Sharp weighted fractional Hardy inequalities

We investigate the weighted fractional order Hardy inequality $${\int }_{\mathrm{\Omega }}{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }f(x)-f(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d+sp}}\text{dist}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{-\alpha }\text{dist}(y,\mathrm{\partial }\mathrm{\Omega }{)}^{-\beta }dydx\ge C{\int }_{\mathrm{\Omega }}\frac{\mathrm{\mid }f(x){\mathrm{\mid }}^p}{\text{dist}(x,\mathrm{\partial }\mathrm{\Omega }{)}^{sp+\alpha +\beta }}dx,$$ for $\mathrm{\Omega }={\mathbb{R}}^{d-1}\times (0,\mathrm{\infty })$, $\mathrm{\Omega }$ being a convex domain or $\mathrm{\Omega }={\mathbb{R}}^d\setminus \{0\}$. 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.