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

TL;DR

This paper improves the series expansion of Hardy's inequality for $L^p$ spaces by adding optimal remainder terms, extending previous results for different ranges of p and addressing open questions.

Contribution

It introduces optimal non-homogeneous improvements to Hardy's inequality series expansion, generalizing and extending prior results for various p ranges.

Findings

For p<n, added an optimal weighted Sobolev norm as a remainder.

For p>n, incorporated an optimal weighted H"older seminorm as a remainder.

Settled an open question from previous research.

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 $1<p\neq n$. 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 H\"{o}lder seminorm, extending the Hardy-Morrey inequality of \cite{Ps} to the series case.