Inclusion Mappings between Orlicz Sequence Space - DiVA

Journal of Functional Analysis 176, 264279 (2000) doi:10.1006jfan.2000.3624, available online at http:www.idealibrary.com on

Inclusion Mappings between Orlicz Sequence Spaces Lech Maligranda Department of Mathematics, Lulea# University of Technology, S-971 87 Lulea# , Sweden E-mail: lechsm.luth.se

and Mieczys*aw Masty*o Faculty of Mathematics and Computer Science, A. Mickiewicz University, Matejki 4849, 60-769 Poznan, Poland E-mail: mastyloamu.edu.pl Communicated by C. Foias Received January 25, 1999; revised June 30, 1999; accepted May 16, 2000

It is shown that if l. is an Orlicz sequence space, then the space l w1 (l. ) of weakly summable sequences in l. is continuously embedded into l.(l2 ) (resp., into l.(l. )) whenever t [ .(- t) is equivalent to a concave function (resp., a convex function and . is a supermultiplicative function). By combining the above results with the interpolation theory we proved continuous inclusions between spaces l w1 (l.0 ) and l,(l.1 ), where l.0 /Ä l.1 and , is a certain Orlicz function depending on . 0 and . 1 . In particular, if . 0 and . 1 are power functions we obtain the well known result on (r, 1)-summability of the inclusion mappings between lp -spaces proved independently by G. Bennett (1973, J. Funct. Anal. 13, 2027) and B. Carl (1974, Math. Nachr. 63, 253360). 2000 Academic Press

INTRODUCTION The well known characterization of absolutely (r, 1)-summing identity operators I : lp /Älq proved independently by Bennett [1] and Carl [4], which itself extends old result of Hardy and Littlewood [9] on continuous bilinear forms on lp _lq (1 p, q), has found many interesting applications in functional analysis, in particular to study multipliers (see [2]). We mention also a remarkable paper by Ovchinnikov [20], where a new interpolation theorem for operators acting on the weighted sequence lp spaces was proved. The proof of this result is based on an application of a corollary of Bennett's theorem saying that the inclusion map lp /Ä l is ( p, 1)-summing. By using this fact together with the reiteration theorem, 264 0022-123600 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved.

265

MAPPINGS BETWEEN ORLICZ SPACES

Ovchinnikov proved a sharp version of the RieszThorinMarcinkiewicz theorem. We refer the reader to [17], where among others an extension of Ovchinnikov's results for a large class of abstract real interpolation spaces was presented via Banach ideals of operators. We also refer to [5], where several results in the theory of ( p, q)-summing operators were improved by using a unified but elementary tensor product concept. In the present paper we study inclusion mappings between Orlicz sequence spaces. Similarly as in the case of lp -spaces, the case l. /Ä l2 is in a sense more delicate. It is shown that if an Orlicz function . is such that t [ .(- t) is equivalent to a concave function, then a certain, independently interesting, inequality holds. By combining the obtained inequality with Khintchine's inequality for convex functions we prove that the Banach space l w1 (l. ) of weakly summable sequences in the Orlicz sequence space l. is continuously embedded into the KotheBochner space l.(l2 ). On the other hand, if a supermultiplicative Orlicz function . is such that t [ .(- t) is equivalent to a convex function, then l w1 (l. ) is continuously embedded into the KotheBochner space l.(l. ). By combining those results with some vector-valued continuous inclusions for CalderonLozanovski@$ spaces, we extend the results mentioned above. As an application we obtain the well known results on (r, 1)-summability of the inclusion mappings between lp -spaces.

1. PRELIMINARIES In order to make the paper as self-contained as possible, we quote in this section some definitions and certain well known facts concerning Orlicz sequence spaces. Let L 0(+) denote the space of all equivalence classes of measurable functions on 0 with the topology of convergence in measure on +-finite sets and let E/L 0(+) be a Banach lattice (on (0, +)). If the unit ball B E = [x # E : &x& E 1] is closed in L 0(+), i.e., E has the Fatou property, then E is a called a maximal Banach lattice. A Banach lattice defined on (N, +) with the counting measure + is called a Banach sequence lattice. If E is a Banach lattice on (0, +), then the Kothe dual E$ of E is a Banach lattice, which can be identified with the space of all functionals possessing an integral representation, that is,

{

E$= y # L 0(+) : &y& E$ = sup &x&E 1

|

=

|xy| d+1 that B= k=1 u k v k >v n . Combining these remarks with the above inequality we obtain f $(x)

