Fractional boundary Hardy inequality for the critical cases

TL;DR

This paper proves a generalized fractional boundary Hardy inequality applicable to various domains and parameter regimes, including the critical case, and extends results to weighted inequalities, solving open problems and broadening understanding of fractional Sobolev spaces.

Contribution

It establishes the fractional boundary Hardy inequality for all regimes, including the critical case, and extends results to weighted inequalities, addressing open questions in the field.

Findings

Proved the inequality for all s and p on Lipschitz domains.

Solved the open problem for the critical case sp=1.

Extended results to weighted inequalities.

Abstract

We establish generalised fractional boundary Hardy-type inequality, in the spirit of Caffarelli-Kohn-Nirenberg inequality for different values of $s$ and $p$ on various domains in $\mathbb{R}^d, ~ d \geq 1$. In particular, for Lipschitz bounded domains any values of $s$ and $p$ are admissible, settling all the cases in subcritical, supercritical and critical regime. In this paper we have solved the open problems posed by Dyda for the critical case $sp =1$. Moreover we have proved the embeddings of $W^{s,p}_{0}(\Omega)$ in subcritical, critical and supercritical uniformly without using Dyda's decomposition. Additionally, we extend our results to include a weighted fractional boundary Hardy-type inequality for the critical case.