Separability, duality and reflexivity of Orlicz-Besov spaces

Series I: COMMENTATIONES MATHEMATICAE XXX (1991) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXX (1991)

Ma r i a n Lis k o w s k i (Poznan)

Separability, duality and reflexivity of Orlicz-Besov spaces

Abstract. We prove the theorem on representation of continuous linear functionals over Orlicz-Besov spaces BkM(Q) generated by a class of N -functions M. We give some sufficient conditions for Bk M(Q) to be separable and reflexive.

1. Preliminaries. Assume that Q is a nonempty, open and convex set in R".

A function M: O x [0 , oo)—►[(), oo) is said to be a (p-function if (i) M(t, 0) = 0 for almost every teQ;

(ii) M(t, ■) is convex and continuous at zero for a.e. t e Q ; (iii) M(-, u) is measurable for every и ^ 0.

A (^-function M which satisfies the condition

(iv) M(t, u)ju —>0 as n->0 and M(t, u)/u-+ oo as h^ oo for a.e. teQ.

is called an N -function.

Moreover, the following condition for (^-functions M will be used:

(v) M(t, u)dt < oo for every bounded set A c Q and every и ^ 0.

We say that M satisfies the condition Д2 if there exists a constant К > 0 such that M(t, 2u) ^ KM(t, и) for a.e. te Q and every и ^ 0 (for consequences, see e.g. [5]).

Denote by X the space of all real-valued and measurable functions defined on Q, with equality almost everywhere on Q.

For any ф-function M we define the Or liez space LM as the set of all f e X such that g(qf) < oo for some a > 0 depending on /, where g(f) =

§QM(x, \f(x)\)dx. The functional q is a convex modular on X (see [5]).

With respect to the Luxemburg norm |j • ||lm, defined on LM by

\\f\\LM = inf{u > 0: Q{f/u) ^ 1}, LM is a Banach function space (see [5]).

If M is an N-function, then we define the complementary N-function N to M by

N(t, и) = su p {uv — M(t, p)| for и ^ 0 and teQ . v > 0

Let к be an arbitrary positive, noninteger number and к = [fc] + A, where [к] denotes the integer part of к, 0 < A < 1. Then, for any (^-function M, we define on X a functional / by

/ ( / ) = I

|a|=S[fc] f M(x, \Daf(x)\)dx + J f M

S i S i S i

Х + У И(х, y)D*f 1 . 2 ’ |x -y |*

dxdy ^

\ x - y \ n ) ’

d(x, y)« = и(х) —u(y), a = (al5 ..., a„) is a multiindex with a , 0, D*f =

^ “'//dx®1 ... 5x“" is the distributional derivative of /, |a| = a,-. The functional I is a convex modular on X.

Further, for any fixed k, we define

Bk,M(Q) = { / e l : I(af) < oo for some a > 0}.

The vector space Bk,M{Q) is called the Orlicz-Besov space (see [4]). The space Bk M{Q) with the Luxemburg norm || • ||Вк,м generated by the convex modular / is a Banach function space ([4]).

2. Lemmas. Define

В = {(x, y )e Q x Q : x = y}.

For any set A in the сг-algebra I of Lebesgue measurable subsets of Q x Q, Q a R", we define the nonnegative measure v by

(1) v(A) = j J and v(B) = 0.

In the following, LM{Q xQ ,v) is the Orlicz space of all real and measurable functions F defined on Q xQ , generated by the modular

J(F) = j f M((x + y)/2, |F(x, y)|)dv(x, y).

Q S i

The functional J is a convex modular in LM(Q x Q, v) and by || • ||j we denote the Luxemburg norm generated by J .

Lemma 2.1. The measure v is separable.

P ro o f. Let P denote a 2n-dimensional rectangle whose centre has rational coordinates and whose edges have rational length, and such that dist(P, В) > 0.

Let IF be a finite sum of such rectangles P. Then the family of all sets W is countable and dense in the family of sets of finite Lebesgue measure | • | on Q xQ . Moreover, v(IF) < oo for all W.