2x .(v n ) .$(v n )& 0 vn vn

\

+

for any x>0. This implies that f is nondecreasing on [0, ). Now, by the inductive assumption, we have f(0)=.(- A)+.(B)&C0, and thus f (u n ) f (0) yields the required inequality. 2 2 n 2 2 If B= n&1 k=1 u k v k 1, then by k=1 u k v k >1 and v n =min[v 1 , ..., v n ] 2 we have B+u n v n >v n . Hence, by the concavity of and 0u n 0. n

K

Corollary 2.2. If . is an Orlicz function such that t [ .(- t) is a concave function, then the following inequality holds, n

u 2k .(v k )4. v2 k=1 k :

\\

n

12

: u 2k k=1

+ +

+4.

\

n

u 2k v k=1 k :

+

for all u j 0, v j >0, j=1, ..., n, and n # N. Proof. Since (t)=.(- t) is a concave function, t [ .(t)t 2 is nonincreasing. This implies that .(2t)4.(t) for all t0. Define an Orlicz function . 0 by . 0(t)=

|

t 0

.(s) ds s

for t0. Since s [ .(s)s is a nondecreasing function, we obtain by the above inequality 1 4

.(t). 0(t).(t).

271


Moreover, the function s [ (s)s is nonincreasing whence t [ . 0(- t)=

|

t 0

.(- s) ds= 2s

|

t 0

(s) ds 2s

is a concave function. Thus, applying Theorem 2.1 to . 0 and the above estimates, we obtain the required inequality. K Remark. Note that if .: R + Ä R + is a nondecreasing function, then t [ .(- t) is equivalent to a concave function if and only if t [ .(t)t 2 is almost nonincreasing function; i.e., there exists a constant c>0 such that .(s)s 2 c.(t)t 2 for all s>t>0. This easily yields that for any Orlicz function . the inequality in Corollary 2.2 with a constant C>0 instead of 4 is equivalent to the statement that t [ .(- t) is equivalent to a concave function. In fact, if 00 and s 0 >0 such that the estimate .(st)C.(s) .(t) holds for all 00, then the identity map I: l. /Ä l. is (., 1)-summing. Proof. First observe that if . # 2 2 , then there exists C . >0 such that for any x i =[x ij ] j=1 # l. with &[x i ]& 1, w 1 we have

: . j=1

\\

n

: |x ij | 2 i=1

12

+ + C

.

for all n # N.

In order to prove this fact we use the following version of Khintchine's inequality (see [8, p. 282] when ., . # 2 2 or [10, Lemma 6.2], [15, Sub* lemma 14.6] when . # 2 2 ) which states that if . is an Orlicz function with

272

MALIGRANDA AND MASTY4O

. # 2 2 and [r n ] is a sequence of Rademacher's functions, then there exists a constant C . >0 such that the following inequality holds .

\\

n

12

: |a i | 2 i=1

+ +

C .

|

1

.

0

\}

n

: a i r i (t) i=1

}+ dt

for any finite sequence [a i ] ni=1 of real numbers. Hence

: . j=1

\\

n

12

: |x ij | 2 i=1

+ +

C .

|

1

\}

: .

0 j=1

C . sup \ . |=i | 1

n

: x ij r i (t) i=1 n

\{}

}+ dt

: = i x ij

}= + . j=1

i=1

Now observe that by &[x i ]& 1, w 1, it follows that sup |=i | 1

"

n

: =i xi i=1

"

=sup .

{

n

=

: |x*(x i )|: &x*& l*. 1 1. i=1

Since &!& . 1 implies \ .(!)1, we obtain sup \ . |=i | 1

\{}

n

: = i x ij i=1

}= + 1. j=1

Combining the above inequalities, we obtain the desired inequality. Suppose that (i) holds. This implies that there exists c>0 such that ct 2 .(t) for all 0t1, which gives continuous embedding l. /Ä l2 . We show that [x i ] # l.(l2 ). By Corollary 2.2 we may assume, without loss of generality, that . is a differentiable Orlicz function. Combining the above remarks and Theorem 2.1, we have for any n # N with &x i & :=&x i & l2 , n

n

: .(&x i &)= : i=1

i=1

j=1

: . j=1

: |x ij | 2 .(&x i &) &x i & &2 = :

