A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces

Lütfi Akın

doi:10.3390/fractalfract5010007

Abstract

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-α integral operator Ma,δc to the norm of the centered fractional maximal diamond-α integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales.

Highlights

Our main purpose in this study is to examine the equivalence of the norms of fractional integral operators
For more than a quarter century, the concept of time scales has taken an important place in the literature
Mathematicians and scientists working in other disciplines have demonstrated many applications of dynamic equations and integral inequalities; for example, transformations, inverse conversions, extensions, wave equations, heat transfer, optics, fluid dynamics, quantum calculus, economy, etc

Summary

Introduction

Citation: Akın, L. A New Approach for the Fractional Integral Operator inTime Scales with Variable ExponentLebesgue Spaces. Fractal Fract. 2021, 5, 7. https://doi.org/10.3390/fractalfract5010007Received: 5 November 2020Accepted: 5 January 2021Published: 8 January 2021Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/ 4.0/).

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