CERTAIN PROPERTIES OF GENERALIZED ORLICZ
SPACES
Generalized Orlicz Spaces
PRITI UPRETI
PANKAJ JAIN
Department of Mathematics
Deshbandhu College (University of Delhi)
Kalkaji, New Delhi - 110 019, India
EMail: pankajkrjain@hotmail.com
Department of Mathematics
Moti Lal Nehru College (University of Delhi)
Benito Juarez Marg, Delhi 110 021, India
Pankaj Jain and Priti Upreti
vol. 10, iss. 2, art. 37, 2009
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Received:
20 March, 2008
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Accepted:
09 October, 2008
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Communicated by:
L.-E. Persson
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2000 AMS Sub. Class.:
26D10, 26D15, 46E35.
Key words:
Banach Function spaces, generalized Orlicz class, generalized Orlicz space, Luxemburg norm, Young function, Young’s inequality, imbedding, convergence, separability.
Abstract:
In the context of generalized Orlicz spaces XΦ , the concepts of inclusion, convergence and separability are studied.
Acknowledgements:
The research of the first author is partially supported by CSIR (India) through the
grant no. 25(5913)/NS/03/EMRII.
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Contents
1
Introduction
3
2
Preliminaries
4
3
Comparison of Generalized Orlicz Spaces
8
Generalized Orlicz Spaces
4
Convergence
15
Pankaj Jain and Priti Upreti
vol. 10, iss. 2, art. 37, 2009
5
Separability
19
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1.
Introduction
In [4], Jain, Persson and Upreti studied the generalized Orlicz space XΦ which is a
unification of two generalizations of the Lebesgue Lp -spaces, namely, the X p -spaces
and the usual Orlicz spaces LΦ . There the authors formulated the space XΦ giving
it two norms, the Orlicz type norm and the Luxemburg type norm and proved the
two norms to be equivalent as is the case in usual Orlicz spaces. It was shown that
XΦ is a Banach function space if X is so and a number of basic inequalities such
as Hölder’s, Minkowski’s and Young’s were also proved in the framework of XΦ
spaces.
In the present paper, we carry on this study and target some other concepts in the
context of XΦ spaces, namely, inclusion, convergence and separability.
The paper is organized as follows: In Section 2, we collect certain preliminaries
which would ease the reading of the paper. The inclusion property in XΦ spaces has
been studied in Section 3. Also, an imbedding has been proved there. In Sections 4
and 5 respectively, the convergence and separability properties have been discussed.
Generalized Orlicz Spaces
Pankaj Jain and Priti Upreti
vol. 10, iss. 2, art. 37, 2009
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2.
Preliminaries
Let (Ω, Σ, µ) be a complete σ-finite measure space with µ(Ω) > 0. We denote
by L0 (Ω), the space of all equivalence classes of measurable real valued functions
defined and finite a.e. on Ω. A real normed linear space X = {u ∈ L0 (Ω) : kukX <
∞} is called a Banach function space (BFS for short) if in addition to the usual norm
axioms, kukX satisfies the following conditions:
P1. kukX is defined for every measurable function u on Ω and u ∈ X if and only
if kukX < ∞; kukX = 0 if and only if, u = 0 a.e.;
Generalized Orlicz Spaces
Pankaj Jain and Priti Upreti
vol. 10, iss. 2, art. 37, 2009
P2. 0 ≤ u ≤ v a.e. ⇒ kukX ≤ kvkX ;
P3. 0 < un ↑ u a.e. ⇒ kukX ↑ kukX ;
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P4. µ(E) < ∞ ⇒ kχE kX < ∞;
R
P5. µ(E) < ∞ ⇒ E u(x)dx ≤ CE kukX ,
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where E ⊂ Ω, χE denotes the characteristic function of E and CE is a constant
depending only on E. The concept of BFS was introduced by Luxemburg [9]. A
good treatment of such spaces can be found, e.g., in [1]
Examples of Banach function spaces are the classical Lebesgue spaces Lp , 1 ≤
p ≤ ∞, the Orlicz spaces LΦ , the classical Lorentz spaces Lp,q , 1 ≤ p, p ≤ ∞, the
generalized Lorentz spaces Λφ and the Marcinkiewicz spaces Mφ .
Let X be a BFS and −∞ < p < ∞, p 6= 0. We define the space X p to be the
space of all measurable functions f for which
1
kf kX p := k|f |p kXp < ∞ .
