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

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 $1<p<\mathrm{\infty }$, 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.