Copias de los espacios de Orlicz en espacios de interpolación

Divulgaciones Matem´ aticas Vol. 16 No. 1(2008), pp. 185–193

Copies of Orlicz sequences spaces in the interpolation spaces Aρ,Φ Copias de espacios de Orlicz de sucesiones en los espacios de Interpolaci´on Aρ,Φ Ventura Echand´ıa (vechandi@euler.ciens.ucv.ve) Escuela de Matem´aticas, Facultad de Ciencias, Universidad Central de Venezuela

Jorge Hern´andez (jorgehernandez@ucla.edu.ve) Dpto. T´ecnicas Cuantitativas, DAC, Universidad Centro-Occidental Lisandro Alvarado Abstract We prove, by using techniques similar to those in [3], that the interpolation space Aρ,Φ contains a copy of the Orlicz sequence space hΦ .

Here ρ is a parameter function and Φ is an Orlicz function.

Key words and phrases:Orlicz spaces, Interpolation spaces, parameter functions.

Resumen En el presente trabajo, usando t´ecnicas an´ alogas a las usadas en [3], demostramos que el espacio de Interpolaci´ on Aρ,Φ contiene una copia del espacio de Orlicz de sucesiones hΦ . ρ denotar´ a una funci´ on par´ ametro y Φ una funci´ on de Orlicz.

Palabras y frases clave: Espacios de Interpolaci´ on, Espacios de Orlicz, funci´ on Par´ ametro

1

Introduction

In [3] it was proved that the classical Interpolation space Aθ,p contains 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.

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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 supΦ(2t)/Φ(t) < ∞. t→0

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

µ Φ

n=1

|αn | µ

¶ < ∞ for some µ > 0,

provided with the norm ( ∞ k{αn }n=1 k`Φ

= ´ınf

µ>0:

∞ X n=1

µ Φ

|αn | µ

¶

) ≤1 ,

is a Banach space called an Orlicz sequence space. ∞

The closed subspace hΦ of `Φ , consists of all scalar sequences {αn }n=1 such that µ ¶ ∞ X |αn | Φ < ∞, for all µ > 0. µ n=1 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 p case Φ(t) = tp , 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 of hΦ .

In the following the next concept is very important.

Definition 3. A function ρ is called a parameter function, or ρ ∈ BK , if ρ is a positive increasing continuous function on (0, ∞) , such that Z

∞

Cρ = 0

dt ρ(st) 1 m´ın(1, )ρ(t) < ∞, where ρ(t) = sup . t t s>0 ρ(t)

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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 µ

X

Φ

m∈Z

|αm | µρ(2m )

¶ < ∞ for some µ > 0,

equipped with the norm ° °{αm }

° °

(

m∈Z `ρ,Φ

= ´ınf

µ>0:

X m∈Z

1.2

µ Φ

|αm | µρ(2m )

)

¶

≤1 .

Interpolation spaces

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

P¡ ¢ The space A = A0 + A1 is endowed with the norm K (1, a), where ¢ ¡ K (t, a) = K t, a; A =

´ınf

a=a0 +a1

©

ª ka0 kA0 + t ka1 kA1 : a0 ∈ A0 , a1 ∈ A1 ,

is the so called Peetre’s K−functional.

For 0 < θ < 1 andP 1≤ ¡ p¢ < ∞, the classical Interpolation space Aθ,p , consists of those a in A , such that p Z∞ dt =  (t−θ K (t, a))p  < ∞. t 

kakθ,p

0

In [2] we introduced the following function norm.

Definition ¡ 6. For¢ρ ∈ BK and Φ an Orlicz function, the function norm Fρ,Φ on (0, ∞), dt is defined by t  

Fρ,Φ

 ¸ Z∞ ·  |u(t)| dt = ´ınf r > 0 : Φ ≤1 ,   rρ(t) t 0

where u is a measurable function on (0, ∞).

