Volume 10 (2009), Issue 2, Article 37, 10 pp.
CERTAIN PROPERTIES OF GENERALIZED ORLICZ SPACES
PANKAJ JAIN AND PRITI UPRETI
D EPARTMENT OF M ATHEMATICS
D ESHBANDHU C OLLEGE (U NIVERSITY OF D ELHI )
K ALKAJI , N EW D ELHI - 110 019
I NDIA
pankajkrjain@hotmail.com
D EPARTMENT OF M ATHEMATICS
M OTI L AL N EHRU C OLLEGE (U NIVERSITY OF D ELHI )
B ENITO J UAREZ M ARG , D ELHI 110 021
I NDIA
Received 20 March, 2008; accepted 09 October, 2008
Communicated by L.-E. Persson
A BSTRACT. In the context of generalized Orlicz spaces XΦ , the concepts of inclusion, convergence and separability are studied.
Key words and phrases: Banach Function spaces, generalized Orlicz class, generalized Orlicz space, Luxemburg norm, Young
function, Young’s inequality, imbedding, convergence, separability.
2000 Mathematics Subject Classification. 26D10, 26D15, 46E35.
1. I NTRODUCTION
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.
The research of the first author is partially supported by CSIR (India) through the grant no. 25(5913)/NS/03/EMRII..
087-08
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PANKAJ JAIN AND P RITI U PRETI
2. P RELIMINARIES
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.;
P2. 0 ≤ u ≤ v a.e. ⇒ kukX ≤ kvkX ;
P3. 0 < un ↑ u a.e. ⇒ kukX ↑ kukX ;
P4. µ(E) < ∞ ⇒ kχ
R E kX < ∞;
P5. µ(E) < ∞ ⇒ E u(x)dx ≤ CE kukX ,
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 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 ,
0
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
Φ(0) = 0 , lim Φ(s) = ∞ .
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 .
We call a Young function an N -function if it satisfies the limit conditions
Φ(s)
=∞
s→∞
s
lim
and
Φ(s)
= 0.
s→0
s
lim
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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G ENERALIZED O RLICZ S PACES
3
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, ∞).
Equality in (2.2) holds if and only if
v = Φ(u) or
(2.3)
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 Orlicz
eΦ consists of all functions u ∈ L0 (Ω) such that
class X
ρX (u, Φ) = kΦ(|u|)kX < ∞ .
eΦ coincides algebraically with the space X p endowed
For the case Φ(t) = tp , 0 < p < ∞, X
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 the norm
It was proved that for a Young function Φ, X
(2.4). Further, on the generalized Orlicz space XΦ , a Luxemburg type norm was defined in the
following way
|u|
0
,Φ ≤ 1 .
(2.5)
kukΦ = inf k > 0 : ρX
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.
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PANKAJ JAIN AND P RITI U PRETI
3. C OMPARISON
OF
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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 ) .
Remark 1. It was proved in [2] that the generalized Orlicz space XΦ contains the generalized
eΦ . Towards the converse, we prove the following:
Orlicz class X
Theorem 3.1. Let Φ be a Young function, X be a BFS satisfying the L-property and u ∈ XΦ
u
eΦ .
be such that kukΦ 6= 0. Then kuk
∈X
Φ
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
The monotonicity of Φ and Ψ gives that the functions Φ kuk1 Φ |u(x)| and Ψ(|v(x)|) are also
1
bounded. Consequently, property (P2) of X yields that Φ kukΦ |u(x)| < ∞ and kΨ(|v(x)|)kX <
X
∞ which by using (2.2) gives:
u·v |u|
≤ Φ
+ Ψ(|v|)
kukΦ kukΦ
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|
≥a (3.2)
Φ kukΦ + kΨ(|v|)kX .
X
u
Applying (3.1) for
, v, we find that
kukΦ
u·v max(ρX (v, Ψ), 1) ≥ kukΦ X
and therefore, by (3.2), we get that
|u| max(ρX (v, Ψ), 1) ≥ a Φ
+ kΨ(|v|)kX .
kukΦ X
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G ENERALIZED O RLICZ S PACES
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Now, if ρX (v, Ψ) > 1, then the above estimate gives
|u| ≤ ρX (v, Ψ) 1 − 1
Φ
kukΦ a
X
and if ρX (v, Ψ) ≤ 1, then
|u| + ρX (v, Ψ) ≤ 1 .
Φ
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].
eΦ .
