Copias de espacios de Orlicz de sucesiones en los espacios de Interpolaci´on A

Copies of Orlicz sequences spaces in the interpolation spaces A _ρ,Φ

Copias de espacios de Orlicz de sucesiones en los espacios de Interpolaci´on A

Ventura Echand´ıa ([email protected])

Escuela de Matem´aticas, Facultad de Ciencias, Universidad Central de Venezuela

Jorge Hern´andez ([email protected])

Dpto. T´ecnicas Cuantitativas, DAC, Universidad Centro-Occidental Lisandro Alvarado

Abstract

We prove, by using techniques similar to those in [3], that the inter- polation spaceAρ,Φcontains a copy of the Orlicz sequence spacehΦ. Hereρis a parameter function and Φ is an Orlicz function.

Key words and phrases:Orlicz spaces, Interpolation spaces, param- eter functions.

Resumen

En el presente trabajo, usando t´ecnicas an´alogas a las usadas en [3], demostramos que el espacio de Interpolaci´onAρ,Φ contiene una copia del espacio de Orlicz de sucesioneshΦ. ρdenotar´a una funci´on par´ame- tro y Φ una funci´on de Orlicz.

Palabras y frases clave:Espacios de Interpolaci´on, Espacios de Or- licz, funci´on Par´ametro

1 Introduction

In [3] it was proved that the classical Interpolation spaceAθ,pcontains a copy of`p. Here we are going to give a similar result for Orlicz spaces, for that we need some concepts.

Received 2006/02/22. Revised 2006/08/08. Accepted 2006/08/23.

MSC (2000): Primary 46E30.

This research has been financed by the project number 03-14-5452-2004 of the CDCH-UCV.

1.1 Orlicz spaces and parameter functions

Definition 1. An Orlicz function Φ is a an increasing, continuous , convex function on [0,∞) such that Φ(0) = 0. Φ is said to satisfy the ∆2-condition at zero if l´ım sup

t→0 Φ(2t)/Φ(t)<∞.

Definition 2. Let Φ be an Orlicz function, the space`Φof all scalar sequences {αn}^∞_n=1 such that

X∞

n=1

Φ µ|αn|

µ

¶

<∞ for some µ >0, provided with the norm

k{αn}^∞_n=1k_`

Φ = ´ınf (

µ >0 : X∞

n=1

Φ µ|αn|

µ

¶

≤1 )

,

is a Banach space called anOrlicz sequence space.

The closed subspace hΦ of `Φ, consists of all scalar sequences {αn}^∞_n=1 such

that X∞

n=1

Φ µ|αn|

µ

¶

<∞, for all µ >0.

Remark 1. if Φ satisfy the ∆2-condition at zero, we have that the spaces`Φ

and hΦ coincide, so the result in this work generalize the one in [3] for the case Φ(t) = ^t_p^p, p >1.

We have the following result proved in [4]

Proposition 1. Let Φ be an Orlicz function. Then hΦ is a closed subspace of `Φ and the unit vectors{en}^∞_n=1 form a symetric basis ofhΦ.

In the following the next concept is very important.

Definition 3. A functionρis called a parameter function, orρ∈BK, if ρis a positive increasing continuous function on (0,∞),such that

Cρ= Z _∞

0

m´ın(1,1 t)ρ(t)dt

t <∞, where ρ(t) = sup

s>0

ρ(st) ρ(t).

Definition 4. Given ρ∈BK and Φ an Orlicz function, we define the weighted Orlicz sequence space `ρ,Φ, as the space of all scalar sequences {αm}_m∈Z such that

X

m∈Z

Φ

µ |αm| µρ(2^m)

¶

<∞ for some µ >0, equipped with the norm

°°{αm}_m∈Z°

°`ρ,Φ= ´ınf (

µ >0 : X

m∈Z

Φ

µ |αm| µρ(2^m)

¶

≤1 )

.

1.2 Interpolation spaces

Definition 5. An Interpolation couple A = (A0, A1) consists of two Banach spacesA0andA1which are continuously embedded into a Haussdorff topological vector space V.

The spaceP ¡ A¢

=A₀+A₁ is endowed with the normK(1, a), where K(t, a) =K¡

t, a;A¢

= ´ınf

a=a0+a1

©ka0k_A

0+tka1k_A

1 :a0∈A0, a1∈A1

ª,

is the so called Peetre’s K−functional.