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Using the function norm Fρ,Φ , we introduced in [2] for aPBanach ¡ ¢ pair A, the interpolation space Aρ,Φ , as the space of all a ∈ A such that Fρ,Φ [K(t, a)] < ∞, endowed with the norm kakρ,Φ = Fρ,Φ [K(t, a)].

Since for a ∈ Aρ,Φ , with ρ ∈ BK and Φ an Orlicz function, we have that 1 Φ 2

µ

K(2m , a) 1 ρ(2m ) ρ(2)

¶

µ

K(2m , a) ln(2)Φ ρ(2m+1 ) µ ¶ K(2m , a) Φ 2 , ρ(2m )

≤ ≤

m+1 2Z

¶ ≤

Φ 2m

µ

K(t, a) ρ(t)

we obtain that, for all a ∈ Aρ,Φ , ° ° ° ° °{K(2m , a)} ° ≤ 2 kakρ,Φ ≤ 4ρ(2) °{K(2m , a)}m∈Z ° m∈Z `ρ,Φ

`ρ,Φ

¶

,

dt t

(1)

which gives a discretization of Aρ,Φ .

In this work we use this discretization to prove that the interpolation space Aρ,Φ contains a copy of hΦ .

2

The main result.

Theorem 1. Let (A0 , A1 ) be a Interpolation couple, ρ ∈ BK and Φ an Orlicz function. We have that if A0 ∩ A1 is not closed in A0 + A1 , then (A0 , A1 )ρ,Φ contains a subspace isomorphic to hΦ . ∞

Let ε > 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. kxn kρ,Φ = 1 ) ( ° ° ´ ³ m P ° ° ,xn ) ε m ≤ 1 = {K(2 , x )} 2. ´ınf Φ K(2 ≤ 2n+2 ° ° m n |m|>N µρ(2 ) n µ>0

( 3. ´ınf

µ>0

|m|>Nn

P |m|≤Nn

³ Φ

K(2m ,xn+1 ) µρ(2m )

)

´ ≤1

`ρ,Φ

° ° ° ° = °{K(2m , xn+1 )}|m|≤Nn °

`ρ,Φ

≤

ε 2n+2 .

For the purpose, supposse we have defined x1 , x2 , ..., xn , N1 , ..., Nn−1 , which satisfies the above conditions. Since {K(2m , xn )} ∈ `ρ,Φ , i.e., ) ( X µ K(2m , xn ) ¶ Φ ≤ 1 < ∞, ´ınf µ > 0 : µρ(2m ) m∈Z

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189

there exists 0 < µ0 < ∞, so that X µ K(2m , xn ) ¶ Φ ≤ 1; µ0 ρ(2m ) m∈Z

thus there exists Nn > Nn−1 , such that µ ¶ µ ¶ X K(2m , xn ) ε 1 Φ ≤ n+2 . µ0 ρ(2m ) 2 µ0 |m|>Nn

Therefore µ

X

Φ

|m|>Nn

K(2m , xn ) ε m 2n+2 ρ(2 )

¶ ≤ 1;

and from this we deduce that ° ° ε ° ° m ≥ {K(2 , x )} . ° ° n |m|>N n 2n+2 `ρ,Φ By using (1) we can find k1 , k2 > 0 so that ° ° ° ° k1 kxkΣ(A) ≤ °{K(2m , x)}|m|>Nn °

`ρ,Φ

≤ k2 kxkΣ(A),

for all x ∈ (A0 , A1 )ρ,Φ .

Let now xn+1 ∈ (A0 , A1 )ρ,Φ be such that kxn+1 kΣ(A) ≤

ε and k2 2n+2

then we have that ° ° ° ° °{K(2m , xn+1 )}|m|≤Nn °

`ρ,Φ

kxn+1 kρ,Φ = 1,

≤ k2 kxn+1 kΣ(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 ¶ ∞ °X ° 3ε ° ° ∞ αn xn ° 1− k{αn }n=1 khΦ ≤ ° ° ° 2 n=1

∞

≤ (1 + ε) k{αn }n=1 khΦ

ρ,Φ

∞

∞

This would mean that {xn }n=1 is equivalent to the basis {en }n=1 of hΦ .

