Q) is a Banach space.
P ro o f. Let {/г}е L(P,^{Q) be a Cauchy sequence,i. e. lim IIf i —fr\\L(p,i)(£3
l, T—»-oo
-> 0. We choose subsequence {fh} such that S\\fik+1 - f i k\\LM{D)< 00•
к
Arguing as in the proof of L p (see [10]), we get fik{x) -+ f(x) almost every
where in Q. Applying Fatou Lemma we have II 4 - / llj* U ) (0, = { / « m | 4 W - / I(WI,’<b +
frS < ->oo
п Pn
+ sup Г/ Qi ’p< inf flim l/г ( x ) - f h(x)\pdx\llPn
*»,5{ U “ « ( « H i ’
< lim { J \fik{ x ) - f h{x)\pdx +
hence fik{x) ->f{x) in l i p,X)(Q). We deduce from the Cauchy condition and the triangle inequality:
^||/;-/||х(г,,д,(п) <ito||4-/||i(J,,4 + lim II4_ /iILcj>,J)(o) = 0-
Z->oo k->oo l, Jc—>co
Properties of the spaces W {p,X){Q) 371
At last, we have
and
ll/lli(>>.4(B) < V - f l k\\L(p,X)fat+\\flh\\i,M(D)< 00 f ,( x ) ^ f ( x ) , f t L ^ '( Q ) ,
which, makes the proof of completeness of L^P,X\Q ) finite. We know from [9] that l J p,x)(Q) = ЪТр,х\ й ) for 0 < Л* < m{, which proves the complete
ness of the space M {P,X)(D ).
Theorem 8. The space is a Banach space with the norm (6).
P ro o f. We use the Gagliardo’s method [10] as applied to the space Wp ( Q). The space W(P,X) (Ü) can, for s = 0, be reduced to the space ATP,X) (Ü) and then it is a Banach space. Let s = 1 and {/z} be a Cauchy sequence in the sense of the norm (6), f te W(p,x)(Ü). Since the space is a Banach space, we can find A + 1 functions g0, ..., gN from the space M(P,X\Q) such that
Wfi 9o\\m(p,z)(q) * 0, dfi_
dXi ~ 9 i
for oo, i — 1, ..., A . After proving that g0e W^P,^(Q) and that the functions g1} ..., gN are derivatives of order one of the functions g0, the proof will be completed. Let D = {а{ < xt < b{, i — 1, ..., A} be the parallelepiped with edges being parallel to the axis, D c Q. Let B t be its face situated on the hiperplanes x{ — aif xi — b{ respectively.
Integrating by parts, we obtain, for f e W[p’ ( i2) : f f d B J f d A t =
Щ Аг-
dD, i = 1, A .
Applying the equality to the function /г of the choosen sequence, we get, passing to the limit
j g Qd B i - f g0dA{ = f gt dD, i = 1, ..., N ,
Si a{ в
and hence the thesis follows for s = 1. Applying the obtained results to the derivatives of order less than s, we get the thesis for s > 1. Making use of Theorem 7 and Theorem 8 we can easily prove the completeness of the space W^,A)(^) for s > 0, non-integer.
References
[1] A. B enedek and R. P anzone, The spaces L p w ith m ixed norm , Duke Math. J.
28 (1961), p . 301-324.
12] P. L. B u tz e r and H. Be rens, S em i-gro up s of operators an d ap p ro x im atio n , Springer-Verlag, Berlin-Heidelberg-New York 1968.
5 — Roczniki PTM — Frace M atem atyczne XVII.
[3] S. C amp an a t о, II teorem a d i im m ersione d i Sobolev p e r u n a classe d i a p e rü non dotati d ella p ro p rieta d i cono, Kicerche di Matem. 11 (1962), p. 103-122.
[4] — P ro p rie ta d i in clu sio n e p e r sp a z i d i M orrey, ibidem, 12 (1963), p. 67-85.
[5] — P ro p rie ta d i H ô ld e ria n ita d i alcu n e c la s s i d i fu n z io n i, Ann. Scuola Norm.
Sup. di Pisa 17 (1963), p. 175-188.
[6] G. M. F ic h te n h o lz , BachuneTc rôzniczkowy i calkow y, t. I ll, PWN, Warszawa 1963.
[7] E. G a g lia rd o , P ro p rie ta d i alcun e c la s s i d i fu n z io n i in p iû v a r ia b ili, Kicerche di Matem. 7 (1958), p. 102-137.
[8] M. Ja ro s z e w s k a , H o ld er's condition fo r some fu n ctio n spaces, Prace Mat. 15 (1971), p. 75-86.
[9] - P rzestrzenie L M { Ü ) i M M { Q ) , Fasc. Math. (1972), p. 79-84.
[10] K. Y o sid a, F u n ctio n a l a n a ly s is , Springer-Yerlag, Berlin-Heidelberg-New York 1968.
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