Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability

Modular Uniform Convexity of Lebesgue Spaces of Variable Integrability

We analyze the modular geometry of the Lebesgue space with variable exponent, <inline-formula> <math display="inline"> <semantics> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>(</mo> <mo>·</mo> <mo>)<...

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Bibliographic Details
Main Authors:	Mostafa Bachar, Osvaldo Mendez, Messaoud Bounkhel
Format:	Article
Language:	English
Published:	MDPI AG 2018-12-01
Series:	Symmetry
Subjects:	fixed point theorem modular uniform convexity modular vector spaces Nakano spaces uniform convexity variable exponent spaces
Online Access:	https://www.mdpi.com/2073-8994/10/12/708

Description
Summary:	We analyze the modular geometry of the Lebesgue space with variable exponent, <inline-formula> <math display="inline"> <semantics> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>(</mo> <mo>·</mo> <mo>)</mo> </mrow> </msup> </semantics> </math> </inline-formula>. Our central result is that <inline-formula> <math display="inline"> <semantics> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo>(</mo> <mo>·</mo> <mo>)</mo> </mrow> </msup> </semantics> </math> </inline-formula> possesses a modular uniform convexity property. Part of the novelty is that the property holds even in the case <inline-formula> <math display="inline"> <semantics> <mrow> <munder> <mo movablelimits="false" form="prefix">sup</mo> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="sans-serif">Ω</mi> </mrow> </munder> <mi>p</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mo>∞</mo> </mrow> </semantics> </math> </inline-formula>. We present specific applications to fixed point theory.
ISSN:	2073-8994