s. s, P P со, On the spaces with mixed norms. I

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIII (1983) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIII (1983)

M. J

(Poznan)

On the spaces E £ ( Q ) with mixed norms. I

1. Introduction, notations. In this paper we introduce and investigate the space E£(Q) which consists of functions / defined and integrable with mixed powers p = (pi, ..., pn) in a product Р,"=1£?г, c R l. In the last part of the paper we give, using the properties of E„(Q)~spaces, some characterization of the function spaces Cm,a(£?) of functions satisfying the product’s Holder condition with exponent a = (a1? ..., a„), with theirs derivatives of certain order. Particular cases of some results of this paper can be found in [2] for m,= 0, in [1] for i = l. Spaces of this type were studied also by G.

Stampacchia, Y. K. Murthy, F. John, L. Nirenberg and others (for references

see [1]). /

The index i runs through 1, ..., n everywhere, unless otherwise stated.

Let R be the set of real numbers and k( > 0, an integer, 1 < p, < со, Л, ^ 0.

In what follows we shall use vector notations, i.e., x = (x1? ..., xn), p

= {pi, ..., pn),

= (Qi, ..., p„) etc. Let be an open, connected, bounded subset of the real Euclidean space R . Let d(£2f) denote the diameter of and let Q = P "=1 Qb Q = Р,"=10 г. Let /(xf, q J cz Rki be the ball with centre at xfeQ, and radius Qh 0 < g, ^ d(Qi) and let St — I(x f, g{)n Q , S

— I(x 0, g )n Û = P "=1 The measure used is always the Lebesgue measure.

We assume that the boundary Qt has Lebesgue measure zero in R ■ and that Qt satisfies the condition:

(A) There exist positive constants A(- such that for every х ? е Ц and e,-e[0, d(Q,)]

R(Sd > A #?.

To simplify the notations, we shall write, for example:

J I/(x)|dx = j ... J I/(x)| d xj. ..dxn,

S Sn Sl

I |/(x)rJx = ||/lt(S1 = ( ( . . . ( J \f(x)f'clxlf 2"’4 x 2. . ^ - ^ x „ .

bS s. s,

Supremum, denoted by sup, runs over x fe Q h 0 < ^

Let /! = 11l /,!=/'! ... 1[.\, \l\ = Mi I + ••• +|/„|, |/,-| - Z i+ .- .+ f t ,, x1

= X! 1 x2 2.. .x^*1", kt = N. Let Dlf (x) denote the generalized derivative of function / (x) in Sobolev sense and let

f ( v)

Dlf ( x ) --- T - - - if |/| > 0, Dÿ-(x) = /(x) if 1 = 0, rx}'1 ...rbc^"

and Pw(x) a polynomial of variable x with real coefficients. Let Pm denote

Л

the set of polynomials P(x) of degree |m| = Z mf, i.e., of degree with

i = 1

respect to xh m, a non-negative integer.

We shall denote by Cm{Q) the space of functions defined in Q, which have continuous partial derivatives of order up to and including m, with respect to xf, m, an integer ^ 0.

We shall denote by Cm,0C(Q), mt an integer ^ 0, a = (al5 ..., an), 0 < a, ^ 1, the subspace of Cm{Q) of functions, the derivative of which of order \m\ satisfies the product’s Holder condition in Q with exponent a.

The space Cm(Q) is a Banach space with the norm

(Ы ) \f\m,n = Z SUP \Dlf\.

|ЛИт. «

1 Г l

i~

The space Стл(й) is a Banach space (compare [9] p. 26) with the norm

\D‘f{x)-D 'f(y)\

(1-2) \f\m,a,Q= Z ^SUp \Dlf\ + ^{SUp SUp} i=

t. x.,y.el2. i г , ,a.

' i . V П k ’-y.-l ‘

i= 1

For f e C m,a(Q) we write

(1.3) [/L ,,a ,ô = SUp SUp_ \Dlf ( x ) - D lf(y)\

|/.| = m. x.,y.eQ . т—г . .a.

