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arXiv · 2308.11956

Fractional boundary Hardy inequality for the critical cases

Adimurthi·Prosenjit Roy·Vivek Sahu
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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.

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Adimurthi, Prosenjit Roy, Vivek Sahu. 2023-08-23. Fractional boundary Hardy inequality for the critical cases. https://doi.org/10.1016/j.jfa.2026.111351

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