For 0< θ <1 and 1≤p <∞,theclassical Interpolation space Aθ,p, consists of those ain P ¡

A¢

, such that

kak_θ,p=



 Z∞

0

(t^−θK(t, a))^pdt t





p

<∞.

In [2] we introduced the following function norm.

Definition 6. For ρ∈BK and Φ an Orlicz function, the function norm Fρ,Φon¡

(0,∞),^dt_t¢

is defined by

Fρ,Φ= ´ınf



r >0 : Z∞

0

Φ

·|u(t)|

rρ(t)

¸dt t ≤1



,

where uis a measurable function on (0,∞).

Using the function normFρ,Φ, we introduced in [2] for a Banach pairA, the interpolation space Aρ,Φ, as the space of all a ∈ P ¡

A¢

such that Fρ,Φ[K(t, a)]<∞,endowed with the normkak_ρ,Φ=Fρ,Φ[K(t, a)].

Since fora∈Aρ,Φ,withρ∈BK and Φ an Orlicz function, we have that 1

2Φ

µK(2^m, a) ρ(2^m)

1 ρ(2)

¶

≤ ln(2)Φ

µK(2^m, a) ρ(2^m+1)

¶

≤

2Z^m+1

2^m

Φ

µK(t, a) ρ(t)

¶dt t

≤ Φ µ

2K(2^m, a) ρ(2^m)

¶ , we obtain that, for alla∈Aρ,Φ,

°°{K(2^m, a)}_m∈Z°

°

`ρ,Φ ≤2kak_ρ,Φ ≤4ρ(2)°

°{K(2^m, a)}_m∈Z°

°

`ρ,Φ, (1) which gives a discretization of Aρ,Φ.

In this work we use this discretization to prove that the interpolation space Aρ,Φcontains a copy ofhΦ.

2 The main result.

Theorem 1. Let(A0, A1)be a Interpolation couple,ρ∈BK andΦan Orlicz function. We have that ifA0∩A1 is not closed inA0+A1 , then(A0, A1)ρ,Φ

contains a subspace isomorphic tohΦ.

Letε >0; we are going to construct a sequence{xn}^∞_n=1 in (A0, A1)ρ,Φ

and a sequence of integer {Nn}^∞_n=1, strictly increasing, which satisfies the following conditions

1. kxnk_ρ,Φ= 1 2. ´ınf

µ>0

( P

|m|>Nn

Φ

³K(2^m,xn) µρ(2^m)

´

≤1 )

=

°°

°{K(2^m, xn)}_|m|>Nn

°°

°`ρ,Φ

≤₂n+2^ε

3. ´ınf

µ>0

( P

|m|≤Nn

Φ

³K(2^m,xn+1) µρ(2^m)

´

≤1 )

=

°°

°{K(2^m, xn+1)}_|m|≤Nn

°°

°`ρ,Φ

≤ ₂n+2^ε . For the purpose, supposse we have defined x1, x2, ..., xn, N1, ..., Nn−1, which satisfies the above conditions. Since{K(2^m, xn)} ∈`ρ,Φ, i.e.,

´ınf (

µ >0 : X

m∈Z

Φ

µK(2^m, xn) µρ(2^m)

¶

≤1 )

<∞,

there exists 0< µ0<∞,so that X

m∈Z

Φ

µK(2^m, xn) µ0ρ(2^m)

¶

≤1;

thus there exists Nn> Nn−1,such that X

|m|>Nn

Φ

µK(2^m, xn) µ0ρ(2^m)

¶

≤ ε 2ⁿ⁺²

µ 1 µ0

¶ . Therefore

X

|m|>Nn

Φ

µK(2^m, xn)

ε 2ⁿ⁺²ρ(2^m)

¶

≤1;

and from this we deduce that ε 2ⁿ⁺² ≥

°°

°{K(2^m, xn)}_|m|>Nn

°°

°`ρ,Φ

. By using (1) we can findk1, k2>0 so that

k1kxk_Σ(^A)≤

°°

°{K(2^m, x)}_|m|>N

n

°°

°`ρ,Φ

≤k2kxk_Σ(^A)^, for allx∈(A0, A1)_ρ,Φ.