Remark 2. In view of the above theorem, for u ∈ XΦ , there exists c > 0 such that cu ∈ X
eΦ with the
In other words, the space XΦ is the linear hull of the generalized Orlicz class X
assumption on X that it satisfies the L-property.
We prove the following useful result:
Proposition 3.2. Let Φ be a Young function satisfying the ∆2 -condition (with T = 0 if µ(Ω) =
eΦ .
∞) and X be a BFS satisfying the L-property. Then XΦ = X
Proof. Let u ∈ XΦ , kukΦ 6= 0. By Theorem 3.1, we have
1
eΦ .
w=
·u∈X
kukΦ
eΦ (Ω) is a linear set, we have
Since X
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 3.3. 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
XΦ1 ⊂ XΦ2
holds.
Proof. Since Φ2 ≺ Φ1 , there exists constants, c > 0, T ≥ 0 such that
(3.3)
Φ2 (t) ≤ Φ1 (ct),
t≥T.
eΦ1 , i.e.,
Let u ∈ XΦ1 . Then in view of Theorem 3.1, there exists k > 0 such that ku ∈ X
ρX (ku; Φ1 ) < ∞. Denote
cT
Ω1 = x ∈ Ω; |u(x)| <
.
k
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PANKAJ JAIN AND P RITI U PRETI
Then for x ∈ Ω \ Ω1 , |u(x)| ≥
cT
,
k
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
which implies that
Φ2 k |u(x)| = Φ2 k |u(x)| χΩ1 + Φ2 k |u(x)| χΩ\Ω1 c
c
c
X
X k
k
≤
Φ2 c |u(x)| χΩ1 + Φ2 c |u(x)| χΩ\Ω1 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 vector space
Consequently, kc u ∈ X
c
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 .
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 inequality
kukΦ2 ≤ kkukΦ1
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
t
Ψ1
≤ Ψ2 (t) for t ≥ c1 T1 .
c1
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
By the property (P4) of BFS, kχΩ kX < ∞. Denote α = (Ψ1 (T1 )kχΩ kX + 1)−1 and k =
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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c1
.
α
t > 0.
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G ENERALIZED O RLICZ S PACES
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eΨ2 be such that ρX (v; Ψ2 ) ≤ 1. Then, using (3.5) for β = α and t =
Now, let v ∈ X
(3.4), we obtain that
v
α|v(x)|
ρX
; Ψ1 = Ψ
1
k
c1
X
|v(x)| ≤ α
Ψ
1
c1
X
≤ αkΨ1 (T1 ) + Ψ2 (|v(x)|)kX
|v(x)|
c1
and
≤ 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
definition of the generalized Orlicz norm, we obtain
kukΦ2 =
v
; Ψ1
k
≤ 1 and consequently using the
k(|u(x)v(x)|)kX
sup
ρ(v;Ψ2 )≤1
v(x)
u(x)
= k sup k X
ρ(v;Ψ2 )≤1
v(x)
u(x)
≤ k sup k
v
X
ρ( k ;Ψ1 )≤1
= k sup k|u(x)w(x)|kX
ρ(w;Ψ1 )≤1
= k · kukΦ1
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.
4. C ONVERGENCE
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→∞
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→∞
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 of Young’s
eΦ , v ∈ X
eΨ
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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PANKAJ JAIN AND P RITI U PRETI
so that
kukΦ ≤ ρX (u; Φ) + 1 .
Now, we prove the following:
Lemma 4.1. 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) < ∞ .
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 } .
Then for x ∈ Ω1 , we get
Φ(2m |u(x)|) ≤ Φ(T )
(4.3)
and for x ∈ Ω \ Ω1 , by repeated applications of the ∆2 -condition, we obtain
Φ(2m |u(x)|) ≤ k m Φ(|u(x)|) .
(4.4)
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
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
or
kukΦ ≤ 2−m r
and we are done.
Let us recall the following result from [4]:
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G ENERALIZED O RLICZ S PACES
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Lemma 4.2. Let u ∈ XΦ . Then
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 4.3. 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 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 4.1 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
kun − ukΦ ≤ c1 .
Then (2.6) gives
kun − uk0Φ ≤ 1
which, in view of Lemma 4.2 and again (2.6), gives that
ρX (un − u; Φ) ≤ kun − uk0Φ
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.
5. S EPARABILITY
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.
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.
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PANKAJ JAIN AND P RITI U PRETI
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 5.1. 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.
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