Sharp Weighted Discrete $p$-Hardy Inequality and Stability

TL;DR

This paper establishes a weighted discrete $p$-Hardy inequality on the half-line, compares sharp constants with continuous cases, and examines the stability of the inequality in the discrete setting.

Contribution

It introduces a new quantitative approach for $p$-Hardy inequalities involving two measures on the discrete half-line, extending prior work and analyzing stability.

Findings

Proved a $p$-Hardy inequality with weights $n^{eta}$ for all $p > 1$.

Compared sharp constants between discrete and continuous cases.

Analyzed the stability of the discrete $p$-Hardy inequality.

Abstract

In this paper, we prove a $p$-Hardy inequality on the discrete half-line with weights $n^{\alpha}$ for all real $p > 1$. Building on the work of Miclo for $p = 2$ and Muckenhoupt in the continuous settings, we develop a quantitative approach for the existence of a $p$-Hardy inequality involving two measures $\mu$ and $\nu$ on the discrete half-line. We also investigate the comparison between sharp constants in the discrete and continuous settings and explore the stability of the inequality in the discrete case.