‘ ‘ i V П l>=i— ‘ .

i = 1

The above is a seminorm in Cm’“(&).

D

. We shall say that the function f ( x ) e L p(Q) belongs to the class L^,A(Q) if

(1-4) III/IL ju = sup ( П е Г 1‘РЛ inf J \f(x)-P(x)\*dx}W- < c c .

r ° c O , P e P s

X j I — 1 m b

0 <e. =$d(Of)

This is a seminorm in Р£л(£2). The space L£A(^) is a Banach space under the

norm

Spaces Ц^{£2) with mixed norms. I 239

(1-5) ll/IL,.A = {ll/lfr( a +lll/ llfcu !'/'”-

The proof proceeds similarly to that of completeness of LP'X(Q) or MP*(Q) given in [6], [8].

2. Let / (x) eL ^ (C ); we can prove analogously as in n. 3 [1] and 1.2. [5]

that for every x? e Ц, e[0, d (£?,)] there exists only one polynomial Pm(x, x0, в , P ,f) such that

(2.1) inf J |/(x)-.P(x)|pdx = J If ( x ) - P m(x, x0, Q, p,f)\pdx.

PePm bS „s

Let P(x) be the polynomial P (x )eP m. We can write it in the form P {x )= £ a .fl/ ir1 П

H,l <», i=i

i = l,...,n

We shall write also Pm(x, x0, q ) or Pw(x, x0, p, p) instead of Pm(x, x0, p, p,/) if there will be no possibility of misunderstanding. We have (2.2) at(x0, q ) = [DlPm(x, x0, p)L=*0.

L emma 2.1. I f f eLF^{Q), then there exists a positive constant Cj(p, A) such that

(23) J |P„(x,x0, e\2h) - P m( x ,x 0, e\2"+lr d x bS"

« c, |l|/llfcM П 2 -> ¥ Л е},-Л

i = 1

for every x fe Q h 0 < p , d (Ц) and positive integer h, where we set

S’, = Цх9, й |2*)пй„ S," = /(x?, ei\2h + ,)n Q h g|2* =te,|2”, .... e„|2”).

Proof. Let х?е Д , 0 < p<^ d(0,), h positive integer. For almost all 2СбР"=15г we have

1ЛЛ*. * 0, Q\2h) - P m(x, ^{X 0 ,} e|2fc+,r

^ 2р‘ |Рж(х, x0, р| 2 й) - / ( х )|р1 + 2Р1|Рт (л:, x0, p|2*)-/(x)|pi.

Integrating both sides of this inequality with respect to all variables x, on S,"

successively and taking a suitable power p i + l \Pi we obtain 1 \pm(x, x0, g\2h) - P J x , x0, fi|2"+1)l"dx

bS"

\Pm( x ,x 0,e\2h) - f ( x t d x + f - J \Pm(x ,x 0,a\2h+' ) - f ( x t d x .

.s’

Then, by (1.4) and (2.1) we have

J 1Лп(*» * 0 , ô\2h) - P m{x, X q , g\2h+1)\pdx

«^-III/III'V a П (Z-WA + ^'III/IICV a п ^(2-н- ₁ ^{Ф " Л}

i= 1 i= 1

= 2p- ( l + n 2 -W -’,)|||/||['V i П (2 -*a )V A

/ = 1 i = 1

Hence the thesis follows.

L

2.2. I f condition (A) is satisfied, f eLP^{Q), then for every pair o f points xf, у ?е Д and for every system o f N numbers / = (/},..., l£n) with

|/,| = mt there holds the inequality

(2.4) |a,(x0, 2 |x0 - y0l) - a, (y0> 2 |x0 - у01)Г"

/А. fc* ч

n я ( — — \т.\/р

* i c 2( l,p ,k ) 2p- + 1 п ² ^Л|||/|||^ П п ' ".

1 = 1 1 =1

Proof. Let xf, у?еЦ-. We set д{ = |xf — y?| and S'-'= S] n S'-

= [I(xf, 2gf) n Д ] n [/(y?, 2 ft)n flj] for almost all x e P ?=1S- we have

\pm(x,

2g)

Pm(y, y0, 2g)\Pl

< 2Pl|Pm(x, x0, 2g)-f{x)\Pl + f 1\Pm(y, y0, 2g)-f(x)\Pl.

