Fractional Operators in  p -adic Variable Exponent Lebesgue Spaces and Application to  p -adic Derivative

Leonardo Fabio Chacón-Cortés,Humberto Rafeiro,Douadi Drihem

doi:10.1155/2021/3096701

Fractional Operators in p -adic Variable Exponent Lebesgue Spaces and Application to p -adic Derivative

Leonardo Fabio Chacón-Cortés, Humberto Rafeiro + Show 1 more

Open Access

https://doi.org/10.1155/2021/3096701

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Abstract

In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the p -adic variable exponent Lebesgue spaces.

Highlights

The field of p-adic numbers are an interesting and useful tool to study phenomena in physics, biology, and medicine, among other sciences; see, e.g., [1,2,3,4] and references therein
Variable exponent Lebesgue spaces generalize the notion of q-integrability in the classical Lebesgue spaces, allowing the exponent to be a measurable function
These spaces were introduced in 1931 by Orlicz [5] but lay essentially dormant for more than 50 years. They received a thrust in the paper [6] and are an active area of research having many known applications, e.g., in the modeling of thermorheological fluids [7] as well as electrorheological fluids [8,9,10,11], in differential equations with nonstandard growth [12, 13], and in the study of image processing [14,15,16,17,18,19,20]

Summary

Introduction

The field of p-adic numbers are an interesting and useful tool to study phenomena in physics, biology, and medicine, among other sciences; see, e.g., [1,2,3,4] and references therein. Variable exponent Lebesgue spaces generalize the notion of q-integrability in the classical Lebesgue spaces, allowing the exponent to be a measurable function These spaces were introduced in 1931 by Orlicz [5] but lay essentially dormant for more than 50 years. We are interested in the boundedness of the fractional integral and maximal fractional operator on the p-adic Lebesgue spaces with a variable exponent. The corresponding result for classical p-adic Lebesgue space is known (cf [25]) These operators play an important role in such areas such as Sobolev spaces, potential theory, PDEs, and integral geometry, to name a few. Ð2Þ is obtained from the boundedness of the fractional maximal operator and Welland’s pointwise inequality tailored for the Journal of Function Spaces p-adic setting; this approach is inspired from [27]. The notation a ≲ b denotes the existence of a constant C for which a ≤ Cb, a≍b means that a ≲ b and b ≲ a

Preliminaries

Main Results

Applications to p-adic Derivative