Let now xn+1∈(A0, A1)_ρ,Φbe such that kx_n+1k_Σ(A)≤ ε

k22ⁿ⁺² and kx_n+1k_ρ,Φ= 1, then we have that

°°

°{K(2^m, xn+1)}_|m|≤Nn

°°

°`ρ,Φ

≤k2kxn+1k_Σ(A)≤ ε 2ⁿ⁺². We have thus contructed the required sequence.

Let us see now that for all sequences{αn}^∞_n=1,such that all but finitely many are zero, we have that

µ 1−3ε

2

¶

k{αn}^∞_n=1k_h

Φ ≤

°°

°°

° X∞

n=1

αnxn

°°

°°

°ρ,Φ

≤(1 +ε)k{αn}^∞_n=1k_h

Φ (2) This would mean that{xn}^∞_n=1is equivalent to the basis{en}^∞_n=1 ofhΦ.

In order to prove the inequality (2) we need the following definitions:

Form∈ Zandx∈Σ(A), put

Hm(x) =K(2^m, x);

Hmis an equivalent norm tok..k_Σ(A), for eachm∈ Z.

Also we put form∈ Z,

Fm=¡ Σ¡

A¢ , Hm

¢,

i.e. Fmis the space Σ(A) provided with the normHm. Let nowF = (⊕m∈ZFm)_`ρ,Φ, i.e.

F=n

{xm}_m∈Z:xm∈Fm,k{Hm(xm)}k_`ρ,Φ<∞o ,

provided with the norm

°°{xm}_m∈Z°

°F =k{Hm(xm)}k_`ρ,Φ.

Given {αn}^∞_n=1a scalar sequence such that all but finitely many are zero, we define X ={Xm}_m∈Z, Y ={Ym}_m∈Z, Zⁿ ={Z_mⁿ}_m∈Z ∈F,in the following way

1. For eachm∈Z, Xm=P_∞

n=1αnxn

2. Ym=

½ α1x1 if |m| ≤N1

αnxn if Nn−1≤ |m| ≤Nn, n≥2

3. Z_m¹ = 0,if|m| ≤N1 and Z_m¹ =α1x1 if|m|> N1

4. Forn≥2,Z_mⁿ =

½ 0, if Nn−1≤ |m| ≤Nn

αnxn, otherwise.

We have then that

X =Y + X∞

n=1

Zⁿ (3)

and that

kXk_F = °

°{Hm(Xm)}_m∈Z°

°`ρ,Φ

=

°°

°°

° (

Hm

Ã_∞ X

n=1

αnxn

!)

m∈Z

°°

°°

°`ρ,Φ

=

°°

°°

° (

K(2^m, X∞

n=1

α_nx_n) )

m∈Z

°°

°°

°

`ρ,Φ

= ´ınf (

λ >0 : X

m∈Z

Φ

µK(2^m,P_∞

n=1αnxn) λρ(2^m)

¶

≤1 )

=

°°

°°

° X∞

n=1

αnxn

°°

°°

°ρ,Φ

.

Moreover, we have that

kZⁿk_F ≤ |αn| ε

2ⁿ⁺¹, for eachn≥1.

In fact , for n= 1,we have that X

m∈Z

Φ

µK(2^m, Z_m¹)

|α1|ρ(2^m)

¶

= X

|m|≥N1

Φ

µK(2^m, x1) ρ(2^m)

¶

< ε 2³ < ε

2²,

then °

°Z¹°

°F ≤ |α1| ε 2². Ifn≥2, we have that

X

m∈Z

Φ

µK(2^m, Z_mⁿ)

|αn|ρ(2^m)

¶

= X

|m|≤N_n−1

Φ

µK(2^m, xn) ρ(2^m)

¶

+ X

|m|>N_n

Φ

µK(2^m, xn) ρ(2^m)

¶

≤ ε

2ⁿ⁺² + ε

2ⁿ⁺² = ε 2ⁿ⁺¹.

i.e.

1≥ 2ⁿ⁺¹ ε

X

m∈Z

Φ

µK(2^m, Z_mⁿ)

|αn|ρ(2^m)

¶

≥X

m∈Z

Φ

µ K(2^m, Z_mⁿ)

|αn|_2n+1^ε ρ(2^m)

¶ , therefore

kZⁿk_F ≤ |αn| ε 2ⁿ⁺¹.

