Modular Uniform Convexity in Every Direction in Lp(·) and Its Applications

Mostafa Bachar,Osvaldo Méndez

doi:10.3390/math8060870

Mostafa Bachar, Osvaldo Méndez

Open Access

https://doi.org/10.3390/math8060870

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Abstract

We prove that the Lebesgue space of variable exponent L p ( · ) ( Ω ) is modularly uniformly convex in every direction provided the exponent p is finite a.e. and different from 1 a.e. The notion of uniform convexity in every direction was first introduced by Garkavi for the case of a norm. The contribution made in this work lies in the discovery of a modular, uniform-convexity-like structure of L p ( · ) ( Ω ) , which holds even when the behavior of the exponent p ( · ) precludes uniform convexity of the Luxembourg norm. Specifically, we show that the modular ρ ( u ) = ∫ Ω | u ( x ) | d x possesses a uniform-convexity-like structure even if the variable exponent is not bounded away from 1 or ∞. Our result is new and we present an application to fixed point theory.

Highlights

The most remote origins of the notion of variable exponent Lebesgue spaces L p(·) can be traced back to [1], where a particular case was introduced as a generalization of the variable exponent sequence spaces
The applicability of the variable exponent spaces was already demonstrated by their role the description of the hydrodynamical behavior of non-Newtonian fluids; see for example [3,4,5] and the model of image restoration [6,7] for more details
It follows by definition that the modular ρ coincides with the L1 norm on any function supported in Ω1 and with the L∞ norm on any function supported on Ω∞

Summary

Introduction

The most remote origins of the notion of variable exponent Lebesgue spaces L p(·) can be traced back to [1], where a particular case was introduced as a generalization of the variable exponent sequence spaces. In view of the result in [8], the condition p− = 1 precludes the uniform convexity for the Luxemburg norm, even if p( x ) > 1 everywhere. In the present work we address the preceding two cases and prove that the modular ρ is uniformly convex in every direction (Definition 1) under the assumptions that p( x ) > 1 almost everywhere and that |Ω∞ | = 0. The novelty here is that the discussed uniform convexity of ρ holds even in the limit-point cases p− = 1 or p+ = ∞. This notion was first introduced by Garkavi [10,11] for the case of a norm. As an application of the geometric property of ρ alluded to above, we present a fixed point result for mappings which are nonexpansive in the modular sense

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Application

Conclusions