Integrating both sides of this inequality with respect to all variables x, on S( <=S’i", taking a suitable power pi + 1/Pi we get, arguing analogously as in the proof of the Lemma 2.1

(2.5) J \Pm(x, x0, 2g ) - P m(y, y0, 2g)\pdx

2P" j \Pm(x, x0,2 g )-f{x )\ pdx + 2Pn J \Pm(y, y0, 2g)-f(x)\pdx

S'

П П /л -

/ = l i = 1

Then, if (2.2) holds, apply Lemma 1 [3] to the polynomial P(x)

= Pm(x, x0, 2g ) - P m(y, y0, 2 q ) and we obtain (2.6) \at (x0, 2g) — ai(y0, 2g)\p”

- й +тУ

< c3 • П Qi Pi ' " j l Pm{x, x0, 2g ) - P m(y, y0, 2g)\pdx.

i = 1

•By (2.5), (2.6) we get the thesis.

Spaces LP^(Q) with mixed norms. I 241

L emma 2.3. I f condition (A) is satisfied, feU ^ (Q ), then there exists a positive constant c4(m, p, q , k, A) such that

л- l n я k

(2.7) la fx о, e )-a ,(x 0, 0|(2Й )| ^ с4 \\\/\\\т>рЛ £ П ( 2 'fe/'* Pj

/ = o j = 1

for every x ° e Ц, 0 < ^ ^ d(Of), positive integer h and |/,-| ^ m,-.

P roo f. We have

h - 1

(2.8) М хо, e )-a ,(x 0, е|2йЖ X la/(*o, e|2*') — ai(x0, e|2i+1)|

i = 0

= £ x0, e l ^ - P ^ x ,

e|2i + 1) ] } ,e , 0|.

i —0

Applying Lemma 1 [4] to the polynomial P( jc ) = Pm(x, x0, g|2')-

— Pm(x, x0, q \2‘ * ’) we deduce (2.9) Я/(Х(Ъ e )-fl,(x 0, e|2f)l •

j= 1 i=0 [ J 1^ и (х , x 0 , e|2f) -

*s'

- P m(x, x0, Q\2+i) fd x ]llP\

After applying (2.3) to the above inequality we get the thesis.

L

2.4. I f condition (A) is satisfied, f e I f f (Q), ki + pi <Ài

^ ki + (Si + l)pf, s( — positive integer, st ^ m,-, then there exists ta system o f functions {vi(x0)}, |/,| ^ Si, defined in Q such that for х ? е й {, 0 < ft < d{Qi),

|/,| ^ Sf, we have

я.

(2.10) |a,(x0, e)-»<(x0)l < c5{X, p, m, k, A)\\\f\\\m>PtX f [ Qjj Pj

\ j= i

and

(2.11) lim a,(x0,

= v{(x0)

e r °

1= 1,

uniformly with respect to x0.

Proof. We fix l,

x0 with |/,| ^ sh x p eQ , 0 < f t ^ d(Qi). Let h and W be two positive integers, for example h > h'; then we apply Lemma 2.3, setting Qj2~h’ instead of Qj and we obtain

(2-!2) |a,(x0, q 2 h’) - a i { x 0, q 2 h)\ ^ c4

h- 1

т,р,Я i = h’ j = 1

Z П (2'e/'

• Prace Matematyczne 23.2

к. Л.

х п -A. -f J / ,| —

By assumption, the series ]T f ] (2lf J 1 ■ Pj is convergent. We deduce, i = h' j = 1

taking into account (2.12) and convergence of this series, that the sequence [иД.Хо, g2~h)} satisfies the Cauchy condition and thus it is convergent with h -> x . We observe that this limit is independent of gh 0 < p, ^ d(Q;). Let gf and of be two real numbers satisfying the relation 0 < gf ^ gl2 ^ d{Q{).

