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 $L^p$-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 $p\ge 2$. 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.

20 pages, added Corollary 4.1 and Corollary 4.2