For 1 < p < ∞, X p is a BFS. Note that for X = L1 , the space X p coincides with Lp
spaces. These spaces have been studied and used in [10], [11], [12]. Very recently
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in [2], [3], Hardy inequalities (and also geometric mean inequalities in some cases)
have been studied in the context of X p spaces. For an updated knowledge of various
standard Hardy type inequalities, one may refer to the monographs [6], [8] and the
references therein.
A function Φ : [0, ∞) → [0, ∞] is called a Young function if
Z s
Φ(s) =
φ(t)dt ,
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Pankaj Jain and Priti Upreti
where φ : [0, ∞) → [0, ∞], φ(0) = 0 is an increasing, left continuous function
which is neither identically zero nor identically infinite on (0, ∞). A Young function
Φ is continuous, convex, increasing and satisfies
vol. 10, iss. 2, art. 37, 2009
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Φ(0) = 0 ,
lim Φ(s) = ∞ .
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s→∞
Moreover, a Young function Φ satisfies the following useful inequalities: for s ≥ 0,
we have
(
Φ(αs) < αΦ(s), if 0 ≤ α < 1
(2.1)
Φ(αs) ≥ αΦ(s), if α ≥ 1 .
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We call a Young function an N -function if it satisfies the limit conditions
Φ(s)
= ∞ and
lim
s→∞
s
Φ(s)
lim
= 0.
s→0
s
Let Φ be a Young function generated by the function φ, i.e.,
Z s
Φ(s) =
φ(t)dt .
0
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Then the function Ψ generated by the function ψ, i.e.,
Z s
Ψ(s) =
ψ(t)dt ,
0
where
ψ(s) = sup t
φ(t)≤s
is called the complementary function to Φ. It is known that Ψ is a Young function
and that Φ is complementary to Ψ. The pair of complementary Young functions Φ,
Ψ satisfies Young’s inequality
(2.2)
u · v ≤ Φ(u) + Ψ(v),
u, v ∈ [0, ∞).
Generalized Orlicz Spaces
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vol. 10, iss. 2, art. 37, 2009
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Equality in (2.2) holds if and only if
(2.3)
v = Φ(u) or
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u = Ψ(v).
A Young function Φ is said to satisfy the ∆2 -condition, written Φ ∈ ∆2 , if there
exist k > 0 and T ≥ 0 such that
Φ(2t) ≤ kΦ(t) for all t ≥ T .
The above mentioned concepts of the Young function, complementary Young
function and ∆2 -condition are quite standard and can be found in any standard book
on Orlicz spaces. Here we mention the celebrated monographs [5], [7].
The remainder of the concepts are some of the contents of [4] which were developed and studied there and we mention them here briefly.
Let X be a BFS and Φ denote a non-negative function on [0, ∞). The generalized
eΦ consists of all functions u ∈ L0 (Ω) such that
Orlicz class X
ρX (u, Φ) = kΦ(|u|)kX < ∞ .
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eΦ coincides algebraically with the space X p
For the case Φ(t) = tp , 0 < p < ∞, X
endowed with the quasi-norm
1
kukX p = k|u|p kXp .
Let X be a BFS and Φ, Ψ be a pair of complementary Young functions. The
generalized Orlicz space, denoted by XΦ , is the set of all u ∈ L0 (Ω) such that
(2.4)
kukΦ := sup k|u · v|kX ,
v
eΨ for which ρX (v; Ψ) ≤ 1.
where the supremum is taken over all v ∈ X
eΦ ⊂ XΦ and that XΦ is a BFS, with
It was proved that for a Young function Φ, X
the norm (2.4). Further, on the generalized Orlicz space XΦ , a Luxemburg type norm
was defined in the following way
|u|
0
(2.5)
kukΦ = inf k > 0 : ρX
,Φ ≤ 1 .
k
It was shown that with the norm (2.5) too, the space XΦ is a BFS and that the two
norms (2.4) and (2.5) are equivalent, i.e., there exists constants c1 , c2 > 0 such that
(2.6)
c1 kuk0Φ ≤ kukΦ ≤ c2 kuk0Φ .
In fact, it was proved that c2 = 2.
Generalized Orlicz Spaces
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vol. 10, iss. 2, art. 37, 2009
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3.
Comparison of Generalized Orlicz Spaces
We begin with the following definition:
Definition 3.1. A BFS is said to satisfy the L-property if for all non-negative functions f, g ∈ X, there exists a constant 0 < a < 1 such that
kf + gkX ≥ a(kf kX + kgkX ) .