Applying Lemma 1 [3] and definition of the class LF^f(Q), setting S'j = /(.vj\ p]|2‘) n Qj, S'- = I(x f, Qj\2) n Qj, we get

(2.13) ^\a{ (x 0, Q ^ t y - a ^ x 0, ^Ql .2ni\ |2‘)|

k. ~L +\l.

^СзП [2,t e i r 1]Pj 4 .f \pm(x, x0, д 1\ 2)~ Р М , x0, д2\2Гс1х}1р"

j=i .S'

<C3 П [ В Д Г 1] " 4 f \PJx x0, g l \2i)~ f(xr\ dx +

j~ ' b^

+ J’ 1ЛЛ* > *

,

bS"

k. k.

1— 2-1 m -- |J.| i(—+|1.|-- l)

> pj Л . П Pj\1/P» ( n i\ pj J . i yPj J pr

< C 3 2 ^• lll/IIL.,., П ( e j Pi + 0/ Pi) IPn(Q}) " ' ■ 2

;= 1

By assumptions of the lemma we get that the latter expression tends to zero for i ■-* r . We set then for x feQ ,, 0 < p, ^ </(&,), |/,| ^ x,

(2.14) vt{x0) = lim a,(x0, q \2%

i -* x where r,(.v0) is defined in Q.

k. A.

x n i ( —^ + I Г| — ^)

By the convergence of the series J] f l 2 Pj 1 Pj and by (2.7) we i=0 /= 1

deduce for every \lj\ < Sj, x JeQ j, 0 < Qj ^ d (Q j, j = 1, ..., n, / ^ 0:

it ki (2.15) M *o , e ) - a i{ x 0, £?|2')| < c6(2, m, k, A) HI/IIL,^ П Q? Pj ' •

j=i From (2.14) and (2.15) we get (2.10) and (2.11).

3. T

3.1. I f condition (A) is satisfied, f e 11^ {Q), Л,- ^ к,- + |/,| p,-, then the functions vt(x) satisfy the Holder condition in Q for |/,| = w, and for every pair o f points x ,y e Q there holds the inequality

Я. k.

M*)-My)| < C7lll/Hlm,p,A П I XJ - y / J Pj

j= 1 n

Proof. Let the integers / = (/}, l\, ..., /^, ..., /у be fixed, £ к,- = JV,

(3.1)

Spaces LPf(Q) with mixed norms. 1 243

|/,.| = nij and let x;, y/eQj be a pair of such points that k^l = |.v, — y,l

^ d(Qi)/2. We have the inequality (3.2) М * ) - М у )1

^ \vl( x ) - a l(x,

+ \v,(y)- a t(>’, 2e)| + |a,(x, 2g )-a ,(y , 2f>)|.

By (2.10) we also have

я. k.

(3.3) Iv ,(x )-a,(x , 2g)\ <- c, ,.„.>.11 (2a r i'= 1

Pi Pi

(3.4) \My)-ai(y, 2^ )K c5 and by (2.4) we get

т,р ,л

П < 2a)

A.

k.

—-- L~ |J.|

Pi Pi '

(3.5) |a,(x, 2 q - at{y, 2^)|

Гр «î + i p

< c 7 "2 П 2A ‘ew\\Un.i П К--У,

1=1 1=1

I Pi Pi к

From (3.2), (3.3), (3.4), (3.5) we get (3.1) for every pair of points ,v,, y;, satisfying the relation |x, — y,| < d (Q f2 . If |x,- — y,| ^ </(Ц) 2 we argue simi­

larly as in Theorem 2 [2].

We set (0) = (0, ..., 0) g Rn, e} = (0, ..., 0, 1, 0, ..., 0 )eR N - the ./-th coordinate is equal to 1, the others are equal zero.

T

3.2. I f condition (A) is satisfied, f e. I f f (Q), ^ 1, •/,• ^ k,+

-h I/,J />,, then for every system o f N positive integers (/}, — /['J, |/,| ^ ml — 1 the function уДх) has partial derivatives o f order one in Q and for every x eQ there holds the equality

(3.6) (-p- = v(l + , (x), j = 1, ..., N,

CXj ]

where Xj denotes the coordinate o f x = (x}, xj?n).

