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One-dimensional discrete Hardy and Rellich inequalities on integers

Shubham Gupta

Abstract

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form $n^α$. We prove the inequality when $α$ is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities and its higher order versions. As a by-product of this work we derive a combinatorial identity using purely analytic methods. This suggests a correlation between combinatorial identities and functional identities.

One-dimensional discrete Hardy and Rellich inequalities on integers

Abstract

In this paper, we consider a weighted version of one-dimensional discrete Hardy inequalities with power weights of the form . We prove the inequality when is an even natural number with the sharp constant and remainder terms. We also find explicit constants in standard and weighted Rellich inequalities and its higher order versions. As a by-product of this work we derive a combinatorial identity using purely analytic methods. This suggests a correlation between combinatorial identities and functional identities.
Paper Structure (9 sections, 12 theorems, 97 equations)

This paper contains 9 sections, 12 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Hardy inequalities
  4. Higher Order Hardy inequalities
  5. Some Auxiliary Results
  6. Proof of Hardy inequalities
  7. Proof of Higher Order Hardy inequalities
  8. Combinatorial identity
  9. Appendix

Key Result

THEOREM 2.1

Let $u \in C_c(\mathbb{Z})$, the space of finitely supported functions, and also assume $u(0)=0$. Then for $k \in {\mathord{\mathbb N}}$, we have where the non-negative constants $\gamma_i^k$ are given by Here $\Gamma(x)$ denotes the Gamma function and ${x \choose y} := \frac{\Gamma(x+1)}{\Gamma(x-y+1) \Gamma(y+1)}$ denotes the binomial coefficient.

Theorems & Definitions (28)

  • Remark 1.1
  • THEOREM 2.1: Improved weighted Hardy inequalities
  • COROLLARY 2.2: Weighted Hardy inequalities
  • THEOREM 2.3: Higher order Hardy inequalities
  • COROLLARY 2.4: Rellich inequality
  • Remark 2.5
  • THEOREM 2.6: Power weight higher order Hardy inequalities
  • Remark 2.7
  • LEMMA 3.1
  • proof
  • ...and 18 more