On necessary and sufficient conditions for the variable exponent Hardy type inequality

TL;DR

This paper establishes necessary and sufficient conditions for the validity of a variable exponent Hardy inequality on the interval (0,1), considering cases where the exponent function is monotonic near zero.

Contribution

It provides new equivalent criteria characterizing when the variable exponent Hardy inequality holds for increasing or decreasing exponents near zero.

Findings

Derived multiple equivalent conditions for the inequality.

Identified the role of the exponent's monotonicity near zero.

Extended understanding of variable exponent Hardy inequalities.

Abstract

We derive a number of equivalent criterions for the variable exponent Hardy type inequality |\frac{1}{x}\int_{0}^{x}f(t)dt|_{L^{p(.)}(0,1)}\leq C|f|_{L^{p(.)}(0,1)}; f\geq 0. to hold, whenever the exponent $p:(0,1)\to (1,\infty)$ is increasing or decreasing near small neighborhood of the origin.