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Open AccessFeature PaperArticle

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

by 1,*,† and 2,*,†
1
Department of Mathematics, College of Sciences, King Saud University, Riyadh 11451, Saudi Arabia
2
Department of Mathematical Sciences, University of Texas at El Paso, 500W University Ave. 124 Bell Hall, El Paso, TX 79968, USA
*
Authors to whom correspondence should be addressed.
These authors contributed equally to this work.
Mathematics 2020, 8(6), 870; https://doi.org/10.3390/math8060870
Received: 23 April 2020 / Revised: 20 May 2020 / Accepted: 22 May 2020 / Published: 28 May 2020
(This article belongs to the Special Issue New Trends in Analysis and Geometry)
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. View Full-Text
Keywords: modular uniform convexity; modular vector spaces; uniform convexity; variable exponent spaces modular uniform convexity; modular vector spaces; uniform convexity; variable exponent spaces
MDPI and ACS Style

Bachar, M.; Méndez, O. Modular Uniform Convexity in Every Direction in Lp(·) and Its Applications. Mathematics 2020, 8, 870. https://doi.org/10.3390/math8060870

AMA Style

Bachar M, Méndez O. Modular Uniform Convexity in Every Direction in Lp(·) and Its Applications. Mathematics. 2020; 8(6):870. https://doi.org/10.3390/math8060870

Chicago/Turabian Style

Bachar, Mostafa; Méndez, Osvaldo. 2020. "Modular Uniform Convexity in Every Direction in Lp(·) and Its Applications" Mathematics 8, no. 6: 870. https://doi.org/10.3390/math8060870

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