Reverse Hardy-type inequalities for supremal operators with measures

TL;DR

This paper characterizes when certain reverse Hardy-type inequalities involving supremal operators and measures hold, providing necessary and sufficient conditions for their validity.

Contribution

It offers a complete characterization of reverse Hardy inequalities with supremal operators and measures, extending classical results to more general measure and weight settings.

Findings

Derived necessary and sufficient conditions for inequalities

Extended classical Hardy inequalities to supremal operators with measures

Provided a framework for analyzing measure-based inequalities

Abstract

In this paper we characterize the validity of the inequalities $\|g\|_{p,(a,b),\lambda} \le c \|u(x) \|g\|_{\infty,(x,b),\mu}\|_{q,(a,b),\nu}$ and $\label{eq.0.1.2} \|g\|_{p,(a,b),\lambda} \le c \|u(x) \|g\|_{\infty,(a,x),\mu}\|_{q,(a,b),\nu} $ for all non-negative Borel measurable functions $g$ on the interval $(a,b)$, where $0 < p \le +\infty$, $0 < q \le +\infty$, $\lambda$, $\mu$ and $\nu$ are non-negative Borel measures on $(a,b)$, and $u$ is a weight function on $(a,b)$.