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
ρ,ΦVentura 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
ofℓp. 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.
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 spaceℓΦof all scalar sequences
{αn}∞n=1 such that
∞
X
n=1
Φ
µ|α n|
µ
¶
<∞ for some µ >0,
provided with the norm
k{αn}
∞
n=1kℓΦ = ´ınf
(
µ >0 :
∞
X
n=1
Φ
µ|α n|
µ
¶
≤1
)
,
is a Banach space called anOrlicz sequence space.
The closed subspace hΦ of ℓΦ, 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 spacesℓΦ
and hΦ coincide, so the result in this work generalize the one in [3] for the
case Φ(t) = tp 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 ℓΦ 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) = sups>0
Definition 4. Given ρ∈BK and Φ an Orlicz function, we define the
weighted Orlicz sequence space ℓρ,Φ, as the space of all scalar sequences
{αm}m∈Z such that
X
m∈Z
Φ
µ
|αm|
µρ(2m) ¶
<∞ for some µ >0,
equipped with the norm
°
°{αm}m∈Z
° °
ℓρ,Φ= ´ınf
(
µ >0 : X
m∈Z
Φ
µ |α m|
µρ(2m) ¶
≤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
©
ka0kA0+tka1kA1 :a0∈A0, a1∈A1ª,
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
kakθ,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,∞),dtt¢ is defined by
Fρ,Φ= ´ınf
r >0 :
∞
Z
0
Φ
·
|u(t)|
rρ(t)
¸
dt t ≤1
,
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 normkakρ,Φ=Fρ,Φ[K(t, a)].
Since fora∈Aρ,Φ,withρ∈BK and Φ an Orlicz function, we have that
1 2Φ
µK(2m, a)
ρ(2m)
1 ρ(2)
¶
≤ ln(2)Φ
µK(2m, a)
ρ(2m+1) ¶
≤
2m+1
Z
2m
Φ
µK(t, a)
ρ(t)
¶dt
t
≤ Φ
µ
2K(2
m, a)
ρ(2m) ¶
,
we obtain that, for alla∈Aρ,Φ,
°
°{K(2m, a)}m∈Z ° °
ℓρ,Φ
≤2kakρ,
Φ ≤4ρ(2)
°
°{K(2m, a)}m∈Z ° °
ℓρ,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. kxnkρ,Φ= 1
2. ´ınf
µ>0 (
P
|m|>Nn
Φ³K(2m,xn)
µρ(2m)
´
≤1
)
=°° °{K(2
m, x
n)}|m|>Nn
° ° °
ℓρ,Φ
≤ ε
2n+2
3. ´ınf
µ>0 (
P
|m|≤Nn
Φ³K(2m,xn+1)
µρ(2m)
´
≤1
)
=°° °{K(2
m, x
n+1)}|m|≤Nn
° ° °
ℓρ,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, x
n)} ∈ℓρ,Φ, i.e.,
´ınf
(
µ >0 : X
m∈Z
Φ
µK(2m, x n)
µρ(2m) ¶
≤1
)
there exists 0< µ0<∞,so that
X
m∈Z
Φ
µ
K(2m, x n)
µ0ρ(2m) ¶
≤1;
thus there exists Nn> Nn−1,such that
X
|m|>Nn
Φ
µK(2m, x n)
µ0ρ(2m) ¶
≤ ε
2n+2 µ 1
µ0 ¶
.
Therefore
X
|m|>Nn
Φ
µK(2m, x n) ε 2n+2ρ(2m)
¶
≤1;
and from this we deduce that
ε 2n+2 ≥
° ° °{K(2
m, x
n)}|m|>Nn
° ° °
ℓρ,Φ .
By using (1) we can findk1, k2>0 so that
k1kxkΣ(A)≤ ° ° °{K(2
m, x)}
|m|>Nn
° ° °
ℓρ,Φ
≤k2kxkΣ(A),
for allx∈(A0, A1)ρ,Φ.
Let now xn+1∈(A0, A1)ρ,Φbe such that
kxn+1kΣ(A)≤
ε k22n+2
and kxn+1kρ,Φ= 1,
then we have that
° ° °{K(2
m, x
n+1)}|m|≤Nn
° ° °
ℓρ,Φ
≤k2kxn+1kΣ(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=1khΦ ≤
° ° ° ° °
∞
X
n=1
αnxn ° ° ° ° °
ρ,Φ
≤(1 +ε)k{αn}
∞
n=1khΦ (2)
In order to prove the inequality (2) we need the following definitions: Form∈ Zandx∈Σ(A), put
Hm(x) =K(2m, x);
Hmis an equivalent norm tok..kΣ(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)ℓρ,Φ, i.e.
F=n{xm}m∈Z:xm∈Fm,k{Hm(xm)}kℓρ,Φ<∞ o
,
provided with the norm
°
°{xm}m∈Z ° °
F =k{Hm(xm)}kℓρ,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 ={Zmn}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. Z1
m= 0,if|m| ≤N1 and Zm1 =α1x1 if|m|> N1
4. Forn≥2,Zn m=
½
0, if Nn−1≤ |m| ≤Nn
αnxn, otherwise.
We have then that
X =Y +
∞
X
n=1
and that
Moreover, we have that
Using the H¨older inequality we get
where Ψ is the complementary function of Φ. Since we have that
kYkF−
we obtain that
kYkF−ε
Now using the fact that
we get, by replacing in (4), that
µ
1−3ε
2
¶ °
°{αn}∞n=1 ° °
hΦ≤ kXkF ≤
³
1 +ε 2
´°
°{αn}∞n=1 ° °
hΦ,
which means
µ
1−3ε
2
¶°°
° ° °
∞
X
n=1
αnen ° ° ° ° °
hΦ ≤
° ° ° ° °
∞
X
n=1
αnxn ° ° ° ° °
ρ,Φ
≤(1 +ε)
° ° ° ° °
∞
X
n=1
αnen ° ° ° ° °
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,pcontain ℓp,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.