Proof. We know from Theorem 3.1 that the functions v^x), |/,| = m, satisfy the Holder condition and hence, in particular, are continuous in Q (see e.g. [10], p. 123-125). We shall prove Theorem 3.2 by induction, if we prove (3.6), by assumption that v{l + e.0)(x) are continuous in Q for ô

= 1, 2, ..., \m\-|/|. Let / be a system of N positive integers such that

|/(| < m(- — 1, let j be a fixed positive integer, 1^ ./ ^ N and suppose that

,;(/ + iy>(*) are continuous in Q for Ô = 1, 2, ..., |m| —1/|. Let x,° еЦ and let

be real numbers sufficiently small such that I(x f, |^,|) d£?(. Let d be a real

number such that d = diam [P,"= ^ (x;, |&|)] <= Q. Taking into account (2.2) we can write

(3.7) M x o + ejd, 2 |e|) — at(x0,'2 1@|)] d ~1

= d~l {Dl [Pm(x, x0 + ejd, 2 \ q \)-Pm(x, x0, 2|e|)]}*=Xo-

- z ( - 1 У Ч ^Г1^ la(l+ôej){x0 + ejd,2\Q\).

1

By Lemma 1 [4] and (2.5) where S{ = I(x f, |e,|) n O,-, we obtain (3.8) \d~x {Dl lP m(x, x0 + ejd, 2\g\)-Pm(x, x0, 2|e|)]}JC=Xo| .

^ C3 П \Qi\ Pi 1 '' { J Ip m(x, x0 + ejd, 2\g\)-Pm(x, x0, 2|e|)|pd x }1/P"

i=l .s

« ^сз П ^{2‘ ■l./l./.} I~1 l^il

—-IM

i= 1 We have for every 1 < S ^ |m| — | (3.9) \a(l + dej)(x0 + ejd, 2\g\)-vu+ôej)(x0)|

^ К + sej) (xo + ejd, 2 |e|) ■- v(l+ôej) (x0 + ejd)\ + \v{l+ôej)(x0 + ejd) - v(l+Sgj) (x0)l.

Applying (2.10) we get

(3.10) K j + ôej) (xo + ejd> 2 |e|) - v(l+dej) (x0 + ejd)|

^ c 5 f l (2|e,-|) 1х.-к.-(\1.\+0)р.]/Р.

'III/IIU p p,k , ' From (3.9), (3.10) and from continuity of the function u(/+e.^(x) for Ô = 1, 2, ..., |m| —1/| it follows

(3.11) \im ail + e0)(x0 +ejd, 2\g\) = vu+ ô)(x0), Ô = 1, 2, \m\-\l\

о

(d -> 0 when Qi -> 0). By (3.7), (3.8), (3.11) we deduce the existence of the finite limes

lim d 1 [a,(x0 + e//, 2\e\)-a,{x0t 2|e|)]

d~+ 0

and there holds

(3.12) lim d~l [al(x0-+eJd, 2\g\)-a,(x0, 2|el)] = vu+ }(x0) d~> 0

uniformly with respect to x0.

Spaces Ц ^ ( й ) with mixed norms. I 245

We have, by (2.10) and by relation \d\n ^ f ] |g,|

(3.13) Id 1 [vl(x0 + eJd ) - a l(x0 + ejdi 2|e|)]| /= l

x. k.

« c5 П 2* " '

i

(3.14) \d~l [uj(x0) —a,(x0, 2|e|)]|

A. k. i

i = 1

П lalp‘ " ' ".

^ C 5

П 2

A. k.

— — L - | M Pi Pi 1

n,p,A

Fl ^\ôi

i = 1

A. k. i

iP; Р,- ' «' и

We observe that, for q ( -> 0, the left-hand side of the above inequalities tends

to zero. We have the equality y

(3.15) d~1 [vl(x0 + ej d ) - v l{x0)'] = d ~1 [vl(x0 + ej d ) - a l(x0 + ejd, 2|e|)] + + d~1[al(x0, 2\Q\)-vl(x0)'] + d~1la l(x0 + ejd, 2| р | )-а,(х0, 2|e|)].