Let £ ^ 0* Let A be tin ctrbitr&ry nieâsurâble subset of* x О such thut v(A)< oo. First, we suppose additionally that A is bounded. Then v (A u W)

< oo. Hence also v(A — W) < oo for any W, where A — W is the symmetric difference.

Since \A\ < oo, by separability of | * | there exists a set Wt such that

\A - W, I < <5(e). Obviously, v is absolutely continuous with respect to | • |.

Accordingly,

If A is unbounded then clearly A is a disjoint union of bounded measurable sets Am, m = 1, 2, ... Since v(A) = ^m = i v(Am) < 00 there exists k0 such that

V( 0

<

m = ko + 1

We conclude that there exists a set W such that

V(A - W) = v( U Am - W) + v( 0 Am) < ie + is = s.

m = 1 m = ko+ 1

This shows that v is separable.

Now, we shall provide an example of a set A such that dist(v4, В) = 0 and v(,4) < oo.

Example. Let (Ak)k=1 be a sequence of measurable and pairwise disjoint subsets of Q x Q , Q c R", such that

dist(>4fc, B) = p \Ak\ = for к = 1, 2, ...

We set A = ( J “=iy4k. It is easily observed that dist(^4, В) = 0 and

v{A) = M \ ^ - y \

dxdy * г r ™ kn

n ^ C £ k "jjd x d y = C ^

к — 1 A h

Lemma 2.2. The measure v is a-jinite.

P roof. Set

A i = {(x, y )e Q x Q : |x — y\ = 1/i}, i = 1, 2 ,...

TakingK t = (B(0, i) — n Q x Q , where B(0, i) denotes the ball with centre at zero and radius i, we have

oo

U * jU В = Q x Q and v(K{) < oo for every i = 1 ,2 ,...

i = 1

From Lemmas 2.1 and 2.2 and the theorem on separability of Orlicz spaces (see e.g. [5]) we obtain

Lemma 2.3. Let M be a (p-function satisfying (v) and the condition Л2. Then

Lm( Qx Q , v) is separable.

We now investigate the problem of continuous linear functionals over LM(Q x fl,v ) and the reflexivity of this space. We first prove

Lemma2.4. Let Q c: R" be such that for all x, уeQ we have j(x + y)efi. If Л е й and \A\ = 0, then

|{(x, y)EÜ x Q: ^(х + у)еЛ}| = 0.

P roof. Set C = {(x, y)EÜ ^x Q: ^(x + y)E A}. We have

|C| = f |CJdu, where Cu = {i;e£2: (и, i>)eC}.

h Then

Cu — {v eQ: j(u + v ) e Л} — {veO: v e 2 A — u } = Q r \ (2A — и) c 2A — и for every ueQ.

Hence |C„| ^ \2A\ = 0 for every ^{u eQ ,} and this implies |C| = 0.

Since M(t, и^{) ,} defined on the product Q x [0, ^{g o) ,} is finite for a.e. tEÜ, in view of Lemma 2.4, the composition M((x, y)/2, u), defined on Q x Q x [0, со), is finite for v -a.e. (x, y )e Q x Q . Hence and from [6] we obtain

Lemma 2.5. Let (Q xQ , I , v) be the measure space with v defined by (1).

Then the function M((x + y)/2, u): £2x£2x[0, ^{g o)} -> [0, oo) satisfies the con­

dition (B) (see [2], [3]).

Lemma 2.5 and Theorem 4.8 in [3] yield immediately

Lemma 2.6. Suppose M((x + y)/2, u): Q x Q x [0, go) ->■ [0, oo) is an N-fun- ction. Then every continuous linear functional on LM{Q x Ü, v) is of the form

F*(G) = J J F(x, y)G(x, y)dv(x, y) n n

for GeL m(Q x Q, v), where F eLn(Q ^x Ü, v) and N is the N-function complemen­

tary to M.

Simultaneously, Lemma 2.5 and Theorem 1.2 in [2] imply

Lemma 2.7. I f M ((x+ y)/2, u): Q x£2x[0, oo ) - [ 0 , oo ) is an N-function, and both complementary functions M and N satisfy the condition A2, then LM(Q x Q, v) is reflexive.