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(2)

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V. Echand´ıa, J. Hern´andez

In order to prove the inequality (2) we need the following definitions: For m ∈ Z and x ∈ Σ(A), put Hm (x) = K(2m , x); Hm is an equivalent norm to k..kΣ(A) , for each m ∈ Z. Also we put for m ∈ Z, ¡ ¡ ¢ ¢ Fm = Σ A , H m , i.e. Fm is the space Σ(A) provided with the norm Hm .

Let now F = (⊕m∈Z Fm )`ρ,Φ , i.e. n o F = {xm }m∈Z : xm ∈ Fm , k{Hm (xm )}k`ρ,Φ < ∞ , provided with the norm ° ° °{xm } ° m∈Z F = k{Hm (xm )}k`ρ,Φ . ∞

Given {αn }n=1 a scalar sequence such that all but finitely many are zero, we n define X = {Xm }m∈Z , Y = {Ym }m∈Z , Z n = {Zm }m∈Z ∈ F, in the following way 1. For each m ∈ Z, Xm = ½ 2. Ym =

P∞ n=1

αn xn

α1 x1 if |m| ≤ N1 αn xn if Nn−1 ≤ |m| ≤ Nn , n ≥ 2

1 1 3. Zm = 0, if |m| ≤ N1 and Zm = α1 x1 if |m| > N1

½ n 4. For n ≥ 2, Zm =

0, if Nn−1 ≤ |m| ≤ Nn αn xn , otherwise.

We have then that X=Y +

∞ X

Zn

n=1

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(3)

Copies of Orlicz sequences spaces

191

and that kXkF

= =

=

= =

° ° °{Hm (Xm )} ° m∈Z `ρ,Φ ° °( Ã∞ !) ° ° X ° ° αn xn ° ° Hm ° ° n=1 m∈Z `ρ,Φ ° °( ) ∞ ° ° X ° ° m αn xn ) ° K(2 , ° ° ° n=1 m∈Z `ρ,Φ ( ) X µ K(2m , P∞ αn xn ) ¶ n=1 ´ınf λ > 0 : Φ ≤1 λρ(2m ) m∈Z ° ° ∞ °X ° ° ° αn xn ° . ° ° ° n=1

ρ,Φ

Moreover, we have that kZ n kF ≤ |αn |

ε , for each n ≥ 1.

2n+1

In fact , for n = 1, we have that µ ¶ X µ K(2m , Z 1 ) ¶ X K(2m , x1 ) ε ε m Φ = Φ < 3 < 2, m m |α1 | ρ(2 ) ρ(2 ) 2 2 m∈Z

|m|≥N1

then

° 1° °Z ° ≤ |α1 | ε .

F 22

If n ≥ 2, we have that X

µ Φ

m∈Z

n K(2m , Zm ) m |αn | ρ(2 )

¶ =

1≥ therefore

Φ

K(2m , xn ) ρ(2m )

ε

ε

|m|≤Nn−1

≤ i.e.

µ

X ε 2n+2

+

2n+2

=

2n+1

¶ +

X |m|>Nn

µ Φ

K(2m , xn ) ρ(2m )

.

¶ µ ¶ X µ n n K(2m , Zm ) K(2m , Zm ) 2n+1 X , Φ ≥ Φ ε ρ(2m ) ε m∈Z |αn | ρ(2m ) |αn | 2n+1 m∈Z kZ n kF ≤ |αn |

ε .

