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The $p$-Hardy-Rellich-Birman inequalities on the half-line

František Štampach, Jakub Waclawek

Abstract

The classical discrete $p$-Hardy inequality establishes a sharp relationship between the $\ell^{p}$-norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer order $\ell \geq 1$, yielding discrete $p$-Rellich ($\ell=2$) and general $p$-Birman ($\ell \geq 3$) inequalities. As a key step in the proof, we deduce a variant of the Copson inequality with a negative exponent, which may be of independent interest. Furthermore, we demonstrate how the continuous $p$-Birman inequality can be recovered from our discrete version, providing an alternative proof of this classical result. All constants in the obtained inequalities are shown to be optimal.

The $p$-Hardy-Rellich-Birman inequalities on the half-line

Abstract

The classical discrete -Hardy inequality establishes a sharp relationship between the -norms of a sequence and its discrete derivative. In this paper, we generalize this inequality to discrete derivatives of arbitrary integer order , yielding discrete -Rellich () and general -Birman () inequalities. As a key step in the proof, we deduce a variant of the Copson inequality with a negative exponent, which may be of independent interest. Furthermore, we demonstrate how the continuous -Birman inequality can be recovered from our discrete version, providing an alternative proof of this classical result. All constants in the obtained inequalities are shown to be optimal.
Paper Structure (13 sections, 6 theorems, 56 equations)

This paper contains 13 sections, 6 theorems, 56 equations.

Table of Contents

  1. Introduction and main results
  2. Introduction
  3. Organization of the paper
  4. Notation for discrete derivatives
  5. Main results
  6. Proofs of the inequalities
  7. An abstract p-Hardy inequality
  8. Proof of the weighted p-Hardy inequality \ref{['eq:p-Hardy_weight']}
  9. Proof of the p-Birman inequality \ref{['eq:p-birman_dis']}
  10. Continuous p-Birman inequality
  11. Optimality
  12. Proofs of optimality of constants
  13. Remarks on optimality of weights

Key Result

Theorem 1

Let $\ell\in\mathbb{N}$ and $p>1$. Then for all $u\in C_{0}(\mathbb{N}_{0})$ such that $u_{n}=0$ for $n<\ell$, the inequality holds with the optimal constant

Theorems & Definitions (18)

  • Remark
  • Theorem 1: discrete $p$-Birman inequality
  • Remark 2
  • Theorem 3
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • proof : Proof of inequality \ref{['eq:p-Hardy_weight']}
  • Remark 6
  • ...and 8 more