\\

j=1 n

i=1

C . + : . j=1

12

: |x ij | 2

\

n

+ +

+: . j=1

+

: |x ij | 2&x i & . i=1

\

n

|x ij | 2 .(&x i &) &x i & 2 i=1 :

n

: |x ij | 2&x i & i=1

+

273


Thus we need only to estimate the second term in the last expression. Let '=[' j ] # (l. )$. Then, by the classical Khintchine's inequality,

n

n

: ' j : |x ij | 2&x i &= : j=1

i=1

: ' j |x ij | |x ij | &x i & &1

i=1

j=1

n

12

\ + \ = : : |' x | \ + - 2 : | : ' x r (t) dt } } : |' j x ij | 2

:

i=1

j=1

n

: |x ij | 2 &x i & &2

j=1

+

12

12

2

j

i=1

ij

j=1 n

1

j

0

i=1

- 2

|

ij j

j=1

n

1

: |(x i , y(t)) | dt,

0 i=1

where y(t)=[' j r j (t)] for 0t1. Since l. is a maximal lattice, i.e., (l. )"=l. , we obtain by taking the supremum over all ' # (l. )$ with &'& (l.)$ 1,

"{

n

: |x ij | 2&x i & i=1

= " j=1

- 2

|

n

1

: |( x i , y(t)) | dt

0 i=1

.

- 2 sup

{

n

=

: |x*(x i )|: &x*& l*. 1 - 2. i=1

This yields, by concavity of the function , that

: . j=1

\

n

+

: |x ij | 2&x i & 2. i=1

Combining the preceding inequalities we obtain n

: .(&x i & l2 )C . +2, i=1

and thus by convexity of ., &[x i ]& l.(l2 ) C . +2. Thus we proved that l w1 (l. ) /Ä l.(l2 ), which is equivalent to the fact that the inclusion map l. /Ä l2 is (., 1)summing.

274


Assume that . satisfies conditions in (ii). Without loss of generality we may assume that (t)=.(- t) is convex on R + . This implies that the function t [ .(t)t 2 is nondecreasing, so .(t)t Ä as t Ä . Thus the Young function . (u)=sup[uv&.(v) : v0] *

for

u0

is finite. Clearly, under the assumption on ., . satisfies the 2 2 -condition and for any u 0 >s 0 there exists C=C(u 0 )>0 such that .(st)C.(s) .(t) holds for 00. Thus we may assume that s 0 =1. &1(1). * Let :=[: j ] # l. be a sequence of positive numbers such that \ . (:)1, * * and hence : j 1. &1(1) for all j # N. Since . # 2 2 , it follows by * . &1(u) . &1(u)2u for all u0 that there exists K>2 such that * . (u)K.(. (u)u) * * for all u>0. Combining these remarks with superadditivity of (by convexity of ) and the inequality &!& . 1+\ .(!) true for all ! # l. , we have n

n

: : i &x i & . = : . (: i ) &: i x i . (: i )& . * * i=1 i=1 n

n

: . (: i )+ : : . (: i ) .(: i |x ij |. (: i )) * * * i=1 i=1 j=1

1+K : j=1

n

: .(. (: i ): i ) .(: i |x ij |. (: i )) * * i=1

n

1+KC &1 : j=1

1+C 1 : . j=1

n

: ( |x ij | 2 )

: .(|x ij | )=1+C 1 : i=1

\\

j=1

n

i=1

12

: ( |x ij | 2 ) i=1

+ + 1+C C =K . 1

.

.

In consequence we conclude that

&[&x i & . ]& . sup \. (:)1 *

: : i &x i & . K . . i=1

Thus, we proved the embedding l w1 (l. ) /Ä l.(l. ), which means that the identity operator I: l. Ä l. is (., 1)-summing. K


275

3. APPLICATIONS In this section we use some vector-valued continuous inclusions for CalderonLozanovski@$ spaces in order to show that under some assumptions on Orlicz functions . 0 and . 1 the inclusion map l.0 /Äl.1 is (,, 1)-summing. Recall that if X =(X 0 , X 1 ) is a couple of Banach lattices on (0, +) and U denotes the set of all concave and positive homogeneous functions of degree one : R + _R + Ä R + such that (0, 0)=0, then the Calderon Lozanovski@ space (X )=(X 0 , X 1 ) consists of all x # L 0(+) such that |x| *(|x 0 |, |x 1 |) +-a.e. for some x j # X j with &x j & Xj 1, j=0, 1. The space .(X ) is a Banach lattice equipped with the norm &x& (X ) =inf[*>0 : |x| *(|x 0 |, |x 1 | ), &x j & Xj 1, j=0, 1] (see [3, 14]). It is easy to see that if . is an Orlicz function and E is a Banach lattice on (0, +), then the CalderonLozanovski@$ space (E, L ) with (s, t)= t. &1(st) for t>0 and 0 if t=0 coincides isometrically with the Banach lattice E . =[x # L 0(+) : .( |x|*) # E