From (3.13), (3.14), (3.15) it follows (3.16) lim d~l \vi(x0 + ejd) — vl(x0)]

d-+ о

= lim d~1[_al(x0 + ejd, 2\Q\)-at(x0t 2 \ q \)]

d-0 and hence (3.6).

The conclusion of Theorems 3.1 and 3.2 is

T

3.3. I f condition (A) is satisfied, f e Iff(Q ), / ( > k( + т,/>, , then the function v(0) (x) e Cm * ( Д , a,- = (xf — /с,- — m,p() p f 1 and

Dlv{0)(x) = i>,(x) for every x eQ , |/f| ^ m,.

Rem ark 3.1. Let condition (A) be satisfied, f e lff fQ ) , m, ^ 1. Assume that there exists an integer, s,-, 0 ^ s, ^ w, - 1 such that the degree of the polynomial Pm(x, x0, q ) is ^ |s| for every х ? е Д and е,е[0, Л(Д)]. Then we deduce that r( 0 )(x) is a polynomial of degree ^ |s|. It is true, since: if at(xo, @) = 0 for |/,| > sh for every x fe Д , 0 ,e[O, Л(Д)], then vt(x) for |/,| > s, are identically equal zero in Q and by Theorem 3.3, we have the conclusion., Rem ark 3.2. If / e LP^(Q), > /c,+(w, + l)p f, then we deduce by (3.1) that vt(x), |/f| =m,, are constant and thus, by Theorem 3.3, u( 0 )(x) is poly­

nomial of degree ^ |m|.

4. ^T

4.1. I f condition (A) is satisfied, f e L p^{Q), /с* + т,р; <

^ kj+ (m, + l)p., then f e C m,a(Q), and there holds the inequality

<4-D [/]„,»< cJI/ IIL ,.,.

I f а , > (m, + 1) P,-, then /'coincides in Q with a polynomial o f degree < \m\.

Proof. Taking into consideration Theorem 3.3 and Remark 3.1, it is sufficient to prove, with assumptions of the theorem, that / (x) coincides, for almost all x e Q , with r(0)(x), i.e., with lim a(0)(x, q ). We have proved in Lemma 4 [2] that for almost all x0eQ , Q.^0

(4.2) lim ... lim f ] [/'№ )]~"Л j l f ( x ) - f ( x or<lx = 0.

0 , ->0 £„-0;=l hs

Let x0eQ be such that (4.2) is satisfied; for almost all x eQ we have l<W*o, 0)-/(*o)|Pl < Сн(р) \\pm(x, x0, Q )-a( 0 )(x0, e)|P‘ +

+ \ p jx ,

р )-/ (х)Г1+ | / (х)-(х0)Г1}.

Integrating both sides of this inequality with respect to all variables x, on Sh and taking a suitable power pi+, /?,• we get

(4.3) |u(0)(x0, p )-/ (x 0)|Pn

n

^С н ( р ) П (Ai6?) PnP‘ f \P,„(x, x0, Q )-a{0){x0, Q)\pdx +

,•=1 „s

+ cH(p )fl (AiQf)~ PjPi J Ip m(x, X0, Q ) - f ( x r d x +

-•= 1 »S

+ cs (p) П J \ f{ x ) - f( x 0)\pdx.

l ~ 1 bS

Consider now the third part of the right-hand side of (4.3). First (see the proof of the Lemma 2 [3]), we have

(4.4) сн(р) П (AiQ/)~PJPi J \Pm(x, x0, e)-fl( 0 )(x0, g)\pdx ' =1

^ cg{A, k, p, m) П (А.-еГо P"/P| X ^)Гл П e!Jflp".

/=1 l^|/K|m| , /=1

Hence and by (2.11) we deduce that (4.4) is infinitesimal with pf. Then, by definition of class Lp,f(Q ), we obtain

(4.5) c„(p) П ( A ,e p fp-,p' J x0> e ) - f ( x r < l x

1=1 .s

W p ) П K a — r ^ lll/ I I L ,..

i = 1 and hence (4.5) is infinitesimal with p(.

