Application of multidimensional Hardy operator and its connection with a certain nonlinear differential equation in weighted variable Lebesgue spaces

TL;DR

This paper establishes a two-weight criterion for the multidimensional geometric mean operator in variable exponent Lebesgue spaces and links it to a nonlinear differential equation, advancing the understanding of weighted inequalities in these spaces.

Contribution

It introduces a new criterion for the geometric mean operator in variable Lebesgue spaces and connects it to a nonlinear integro-differential equation, providing novel insights.

Findings

Established a two-weight criterion for the geometric mean operator

Linked Hardy inequality to a nonlinear integro-differential equation

Extended weighted inequality theory in variable Lebesgue spaces

Abstract

In this paper a two weight criterion for multidimensional geometric mean operator in variable exponent Lebesgue space is proved. Also, we found a criterion on weight functions expressing one-dimensional Hardy inequality via a certain nonlinear differential equation. In particular, considered nonlinear differential equation is nonlinear integro-differential equation.