Sharp weighted fractional Hardy inequalities

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.