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Remark 1. It was proved in [2] that the generalized Orlicz space XΦ contains the
eΦ . Towards the converse, we prove the following:
generalized Orlicz class X
vol. 10, iss. 2, art. 37, 2009
Theorem 3.2. Let Φ be a Young function, X be a BFS satisfying the L-property and
u
eΦ .
u ∈ XΦ be such that kukΦ 6= 0. Then kuk
∈X
Φ
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Proof. Let u ∈ XΦ . Using the modified arguments used in [7, Lemma 3.7.2], it can
be shown that
(
kukΦ ;
for ρX (v; Ψ) ≤ 1,
(3.1)
ku · vkX ≤
kukΦ ρX (v; Ψ) ; for ρX (v; Ψ) > 1 .
Let E ⊂ Ω be such that µ(E) < ∞. First assume that u ∈ XΦ (Ω) is bounded and
that u(x) = 0 for x ∈ Ω \ E. Put
1
v(x) = φ
|u(x)| .
kukΦ
1
The monotonicity of Φ and Ψ gives that the functions Φ kukΦ |u(x)| and Ψ(|v(x)|)
are also bounded. Consequently, property (P2) of X yields that Φ kuk1 Φ |u(x)| <
X
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∞ and kΨ(|v(x)|)kX < ∞ which by using (2.2) gives:
u·v |u|
kukΦ ≤ Φ kukΦ + Ψ(|v|)
X
X
|u| ≤
Φ kukΦ + kΨ(|v|)kX
X
< ∞.
On the other hand, using the L-property of X and (2.3), we get that for some a > 0
u·v |u|
kukΦ = Φ kukΦ + Ψ(|v|)
X
X
|u|
(3.2)
≥a Φ kukΦ + kΨ(|v|)kX .
X
Applying (3.1) for
u
, v, we find that
kukΦ
u·v max(ρX (v, Ψ), 1) ≥ kukΦ X
and therefore, by (3.2), we get that
|u| + kΨ(|v|)kX .
max(ρX (v, Ψ), 1) ≥ a Φ
kukΦ X
Now, if ρX (v, Ψ) > 1, then the above estimate gives
1
|u|
≤ ρX (v, Ψ)
Φ
−1
kukΦ X
a
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vol. 10, iss. 2, art. 37, 2009
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and if ρX (v, Ψ) ≤ 1, then
|u| Φ
+ ρX (v, Ψ) ≤ 1 .
kukΦ X
a
In any case
|u|
Φ
<∞
kukΦ X
and the assertion is proved for bounded u. For general u, we can follow the modified
idea of [10, Lemma 3.7.2].
Remark 2. In view of the above theorem, for u ∈ XΦ , there exists c > 0 such that
eΦ . In other words, the space XΦ is the linear hull of the generalized Orlicz
cu ∈ X
eΦ with the assumption on X that it satisfies the L-property.
class X
We prove the following useful result:
Proposition 3.3. Let Φ be a Young function satisfying the ∆2 -condition (with T = 0
eΦ .
if µ(Ω) = ∞) and X be a BFS satisfying the L-property. Then XΦ = X
Proof. Let u ∈ XΦ , kukΦ 6= 0. By Theorem 3.2, we have
w=
1
eΦ .
·u∈X
kukΦ
eΦ (Ω) is a linear set, we have
Since X
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vol. 10, iss. 2, art. 37, 2009
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eΦ ,
kukΦ · w = u ∈ X
i.e.,
eΦ .
XΦ ⊂ X
The reverse inclusion is obtained in view of Remark 1 and the assertion follows.
Let Φ1 and Φ2 be two Young functions. We write Φ2 ≺ Φ1 if there exists constants c > 0, T ≥ 0 such that
Φ2 (t) ≤ Φ1 (ct),
t≥T.
Now, we prove the following inclusion relation:
Theorem 3.4. Let X be a BFS satisfying the L-property and Φ1 , Φ2 be two Young
functions such that Φ2 ≺ Φ1 and µ(Ω) < ∞. Then the inclusion
Generalized Orlicz Spaces
XΦ1 ⊂ XΦ2
vol. 10, iss. 2, art. 37, 2009
Pankaj Jain and Priti Upreti
holds.
