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, ∞). Divulgaciones Matem´ aticas Vol. 16 No. 1(2008), pp. 185–193
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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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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 Matem´ aticas Vol. 16 No. 1(2008), pp. 185–193
(2)
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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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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
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¶
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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
Divulgaciones Matem´ aticas 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 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.
Divulgaciones Matem´ aticas Vol. 16 No. 1(2008), pp. 185–193