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

Nurgissa Yessirkegenov, Amir Zhangirbayev

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{č}i{ř}{\'ı}k-Štampach [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.

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

Abstract

In this paper, we provide a sharp remainder term for the general weighted discrete -Hardy inequality. By simply choosing weights and specifying , we are able to recover the identity by Krej{č}i{ř}{\'ı}k-Štampach [KS22, Theorem 1], obtain the sharp form of the -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 -Hardy inequality must approach the family of non-trivial minimizers.

Paper Structure

This paper contains 6 sections, 11 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Preliminary results--Proofs of Lemma \ref{['barki almost']} and Lemma \ref{['prop cp identity']}
  5. The general weighted discrete $p$-Hardy identity and Stability--Proofs of Theorem \ref{['thm hardy identity']} and Theorem \ref{['thm hardy stability']}
  6. Some consequences of the main results--Proof of Corollary \ref{['cor 1']}

Key Result

Lemma 2.2

Let $p\geq2$ and $n\geq 1$. Then, for $\xi,\eta\in\mathbb{C}^n$, we have where $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (19)

  • Definition 2.1
  • Lemma 2.2: cazacu2024hardy
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Theorem 3.1
  • Corollary 3.2
  • Corollary 3.3
  • Corollary 3.4
  • ...and 9 more