Proof. Since Φ2 ≺ Φ1 , there exists constants, c > 0, T ≥ 0 such that
(3.3)
Φ2 (t) ≤ Φ1 (ct),
cT
,
k
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t≥T.
e Φ1 ,
Let u ∈ XΦ1 . Then in view of Theorem 3.2, there exists k > 0 such that ku ∈ X
i.e., ρX (ku; Φ1 ) < ∞. Denote
cT
Ω1 = x ∈ Ω; |u(x)| <
.
k
Then for x ∈ Ω \ Ω1 , |u(x)| ≥
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i.e.,
k
|u(x)| ≥ T
c
so that the inequalities (3.3) with t replaced by kc |u(x)| gives
k
Φ2
|u(x)| ≤ Φ1 (k|u(x)|)
c
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which implies that
k
k
k
Φ2
|u(x)| =
Φ2
|u(x)| χΩ1 + Φ2
|u(x)| χΩ\Ω1 c
c
c
X
X k
+ Φ2 k |u(x)| χΩ\Ω1 ≤
Φ
|u(x)|
χ
2
Ω
1
c
c
X
X
≤ Φ2 (T )kχΩ1 kX + kΦ1 (k|u(x)|)χΩ\Ω1 kX
= Φ2 (T )kχΩ1 kX + ρX (ku; Φ1 )
< ∞.
eΦ2 ⊂ XΦ2 , i.e., k u ∈ XΦ2 . But since XΦ2 is in particular a
Consequently, kc u ∈ X
c
vector space we find that u ∈ XΦ2 and we are done.
The above theorem states that Φ2 ≺ Φ1 is a sufficient condition for the algebraic inclusion XΦ1 ⊂ XΦ2 . The next theorem proves that the condition, in fact, is
sufficient for the continuous imbedding XΦ1 ,→ XΦ2 .
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vol. 10, iss. 2, art. 37, 2009
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Theorem 3.5. Let X be a BFS satisfying the L-property and Φ1 , Φ2 be two Young
functions such that Φ2 ≺ Φ1 and µ(Ω) < ∞. Then the inequality
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kukΦ2 ≤ kkukΦ1
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holds for some constant k > 0 and for all u ∈ XΦ1 .
Proof. Let Ψ1 and Ψ2 be the complementary functions respectively to Φ1 and Φ2 .
Then Φ2 ≺ Φ1 implies that Ψ1 ≺ Ψ2 , i.e., there exists constants c1 , T1 > 0 such that
Ψ1 (t) ≤ Ψ2 (c1 t) for t ≥ T1
or equivalently
Ψ1
t
c1
≤ Ψ2 (t) for t ≥ c1 T1 .
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Further, if t ≤ c1 T1 , then the monotonicity of Ψ gives
t
Ψ1
≤ Ψ1 (T1 ).
c1
The last two estimates give that for all t > 0
t
(3.4)
Ψ1
≤ Ψ1 (T1 ) + Ψ2 (t) .
c1
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By the property (P4) of BFS, kχΩ kX < ∞. Denote α = (Ψ1 (T1 )kχΩ kX + 1)−1 and
k = cα1 . Clearly 0 < α < 1. We know that for a Young function Φ and 0 < β < 1,
(3.5)
Φ(βt) ≤ βΦ(t),
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t > 0.
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eΨ2 be such that ρX (v; Ψ2 ) ≤ 1. Then, using (3.5) for β = α and
Now, let v ∈ X
|v(x)|
t = c1 and (3.4), we obtain that
v
α|v(x)| ρX
; Ψ1 = Ψ1
k
c1
X
|v(x)| ≤ α
Ψ1
c1
X
≤ αkΨ1 (T1 ) + Ψ2 (|v(x)|)kX
≤ α(Ψ1 (T1 )kχΩ kX + kψ2 (|v(x)|)kX )
= α(Ψ1 (T1 )kχΩ kX + ρX (v; Ψ2 ))
≤ αα−1 = 1 .
Thus we have shown that ρX (v; Ψ2 ) ≤ 1 implies ρX
v
; Ψ1
k
≤ 1 and consequently
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using the definition of the generalized Orlicz norm, we obtain
kukΦ2 =
k(|u(x)v(x)|)kX
v(x)
= k sup u(x) k ρ(v;Ψ2 )≤1
X
u(x) v(x) ≤ k sup k X
ρ( kv ;Ψ1 )≤1
sup
ρ(v;Ψ2 )≤1
=k
sup
k|u(x)w(x)|kX
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vol. 10, iss. 2, art. 37, 2009
ρ(w;Ψ1 )≤1
= k · kukΦ1
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and the assertion is proved.
