Divulgaciones Matemáticas 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 Interpolación Aρ,Φ
Ventura Echandı́a (vechandi@euler.ciens.ucv.ve)
Escuela de Matemáticas, Facultad de Ciencias,
Universidad Central de Venezuela
Jorge Hernández (jorgehernandez@ucla.edu.ve)
Dpto. Técnicas 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 técnicas análogas a las usadas en [3],
demostramos que el espacio de Interpolación Aρ,Φ contiene una copia
del espacio de Orlicz de sucesiones hΦ . ρ denotará una función parámetro y Φ una función de Orlicz.
Palabras y frases clave: Espacios de Interpolación, Espacios de Orlicz, función Parámetro
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.
186
1.1
V. Echandı́a, J. Hernández
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 lı́m 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
= ı́nf
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
mı́n(1, )ρ(t) < ∞, where ρ(t) = sup
.
t
t
s>0 ρ(t)
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 185–193
Copies of Orlicz sequences spaces
187
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
= ı́nf
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 =
ı́nf
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
= ı́nf r > 0 : Φ
≤1 ,


rρ(t) t
0
where u is a measurable function on (0, ∞).
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 185–193
188
V. Echandı́a, J. Hernández
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. ı́nf
Φ K(2
≤ 2n+2
°
°
m
n
|m|>N
µρ(2 )
n
µ>0
(
3. ı́nf
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 < ∞,
ı́nf µ > 0 :
µρ(2m )
m∈Z
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 185–193
Copies of Orlicz sequences spaces
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 .
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 185–193
(2)
190
V. Echandı́a, J. Hernández
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
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 185–193
(3)
Copies of Orlicz sequences spaces
191
and that
kXkF
=
=
=
=
=
°
°
°{Hm (Xm )}
°
m∈Z `ρ,Φ
°
°(
̰
!)
°
°
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
ı́nf 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 Matemáticas Vol. 16 No. 1(2008), pp. 185–193
¶
192
V. Echandı́a, J. Hernández
Using the Hölder 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
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 185–193
,
`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 Réel, Topologie et Géometrie, Lecture
Notes in Math., 666, Springer-Verlag, 1978.
[2] Echandı́a 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 réel (A0 , A1 )θ,p contain `p , C. R. Acad. Sci.
Paris Sér. 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.
Divulgaciones Matemáticas Vol. 16 No. 1(2008), pp. 185–193