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

TL;DR

This paper extends weighted Hardy and Rellich inequalities with sharp remainder terms to all p in (1, ∞), providing new identities even for classical Laplacian, thus broadening the scope of these fundamental inequalities.

Contribution

It generalizes weighted Hardy and Rellich inequalities with sharp remainder terms to all p in (1, ∞), including new identities for the classical Laplacian.

Findings

Extended inequalities to all p in (1, ∞)

Derived new identities for classical Laplacian

Established sharp remainder terms for degenerate elliptic operators

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.