Volume 8 (2007), Issue 4, Article 115, 3 pp.
A GENERALIZATION OF THE MALIGRANDA - ORLICZ LEMMA
RENÉ ERLÍN CASTILLO AND EDUARD TROUSSELOT
D EPARTAMENTO DE M ATEMÁTICAS
U NIVERSIDAD DE O RIENTE
6101 C UMANÁ , E DO . S UCRE , V ENEZUELA
rcastill@math.ohiou.edu
eddycharles2007@hotmail.com
Received 23 August, 2007; accepted 3 December, 2007
Communicated by S.S. Dragomir
A BSTRACT. 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.
Key words and phrases: Banach algebra, Maligranda-Orlicz.
2000 Mathematics Subject Classification. 46J10.
1. I NTRODUCTION
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
kf k1 = kf k∞ + kf k
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.
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R ENÉ E RLÍN C ASTILLO AND E DUARD T ROUSSELOT
Next, we claim that Lip[a, b] is a Banach algebra. To see this , we just need to check (1.1)
from Lemma 1.1. Indeed,
g(x) − g(y) f g(x) − f 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)
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
(1.4)
kf k∞ ≤ max{1, b − a}kf kLip[a,b] .
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
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] .
2. M AIN R ESULT
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
+ Kkf kkgk∞ + K 2 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.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 115, 3 pp.
http://jipam.vu.edu.au/
G ENERALIZATION OF THE M ALIGRANDA - O RLICZ L EMMA
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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 .
Hence
K k·k ≤ k·k1 ≤ (L + K) k·k .
This completes the proof of Theorem 2.1.
R EFERENCES
[1] L. MALIGRANDA AND W. ORLICZ, On some properties of functions of generalized variation,
Monastsh Math., 104 (1987), 53–65.
J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 115, 3 pp.
http://jipam.vu.edu.au/