Sharp weighted fractional Hardy inequalities

We investigate the weighted fractional order Hardy inequality $\int_Ω\int_Ω\frac{|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}\text{dist}(x,\partialΩ)^{-α}\text{dist}(y,\partialΩ)^{-β}\,dy\,dx\geq C\int_Ω\frac{|f(x)|^{p}}{\text{dist}(x,\partialΩ)^{sp+α+β}}\,dx,$ for $Ω=\mathbb{R}^{d-1}\times(0,\infty)$, $Ω$ being a convex domain or $Ω=\mathbb{R}^d\setminus\{0\}$. Our work focuses on finding the best (i.e. sharp) constant $C=C(d,s,p,α,β)$ 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.

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