3. Separability, duality and reflexivity of Bk,M(Q). Let i

1= X 1 and LFM = Y \(Lm(Q)xLm(QxQ, v)).

|a|<M i= 1

For / = F{),-=i eJFm we define

6( f ) = Z U M (x ’ Ifi(x)\)dx + j J M((x + y)/2, IFf x , y)\)dv(x, y)}.

i = 1 П Q Q

Obviously, q is a convex modular in f£ M. Let Ц-Ц^м denote the Luxemburg norm in f£ M. With this norm is a Banach space.

Suppose that the l multiindices a satisfying |a| ^ [Zc] are linearly ordered so that with each / e B k M(Q) we may associate a well-defined vector P f in S£M given by

P f = Daf ;D °f(x)-D *f(y) l * - y |A |«I*M

Then \\f\\Bk,M = \\Pf\\<r« for any / e B kM(Q). So P is an isometric isomorphism of Bk,M(Q) onto a subspace of f£M.

3.1. Separability. We shall prove the following

Theorem 3.1. I f M is a tp-function satisfying conditions Л2 and (v) then Bk,M(Q) is separable.

P ro o f. S£M is separable as the product of a finite number of separable spaces. Since the operator P is an isomorphism of Bk M onto P(Bk,M) c: <£M and is complete, P(Bk,M) is a closed subspace of f£M. Thus P{Bk,M), and hence Bk M, is separable.

3.2. Duality. From Theorem 4.8 in [3] and Lemma 6 we deduce immediately

Lemma 3.2. I f M is an N -function satisfying condition Л2, then every continuous linear functional over f£ M is of the form

f*(g) = Z j f fi(x)gt{x)dx + $ j F fx, y)Gf(x, у)

f=i Lq qq Iх / Ij

for g = (g{, Gf)i=1 eJS?M, where f = ( / f, Т;)-=1е=^^ and N is complementary to M.

Thus each v e f£ N defines a continuous linear functional over Bk,M of the form

L{u)= { | £ аи(х)г;а(х)Лх + | J

|a|^[k] KQ П Q

where v = (va, VJ]a^ [k], for ue Bk,M(Q).

D*u(x)-Dau(y) dxdy | l x - #

Theorem 3.2. Let M be an N -function satisfying the condition A2. Then for every continuous linear functional L over Bk,M(Q) there exists v e £ f N, v — (u«> ^)|«И[к]5 such that

L(w)= Z j j Dau(x)va{x)dx + $ $

|a|<[k] (.« Q Q

for u e Bk,M(Q).

Dau(x)-D*u(y)

|х - у |я Va(X> У) dxdy I

\x -y \* l

U — Comment. Math. 30.2

P ro o f. By using Lemma 3.2, the proof is analogous to the proof of the respective theorem for the Sobolev space Wk(Q), 1 < p < oo (Theorem 3.8 in [1]).

3.3. Reflexivity.

Theorem 3.3. Let M be an N-function. I f M and its complementary function satisfy the condition Д2, then Bk,M(Q) is reflexive.

P ro o f. f£ M is reflexive, by Lemma 2.7. Since Bk,M(Q) is isomorphic to the closed subspace P(Bk,M) of f£M, it is also reflexive.

References

[1] R. A. A d a m s, Sobolev Spaces, Academic Press, 1975.

[2] H. H u d z ik , On some equivalent conditions in Musielak-Orlicz spaces, Comment. Math. 24 (1984), 57-64.

[3] A. K o z e k , Orlicz spaces o f functions with values in Banach spaces, ibid. 19 (1977), 259-288.

[4] M. L is k o w s k i, On generalized Besov spaces Bk,M(£2), Funct. Approx. Comment. Math. 15 (1986), 175-184.

[5] J. M u s ie la k , Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer, Berlin 1983.

[6] M. W is la , Some remarks on the Kozek Condition B, Bull. Polish Acad. Sci. Math. 32 (1984), 407-415.

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