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Sharp remainder formulae for general weighted Hardy and Rellich type inequalities for $1<p<\infty$

Yerkin Shaimerdenov, Nurgissa Yessirkegenov, Amir Zhangirbayev

Abstract

Inspired by the work of Cossetti and D'Arca [CD25], we show that the general weighted $L^{p}$-Hardy type inequalities [CD25, Theorems 1.1 and 1.2] and the corresponding identities hold for all $1<p<\infty$, thus extending their results beyond the case $p\geq 2$. In addition, we present a general weighted $L^{p}$-Rellich type inequality with a sharp remainder term for quasilinear second order degenerate elliptic differential operators. In particular, even for the classical Laplacian, these identities appear to be new.

Sharp remainder formulae for general weighted Hardy and Rellich type inequalities for $1<p<\infty$

Abstract

Inspired by the work of Cossetti and D'Arca [CD25], we show that the general weighted -Hardy type inequalities [CD25, Theorems 1.1 and 1.2] and the corresponding identities hold for all , thus extending their results beyond the case . In addition, we present a general weighted -Rellich type inequality with a sharp remainder term for quasilinear second order degenerate elliptic differential operators. In particular, even for the classical Laplacian, these identities appear to be new.
Paper Structure (8 sections, 12 theorems, 105 equations)

This paper contains 8 sections, 12 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Applications
  5. Extension of Cossetti and D’Arca results.
  6. Special cases
  7. Proofs of Theorems \ref{['thm1']} and \ref{['thm2']}
  8. Proofs of Proposition \ref{['prop rellich']}, Theorem \ref{['thm rellich']} and Corollary \ref{['cor rellich classic']}

Key Result

Proposition 2.1

Let $1<p<\infty$ and $\Omega\subseteq \mathbb{R}^N$. Then, for all complex-valued $u$ on $\Omega$ and non-trivial real-valued $\phi\in C^{2}(\Omega)$, we have

Theorems & Definitions (31)

  • Proposition 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Lemma 2.6: cazacu2024hardy
  • Lemma 2.7: CT24
  • Lemma 2.8: CT24
  • Proposition 2.9
  • Theorem 3.1
  • ...and 21 more