Optimal non-homogeneous improvements for the series expansion of Hardy's inequality

Abstract

We consider the series expansion of the $L^p$-Hardy inequality of \cite{BFT2}, in the particular case where the distance is taken from an interior point of a bounded domain in ${\mathbb{R}}^n$ and $1<p\ne n$. For $p<n$ we improve it by adding as a remainder term an optimally weighted critical Sobolev norm, generalizing the $p=2$ result of \cite{FT} and settling the open question raised in \cite{BFT1}. For $p>n$ we improve it by adding as a remainder term the optimally weighted Hölder seminorm, extending the Hardy-Morrey inequality of \cite{Ps} to the series case.