A GENERALIZATION OF THE MALIGRANDA ORLICZ LEMMA
RENÉ ERLÍN CASTILLO AND EDUARD TROUSSELOT
Departamento de Matemáticas
Universidad de Oriente
6101 Cumaná, Edo. Sucre, Venezuela
EMail: rcastill@math.ohiou.edu
eddycharles2007@hotmail.com
Generalization of the Maligranda Orlicz Lemma
René Erlín Castillo and
Eduard Trousselot
vol. 8, iss. 4, art. 115, 2007
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Received:
23 August, 2007
Accepted:
3 December, 2007
Communicated by:
JJ
II
S.S. Dragomir
J
I
2000 AMS Sub. Class.:
46J10.
Page 1 of 7
Key words:
Banach algebra, Maligranda-Orlicz.
Abstract:
In their 1987 paper, L. Maligranda and W. Orlicz gave a lemma which supplies
a test to check that some function spaces are Banach algebras. In this paper we
give a more general version of the Maligranda - Orlicz lemma.
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Contents
1
Introduction
3
2
Main Result
5
Generalization of the Maligranda Orlicz Lemma
René Erlín Castillo and
Eduard Trousselot
vol. 8, iss. 4, art. 115, 2007
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1.
Introduction
The following lemma is due to L. Maligranda and W. Orlicz (see [1]).
Lemma 1.1. Let (X, k·k) be a Banach space whose elements are bounded functions, which is closed under pointwise multiplication of functions. Let us assume
that f · g ∈ X and
(1.1)
kf gk ≤ kf k∞ · kgk + kf k · kgk∞
for any f, g ∈ X. Then the space X equipped with the norm
Generalization of the Maligranda Orlicz Lemma
René Erlín Castillo and
Eduard Trousselot
vol. 8, iss. 4, art. 115, 2007
kf k1 = kf k∞ + kf k
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is a normed Banach algebra. Also, if X ,→ B[a, b], then the norms k·k1 and k·k are
equivalent. Moreover, if kf k∞ ≤ M kf k for f ∈ X, then (X, k·k2 ) is a normed
Banach algebra with kf k2 = 2M kf k, f ∈ X and the norms k·k2 and k·k are
equivalent.
At least one easy example might be enlightening here. Recall that the Lipschitz
function space (denoted by Lip[a, b]) equipped with the norm
k·kLip[a,b] = |f (a)| + Lip(f ) f ∈ Lip[a, b],
f (x)−f (y) where Lip(f ) = supx6=y x−y , is a Banach space, which is closed under the
usual pointwise multiplication.
Next, we claim that Lip[a, b] is a Banach algebra. To see this , we just need to
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check (1.1) from Lemma 1.1. Indeed,
f g(x) − f g(y) g(x) − g(y) (1.2)
≤ |f (x)| x − y x−y
f (x) − f (y) , x 6= y
+ |g(y)| x−y ≤ kf k∞ Lip(g) + kgk∞ Lip(f ),
since kf gkLip[a,b] = |f g(a)| + Lip(f g).
By (1.2) we have
(1.3)
Generalization of the Maligranda Orlicz Lemma
René Erlín Castillo and
Eduard Trousselot
vol. 8, iss. 4, art. 115, 2007
kf gkLip[a,b] ≤ 2|f (a)||g(a)| + kf k∞ Lip(g)
+ kgk∞ Lip(f )
≤ kf k∞ |g(a)| + |f (a)| + |f (a)|kgk∞
+ kf k∞ Lip(g) + kgk∞ Lip(f ).
Thus
kf gkLip[a,b] ≤ kf k∞ kgkLip[a,b] + kgk∞ kgkLip[a,b] .
On the other hand, since BV [a, b] ,→ B[a, b] it is not hard to see that
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kf k∞ ≤ max{1, b − a}kf kLip[a,b] .
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Then by (1.3) and (1.4) we can invoke Lemma 1.1 to conclude that Lip[a, b] is
a Banach algebra either with the norm
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(1.4)
k·k1 = k·k∞ + k·kLip[a,b]
or
k·k2 = 2 max{1, b − a} k·kLip[a,b]
which are equivalent to the norm k·kLip[a,b] .
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2.
Main Result
Theorem 2.1. Let (X, k·k) be a Banach space whose elements are bounded functions, which is closed under pointwise multiplication of functions. Let us assume
that f · g ∈ X such that
kf gk ≤ kf k∞ kgk + kf kkgk∞ + Kkf kkgk,
K > 0.
Then (X, k·k1 ) equipped with the norm
kf k1 = kf k∞ + Kkf k,
f ∈ X,
is a Banach algebra. If X ,→ B[a, b], then k·k1 and k·k are equivalent.
Proof. First of all, we need to show that kf gk1 ≤ kf k1 kgk1 for all f, g ∈ X. In
fact,
kf gk1 = kf gk∞ + Kkf gk
≤ kf k∞ kgk∞ + Kkf k∞ kgk
2
+ Kkf kkgk∞ + K kf k·k gk
= (kf k∞ + Kkf k)(kgk∞ + Kkgk)
= kf k1 kgk1 .
This tells us that (X, k·k) is a Banach algebra. It only remains to show that k·k1
and k·k are equivalent norms.
Indeed, since X ,→ B[a, b], there exists a constant L > 0 such that
k·k∞ ≤ L k·k .
Thus
K k·k ≤ k·k∞ + K k·k = k·k1
≤ L k·k + K k·k = (L + K) k·k .
Generalization of the Maligranda Orlicz Lemma
René Erlín Castillo and
Eduard Trousselot
vol. 8, iss. 4, art. 115, 2007
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Hence
K k·k ≤ k·k1 ≤ (L + K) k·k .
This completes the proof of Theorem 2.1.
Generalization of the Maligranda Orlicz Lemma
René Erlín Castillo and
Eduard Trousselot
vol. 8, iss. 4, art. 115, 2007
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References
[1] L. MALIGRANDA AND W. ORLICZ, On some properties of functions of generalized variation, Monastsh Math., 104 (1987), 53–65.
Generalization of the Maligranda Orlicz Lemma
René Erlín Castillo and
Eduard Trousselot
vol. 8, iss. 4, art. 115, 2007
Title Page
Contents
JJ
II
J
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