Sharp weighted fractional Hardy inequalities

TL;DR

This paper establishes sharp weighted fractional Hardy inequalities for specific domains, determining the best constants and extending to Hardy-Sobolev-Maz'ya inequalities using advanced mathematical techniques.

Contribution

It provides the first sharp constants for weighted fractional Hardy inequalities in convex and exterior domains, extending the theory with new inequalities.

Findings

Derived sharp constants for weighted fractional Hardy inequalities.

Extended inequalities to Hardy-Sobolev-Maz'ya type.

Applied ground state representation techniques for proofs.

Abstract

We investigate the weighted fractional order Hardy inequality $$ \int_{\Omega}\int_{\Omega}\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partial\Omega)^{-\alpha}\text{dist}(y,\partial\Omega)^{-\beta}\,dy\,dx\geq C\int_{\Omega}\frac{|f(x)|^{p}}{\text{dist}(x,\partial\Omega)^{sp+\alpha+\beta}}\,dx, $$ for $\Omega=\mathbb{R}^{d-1}\times(0,\infty)$, $\Omega$ being a convex domain or $\Omega=\mathbb{R}^d\setminus\{0\}$. Our work focuses on finding the best (i.e. sharp) constant $C=C(d,s,p,\alpha,\beta)$ in all cases. We also obtain weighted version of the fractional Hardy-Sobolev-Maz'ya inequality. The proofs are based on general Hardy inequalities and the non-linear ground state representation, established by Frank and Seiringer.