for some

*>0]

equipped with the norm &x& E. =inf[*>0 : &.( |x|*)& E 1]. In particular, (l1 , l ) coincides isometrically with the Orlicz sequence space l. . In the sequel we will use the well known formula (see [19, pp. 460461]; see also [15, pp. 178180]) l, =(l.0 , l.1 ), which is true for any Orlicz functions . 0 , . 1 and any # U with , &1 = &1 (. &1 0 , . 1 ). We denote by U(., . 0 , . 1 ) a subset of all functions # U such that there are Orlicz functions ., . 0 , . 1 and positive constants C 1 , C 2 , $ such that .(C 1 (s, t))C 2(. 0(s)+. 1(t)) holds for all 00]. We will need the following result from [6].

276


Theorem 3.1. Let X =(X 0 , X 1 ) be any couple of Banach lattices on (0, +) and let E /Ä l be any Banach sequence lattice. If # U(., . 0 , . 1 ), then (E .0(X 0 ), E .1(X 1 )) /ÄE .((X 0 , X 1 )). Corollary 3.2. Let # U be a submultiplicative function; i.e., there exists a constant C>0 such that (1, st)C(1, s) (1, t) for all s, t>0. Then, for any couple (X 0 , X 1 ) of Banach lattices, the following continuous inclusion holds (l.0(X 0 ), l.1(X 1 )) /Äl,((X 0 , X 1 )), &1 where , &1(t)=(. &1 0 (t), . 1 (t)) for t0.

Proof. Fix s, t>0 and put v=max[. 0(s), . 1(t)]. Since is nondecreasing in each variable, we have , &1(v)(s, t). By the assumptions on , it follows that (s, t)C(s, t) for all s, t>0. Combining these remarks, we get # U(,, . 0 , . 1 ). Thus the result follows from Theorem 3.1. K An immediate consequence of the preceeding results is the following ) for two Orlicz theorem, where the relation .O (resp., equivalence . Ä functions means that there exist C>0 and t 0 >0 such that .(t)C(t) for all 00 such that

\

n

: &x j & 2.0

j=1

+

12

C

|

1

0

"

n

: r j (t) x j j=1

for every x 1 , ..., x n # l.0 and n # N (see [12]).

"

dt .0


277

Let [x n ] # l w1 (l.0 ). We conclude by the above that [x n ] # l2(l.0 ). We also have [x n ] # l.0(l2 ), by Theorem 2.3. In consequence we obtain [x n ] # l2(l.0 ) & l.0(l2 ) /Ä(l2(l.0 ), l.0(l2 )) for any # U. If we assume that is a submultiplicative function, then by Corollary 3.2 it follows that [x n ] # l,((l.0 , l2 ))=l,(l.1 ). This shows that the inclusion map l.0 /Äl.1 is (,, 1)-summing. (ii) If t [ . 0(- t) is equivalent to a concave function and . 1 O t 2, then the identity map I : l.0 /Ä l.1 regarded as the composition of two continuous inclusions l.0 /Äl2 and l2 /Äl.1 (by . 1 Ot 2 ) is (. 0 , 1)-summing, by Theorem 2.3 and the ideal property of 6 l. , 1 . If t [ . 0(- t) is equivalent 0 to a convex function and . 0 is supermultiplicative with . 1 O . 0 , then I: l.0 /Ä l.1 , regarded as the composition of the two continuous inclusions l.0 /Ä l.0 and l.0 /Ä l.1 , is (. 0 , 1)-summing, by Theorem 2.3. K An application of the above theorem yields immediately the result proved independently by Bennett [1] and Carl [4]. Corollary 3.4. (i) If 1 pq2, then lp /Ä lq is (r, 1)-summing, where 1r=1 p&1q+12. (ii) If either 1 p2q or 2 pq, then lp /Älq is ( p, 1)-summing. Proof. Let . 0(t)=t p and . 1(t)=t q for t0. By Theorem 3.3 we need only to consider the case 1 p