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

Mostafa Bachar and Osvaldo Méndez
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Mostafa Bachar: Department of Mathematics, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia
Osvaldo Méndez: Department of Mathematical Sciences, University of Texas at El Paso, 500W University Ave. 124 Bell Hall, El Paso, TX 79968, USA

Mathematics, 2020, vol. 8, issue 6, 1-12

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.

Keywords: modular uniform convexity; modular vector spaces; uniform convexity; variable exponent spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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