A GENERALIZATION OF THE MALIGRANDA - ORLICZ LEMMA

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Generalization of the Maligranda - Orlicz Lemma René Erlín Castillo and

Eduard Trousselot vol. 8, iss. 4, art. 115, 2007

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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

Received: 23 August, 2007

Accepted: 3 December, 2007 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 46J10.

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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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 func- tions, which is closed under pointwise multiplication of functions. Let us assume that f ·g ∈X and

(1.1) kf gk ≤ kfk_∞· kgk+kfk · kgk_∞ for any f, g∈X.Then the spaceX equipped with the norm

kfk₁ =kfk∞+kfk

is a normed Banach algebra. Also, ifX ,→B[a, b],then the normsk·k₁ andk·kare equivalent. Moreover, if kfk∞ ≤ Mkfk for f ∈ X, then(X,k·k₂)is a normed Banach algebra withkfk₂ = 2Mkfk, 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 byLip[a, b]) equipped with the norm

k·k_Lip[a,b]=|f(a)|+ Lip(f) f ∈Lip[a, b], where Lip(f) = sup_x6=y

f(x)−f(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 Lemma1.1. Indeed,

f g(x)−f g(y) x−y

≤ |f(x)|

g(x)−g(y) x−y

(1.2)

+|g(y)|

f(x)−f(y) x−y

, x6=y

≤ kfk∞Lip(g) +kgk∞Lip(f), sincekf gk_Lip_[a,b] =|f g(a)|+ Lip(f g).

By (1.2) we have

kf gk_Lip_[a,b] ≤2|f(a)||g(a)|+kfk∞Lip(g) (1.3)

+kgk_∞Lip(f)

≤ kfk∞|g(a)|+|f(a)|+|f(a)|kgk∞

+kfk∞Lip(g) +kgk∞Lip(f).

Thus

kf gk_Lip[a,b]≤ kfk∞kgk_Lip_[a,b] +kgk∞kgk_Lip[a,b]. On the other hand, since BV[a, b],→B[a, b] it is not hard to see that (1.4) kfk∞≤max{1, b−a}kfkLip_[a,b].

Then by (1.3) and (1.4) we can invoke Lemma1.1 to conclude that Lip[a, b] is a Banach algebra either with the norm

k·k₁ =k·k_∞+k·k_Lip

[a,b]

or

k·k₂ = 2 max{1, b−a} k·k_Lip

[a,b]

which are equivalent to the norm k·k_Lip

[a,b].

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2. Main Result

Theorem 2.1. Let (X,k·k) be a Banach space whose elements are bounded func- tions, which is closed under pointwise multiplication of functions. Let us assume that f ·g ∈X such that

kf gk ≤ kfk∞kgk+kfkkgk∞+Kkfkkgk, K >0.

Then (X,k·k₁) equipped with the norm

kfk₁ =kfk∞+Kkfk, 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 ≤ kfk₁kgk₁ for all f, g ∈ X. In fact,

kf gk₁ =kf gk_∞+Kkf gk

≤ kfk∞kgk∞+Kkfk∞kgk +Kkfkkgk∞+K²kfk·kgk

= (kfk∞+Kkfk)(kgk∞+Kkgk)

=kfk₁kgk₁.

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_∞ ≤Lk·k.

Thus

Kk·k ≤ k·k_∞+Kk·k=k·k₁

≤Lk·k+Kk·k= (L+K)k·k.

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Hence

Kk·k ≤ k·k₁ ≤(L+K)k·k. This completes the proof of Theorem2.1.

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References

[1] L. MALIGRANDAANDW. ORLICZ, On some properties of functions of gen- eralized variation, Monastsh Math., 104 (1987), 53–65.