The third part of the right-hand side of (4.3) is infinitesimal with p,, by

Spaces LP^X{Q) with mixed norms. I 247

(4.2). Then (4.3) follows, for almost all

and lim a(0)(x0, q ) = f { x 0).

* r °

By (3.1) and by the equality f(x ) = v{0)(x), which holds almost everywhere in Q, we deduce (4.1).

Applying Theorem 4.1 and Theorem 1 [7], we can deduce part (a) of the following

T

4.2 (a) I f condition (A) is satisfied, / e L p;A(Q), ш, - positive integers, kf + s ^ < Af < fc,- + sf + pf, 0 ^ s, ^ mh then f e C s,ac(Q), а,- = (А, — k{ —

— SjPii/Pi and there holds the inequality

(4-6) [ / l.a < ^olH/HImjU.

(b) I f in addition we suppose that Q( is convex, then the space U„ (Q) is isomorphic to the space Cs,a{Q).

P ro o f (b). Let / eC s,a(Q), x?eQ h 0 < ^ d (Ц), х.-.еI (x,°, g,) n Q,.

Taking into account that the derivatives of order |s| of3he function / satisfy Holder condition we can use the Taylor formula

(4.7) |/(x)- £ D'f(x0)(\l\l)-' П (x ,-x ,0)''|

|/.| <m,- 1 i = 1

= | I w w - i y f ( x 0m m - 1 f i °)'l i=i

^ I ( 1 'i r 1 П

|i.| = s. i - l

where ç is the suitable point of the segment [x0, x]. Integrating both sides of the above inequality with respect to all variables x, on St, taking a suitable power Pi + i/Pi we obtain

(4.8) inf J \f(x)-P(x)\pdx ^ cn [/]£« П e ^ JPi-

psPSbs « = 1

By (4.8) and (6) [7] we obtain

(4.9) lll/llfep,^ ci i [/Is,*-

We c a n re p la ce (see (1.1), (1.2)) th e exp ression j/|s = £ su p \Dlj\ by У \\DlfW.„n an d thu s, by (4*9) we d ed u ce th at

\\j IL.P.À ^ c12 l/is,a-

Hence the thesis follows.

Note that we obtain some characterization of the space Cs,3t(£>) using the properties of LP^X (£2)-spaces. Application of this fact, to some generaliza­

tion of the Marcinkiewicz theorem on interpolation, we can find in [3]

and [11].

References

[1 ] S. C a m p a n a to , P roprieta di una fam iqlia di spazi funzionali, Ann. Scuola Norm. Sup.

Pisa 18 (1964), p. 137-160.

[2 ] M. J a r o s z e w s k a , H older’s condition f o r som e function spaces, Comm. Math. 15 (1971), p.

75-86.

[3 ] —, On interpolation in the L p,A-spaces with mixed norms, Boll. B. U. M. I. 5. 14-B (1977), p. 149-159.

[4 ] —, On interpolation in the E'^-spaces with mixed norm, Functiones et approx. 8 (1979), p.

119-128.

[5 ] —, On som e function spaces integrbted with mixed norms. Doctoral dissertation (in polish), Poznan 1971.

[6 ] —, On the spaces MP,X(Q) with m ixed norm, Functiones et approx. 3 (1976), p. 93-99.

[7 ] —, On the spaces E^(£2) with mixed norm. II, Demonstratio Math, (to appear).

[8 ] —, Som e properties o f the spaces WP,X(Q), Comm. Math. 17 (1974), p. 359-372.

[9 ] A. K u fn e r, O. Jo h n , S. F u c ik , Functional spaces, Prague 1977.

[1 0 ] R. S ik o rs k i, R eal functions. I, Warsaw 1958.

[1 1 ] G. S t a m p a c c h ia , E ,x-spaces and interpolation, Comm. Pure Appl. Math. 17 (1974), p.

293-306.