Using the H¨older inequality we get X∞

n=1

kZⁿk_F ≤ε 2

X∞

n=1

|αn| 2ⁿ ≤ε

2

°°{αn}^∞_n=1°

°h_Φ

°°

°°

½ 1 2ⁿ

¾_∞

n=1

°°

°°

h_Ψ

≤ ε 2

°°{αn}^∞_n=1°

°h_Φ,

where Ψ is the complementary function of Φ.

Since we have that kYk_F−

X∞ n=1

kZⁿk_F ≤ kXk_F ≤ kYk_F + X∞ n=1

kZⁿk_F,

we obtain that kYk_F−ε

2

°°{αn}^∞_n=1°

°h_Φ≤ kXk_F ≤ kYk_F +ε 2

°°{αn}^∞_n=1°

°h_Φ. (4) For n= 1 we have that

1 = °

°{K(2^m, x1)}_m∈Z°

°

≤

°°

°{K(2^m, x1)}_|m|≤N

1

°°

°`_ρ,Φ+

°°

°{K(2^m, x1)}_|m|>N

1

°°

°`_ρ,Φ

≤

°°

°{K(2^m, x1)}_|m|≤N

1

°°

°`_ρ,Φ+ ε 2², i.e.

1− ε 2² ≤

°°

°{K(2^m, x1)}_|m|≤N

1

°°

°`_ρ,Φ ≤1, and forn≥2 we have that

1 = °°{K(2^m, xn)}_m∈Z°°

`_ρ,Φ

=

°°

°{K(2^m, xn)}_|m|≤Nn−1+{K(2^m, xn)}_{Nn−1≤|m|≤Nn}+{K(2^m, xn)}_|m|>Nn

°°

°`_ρ,Φ

≤ ε

2ⁿ⁺¹+

°°

°{K(2^m, xn)}_{Nn−1≤|m|≤Nn}

°°

°`_ρ,Φ+ ε 2ⁿ⁺², i.e.

1−3ε

2² ≤1− 3ε 2ⁿ⁺² ≤

°°

°{K(2^m, xn)}_N

n−1≤|m|≤Nn

°°

°`_ρ,Φ≤1.

Now using the fact that

kYk_F = °

°{Hm(Ym)}_m∈Z°

°`ρ,Φ

=

°°

°©

K(2^m, Ym)ª

m∈Z

°°

°`ρ,Φ

=

°°

°°

°

©K(2^m, α1x1)ª

|m|≤N1+ X∞ n=2

µ©

K(2^m, αnxn)ª

Nn−1≤|m|≤Nn

¶°°°°

°

`ρ,Φ

≤

°°

°©

K(2^m, α1x1)ª

|m|≤N1

°°

°

`ρ,Φ

+

°°

°°

° X∞ n=2

µ©

K(2^m, αnxn)ª

Nn−1≤|m|≤Nn

¶°°°

°°

`ρ,Φ

=

°°

°©

|α1|K(2^m, x1)ª

|m|≤N1

°°

°

`ρ,Φ

+

°°

°°

° X∞ n=2

µ©

|αn|K(2^m, xn)ª

Nn−1≤|m|≤Nn

¶°°°°°

`ρ,Φ

,

we get, by replacing in (4), that µ

1−3ε 2

¶°

°{αn}^∞_n=1°

°h_Φ≤ kXk_F ≤

³ 1 +ε

2

´ °°{αn}^∞_n=1°

°h_Φ,

which means µ

1−3ε 2

¶ °°

°°

° X∞ n=1

αnen

°°

°°

°h_Φ

≤

°°

°°

° X∞ n=1

αnxn

°°

°°

°ρ,Φ

≤(1 +ε)

°°

°°

° X∞ n=1

αnen

°°

°°

°h_Φ

, as desired.

References

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[2] Echand´ıa V., Finol C., Maligranda L., Interpolation of some spaces of Orlicz type, Bull. Polish Acad. Sci. Math., 38, 125–134, 1990.

[3] Levy M., L’espace d’Interpolation r´eel(A0, A1)_θ,pcontain `^p,C. R. Acad. Sci.

Paris S´er. A 289, 675–677, 1979.

[4] Lindenstrauss J., Tzafriri L.,Classical Banach Spaces, Vol. I, Springler-Verlag, 1977.

[5] Peetre, J., A theory of interpolation of normed spaces, Notas de Matematica Brazil 39, 1–86, 1968.