Sharp forms and quantitative stability for general weighted discrete $p$-Hardy inequalities

TL;DR

This paper establishes a sharp remainder term for weighted discrete p-Hardy inequalities, recovering and generalizing key results, and proves a stability result showing minimizers approach the optimal family.

Contribution

It provides a sharp remainder term for the general weighted discrete p-Hardy inequality and proves a quantitative stability result for minimizers.

Findings

Recovered the identity by Krej{c}i{br}k-ampach [KS22]

Obtained the sharp form of the p-Hardy inequality by Fischer-Keller-Pogorzelski [FKP23]

Generalized Gupta's weighted inequality with sharp remainder

Abstract

In this paper, we provide a sharp remainder term for the general weighted discrete $p$-Hardy inequality. By simply choosing weights and specifying $1<p<\infty$, we are able to recover the identity by Krej{\v{c}}i{\v{r}}{\'\i}k-\v{S}tampach [KS22, Theorem 1], obtain the sharp form of the $p$-Hardy inequality by Fischer-Keller-Pogorzelski [FKP23, Theorem 1] and generalize the power weighted inequality by Gupta [Gup22, Theorem 2.1]{gupta2022discrete} with sharp remainder. In addition, we prove a quantitative stability result, thereby showing that any minimizing sequence of the discrete $p$-Hardy inequality must approach the family of non-trivial minimizers.