2n+1

Divulgaciones Matem´ aticas Vol. 16 No. 1(2008), pp. 185–193

¶

192

V. Echand´ıa, J. Hern´andez

Using the H¨ older inequality we get ∞ X

kZ n kF ≤

n=1

°½ ¾∞ ° ∞ ° ° ° ° ° ε° ε X |αn | °{αn }∞ ° ° 1 ° ≤ ε °{αn }∞ ° , ≤ n=1 n=1 hΦ ° hΦ ° 2n 2 n=1 2n 2 2 n=1 h Ψ

where Ψ is the complementary function of Φ.

Since we have that ∞ ∞ X X kZ n kF , kZ n kF ≤ kXkF ≤ kY kF + kY kF − n=1

n=1

we obtain that

° ° ° ε° °{αn }∞ ° ≤ kXk ≤ kY k + ε °{αn }∞ ° . n=1 hΦ F F n=1 hΦ 2 2 For n = 1 we have that ° ° 1 = °{K(2m , x1 )}m∈Z ° ° ° ° ° ° ° ° ° ≤ °{K(2m , x1 )}|m|≤N1 ° + °{K(2m , x1 )}|m|>N1 ° `ρ,Φ `ρ,Φ ° ° ε ° ° m ≤ °{K(2 , x1 )}|m|≤N1 ° + 2, 2 `ρ,Φ kY kF −

(4)

i.e.

° ° ε ° ° ≤ °{K(2m , x1 )}|m|≤N1 ° ≤ 1, 2 2 `ρ,Φ and for n ≥ 2 we have that 1−

° ° 1 = °{K(2m , xn )}m∈Z °` ρ,Φ ° ° ° ° m = °{K(2 , xn )}|m|≤Nn−1 + {K(2m , xn )}Nn−1 ≤|m|≤Nn + {K(2m , xn )}|m|>Nn ° `ρ,Φ ° ° ε ε ° ° m ≤ + n+2 , + °{K(2 , xn )}Nn−1 ≤|m|≤Nn ° `ρ,Φ 2n+1 2

i.e.

° ° 3ε 3ε ° ° ≤ 1 − n+2 ≤ °{K(2m , xn )}Nn−1 ≤|m|≤Nn ° ≤ 1.

2 2 2 `ρ,Φ Now using the fact that 1−

kY kF

=

° °{Hm (Ym )}

=

°© ° ª ° ° m ° K(2 , Ym ) m∈Z °

=

m∈Z

° °

`ρ,Φ

`ρ,Φ

° ¶° ∞ µ ° °© X ª © ª ° ° m m K(2 , αn xn ) N ° ° K(2 , α1 x1 ) |m|≤N + 1 ° n−1 ≤|m|≤Nn ° n=2

≤

=

° °© ª ° ° m ° K(2 , α1 x1 ) |m|≤N ° 1

`ρ,Φ

° °© ª ° ° m ° |α1 | K(2 , x1 ) |m|≤N ° 1

`ρ,Φ

° ¶° ∞ µ °X ° © ª ° ° m +° K(2 , αn xn ) N ° ° n−1 ≤|m|≤Nn °

`ρ,Φ

n=2

`ρ,Φ

° ¶° ∞ µ ° °X © ª ° ° m |αn | K(2 , xn ) N +° ° ° n−1 ≤|m|≤Nn ° n=2

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, `ρ,Φ

Copies of Orlicz sequences spaces

193

we get, by replacing in (4), that µ ¶ ³ ´° ° ° 3ε ° °{αn }∞ ° ≤ kXk ≤ 1 + ε °{αn }∞ ° , 1− n=1 hΦ n=1 hΦ F 2 2 which means ° µ ¶° °∞ ° 3ε ° X ° 1− αn en ° ° ° 2 °n=1

hΦ

°∞ ° °X ° ° ° ≤° αn xn ° ° ° n=1

ρ,Φ

°∞ ° °X ° ° ° ≤ (1 + ε) ° αn en ° ° ° n=1

, hΦ

as desired.

References [1] Beauzamy, B., Espaces d’Interpolation R´eel, Topologie et G´eometrie, Lecture Notes in Math., 666, Springer-Verlag, 1978.

[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,p contain `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.

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