Remark 3. If Φ1 and Φ2 are equivalent Young functions (i.e., Φ1 ≺ Φ2 and Φ2 ≺ Φ1 )
then the norms k·kΦ1 and k·kΦ2 are equivalent.
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4.
Convergence
Following the concepts in the Orlicz space LΦ (Ω), we introduce the following definitions.
Definition 4.1. A sequence {un } of functions in XΦ is said to converge to u ∈ XΦ ,
written un → u, if
lim kun − ukΦ = 0 .
n→∞
Pankaj Jain and Priti Upreti
Definition 4.2. A sequence {un } of functions in XΦ is said to converge in Φ-mean
to u ∈ XΦ if
lim ρX (un − u; Φ) = lim kΦ(|un − u|)kX = 0 .
n→∞
Generalized Orlicz Spaces
n→∞
We proceed to prove that the two convergences above are equivalent. In the sequel, the following remark will be used.
Remark 4. Let Φ and Ψ be a pair of complementary Young functions. Then in view
eΦ , v ∈ X
eΨ
of Young’s inequality (2.2), we obtain for u ∈ X
k|uv|kX ≤ kΦ(|u|)kX + kΨ(|v|)kX
= ρX (u; Φ) + ρX (v; Ψ)
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so that
kukΦ ≤ ρX (u; Φ) + 1 .
Now, we prove the following:
Lemma 4.3. Let Φ be a Young function satisfying the ∆2 -condition (with T = 0 if
µ(Ω) = ∞) and r be the number given by
(
2
if µ(Ω) = ∞,
(4.1)
r=
Φ(T )kχΩ kX + 2 if µ(Ω) < ∞ .
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If there exists an m ∈ N such that
ρX (u; Φ) ≤ k −m ,
(4.2)
where k is the constant in the ∆2 -condition, then
kukΦ ≤ 2−m r .
Proof. Let m ∈ N be fixed. Consider first the case when µ(Ω) < ∞ and denote
Ω1 = {x ∈ Ω : 2m |u(x)| ≤ T } .
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vol. 10, iss. 2, art. 37, 2009
Then for x ∈ Ω1 , we get
(4.3)
Φ(2m |u(x)|) ≤ Φ(T )
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and for x ∈ Ω \ Ω1 , by repeated applications of the ∆2 -condition, we obtain
(4.4)
Φ(2m |u(x)|) ≤ k m Φ(|u(x)|) .
Consequently, we have using (4.3) and (4.4)
kΦ(2m |u(x)|)kX = kΦ(2m (|u(x)|))χΩ1 + Φ(2m |u(x)|)χΩ\Ω1 kX
≤ kΦ(2m (|u(x)|))χΩ1 kX + kΦ(2m |u(x)|)χΩ\Ω1 kX
≤ Φ(T )kχΩ1 kX + k m kΦ(|u(x)|)kX
≤ Φ(T )kχΩ kX + k m ρX (u; Φ)
≤ Φ(T )kχΩ kX + 1
= r − 1.
In the case µ(Ω) = ∞ we take Ω1 = φ and then (4.4) directly gives
kΦ(2m |u(x)|)kX ≤ 1 ≤ r − 1
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since r = 2 for µ(Ω) = ∞. Thus in both cases we have
kΦ(2m |u(x)|)kX ≤ r − 1
which further, in view of Remark 4 gives
k2m u(x)kΦ ≤ r
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or
kukΦ ≤ 2−m r
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vol. 10, iss. 2, art. 37, 2009
and we are done.
Let us recall the following result from [4]:
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Lemma 4.4. Let u ∈ XΦ . Then
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ρX (u; Φ) ≤ kuk0Φ
if kuk0Φ ≤ 1
and
ρX (u; Φ) ≥
kuk0Φ
if
kuk0Φ
> 1,
where kuk0Φ denotes the Luxemburg type norm on the space XΦ given by (2.5).
Now, we are ready to prove the equivalence of the two convergence concepts
defined earlier in this section.
Theorem 4.5. Let Φ be a Young function satisfying the ∆2 -condition. Let {un } be
a sequence of functions in XΦ . Then un converges to u in XΦ if and only if un
converges in Φ-mean to u in XΦ .
Proof. First assume that un converges in Φ-mean to u. We shall now prove that
un → u. Given ε > 0, we can choose m ∈ N such that ε > 2−m r, where r is as
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given by (4.1). Now, since un converges in Φ-mean to u, for this m, we can find an
M such that
ρX (un − u; Φ) ≤ k −m for n ≥ M
which by Lemma 4.3 implies that
kun − ukΦ ≤ 2−m r < ε for n ≥ M
and we get that un → u.
Conversely, first note that the two norms k·kΦ and k·k0Φ on the space XΦ are
equivalent and assume, in particular, that the constants of equivalence are c1 , c2 , i.e.,
(2.6) holds.
Now, let un , u ∈ XΦ so that
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vol. 10, iss. 2, art. 37, 2009
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kun − ukΦ ≤ c1 .
Then (2.6) gives
kun −
uk0Φ
≤1
which, in view of Lemma 4.4 and again (2.6), gives that
uk0Φ
ρX (un − u; Φ) ≤ kun −
1
≤ kun − ukΦ .
c1
The Φ-mean convergence now, immediately follows from the convergence in XΦ .
Remark 5. The fact that the Φ-mean convergence implies norm convergence does
not require the use of ∆2 -conditions. It is required only in the reverse implication.
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5.
Separability
Remark 6. It is known, e.g., see [7, Theorem 3.13.1], that the Orlicz space LΦ (Ω)
is separable if Φ satisfies the ∆2 -condition (with T = 0 if µ(Ω) = 0). In order to
obtain the separability conditions for the generalized Orlicz space XΦ , we can depict
the same proof with obvious modifications except at a point where the Lebesgue
dominated convergence theorem has been used.
Generalized Orlicz Spaces
In the framework of general BFS, the following version of the Lebesgue dominated convergence theorem is known, see e.g. [1, Proposition 3.6].
Definition 5.1. A function f in a Banach function space X is said to have an absolutely continuous norm in X if kf χEn kX → 0 for every sequence {En }∞
n=1 satisfying
En → φ µ-a.e.
Proposition A. A function f in a Banach function space X has an absolutely continuous norm iff the following condition holds; whenever fn {n = 1, 2, . . .} and g
are µ-measurable functions satisfying |fn | ≤ |f | for all n and fn → g µ-a.e., then
kfn − gkX → 0.
Now, in view of Remark 6 and Proposition A we have the following result.
Theorem 5.2. Let X be a BFS having an absolutely continuous norm and Φ be a
Young function satisfying the ∆2 -condition (with T = 0 if µ(Ω) = 0). Then the
generalized Orlicz space XΦ is separable.
Pankaj Jain and Priti Upreti
vol. 10, iss. 2, art. 37, 2009
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References
[1] C. BENNETT AND R. SHARPLEY, Interpolation of Operators, Academic
Press, London, 1988.
[2] P. JAIN, B. GUPTA AND D. VERMA, Mean inequalities in certain Banach
function spaces, J. Math. Anal. Appl., 334(1) (2007), 358–367.
[3] P. JAIN, B. GUPTA AND D. VERMA, Hardy inequalities in certain Banach
function spaces, submitted.
[4] P. JAIN, L.E. PERSSON AND P. UPRETI, Inequalities and properties of some
generalized Orlicz classes and spaces, Acta Math. Hungar., 117(1-2) (2007),
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[5] M.A. KRASNOSEL’SKII AND J.B. RUTICKII, Convex Functions and Orlicz
Spaces, Noordhoff Ltd. (Groningen, 1961).
[6] A.KUFNER, L.MALIGRANDA AND L.E.PERSSON, The Hardy Inequailty About its History and Some Related Results, (Pilsen, 2007).
[7] A. KUFNER, J. OLDRICH AND F. SVATOPLUK, Function Spaces, Noordhoff
Internatonal Publishing (Leydon, 1977).
[8] A. KUFNER AND L.E. PERSSON, Weighted Inequalities of Hardy Type, World
Scientific, 2003.
[9] W.A.J. LUXEMBURG, Banach Function Spaces, Ph.D. Thesis, Technische
Hogeschoo te Delft (1955).
[10] L. MALIGRANDA AND L.E. PERSSON, Generalized duality of some Banach
fucntion spaces, Proc. Konin Nederlands, Akad. Wet., 92 (1989), 323–338.
Generalized Orlicz Spaces
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[11] L.E. PERSSON, Some elementary inequalities in connection with X p -spaces,
In: Constructive Theory of Functions, (1987), 367–376.
[12] L.E. PERSSON, On some generalized Orlicz classes and spaces, Research
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(1988).
[13] M.M. RAO AND Z.D. REN, Theory of Orlicz spaces, Marcel Dekker Inc. (New
York, Basel, Hong Kong, 1991).
Generalized Orlicz Spaces
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