Sharp weighted fractional Hardy inequalities
Volume 274 / 2024
Abstract
We investigate the weighted fractional order Hardy inequality $$\int _{\Omega }\int _{\Omega }\frac {|f(x)-f(y)|^{p}}{|x-y|^{d+sp}}{\rm dist} (x,\partial \Omega )^{-\alpha }{\rm dist} (y,\partial \Omega )^{-\beta }\,dy\,dx\geq C\int _{\Omega }\frac {|f(x)|^{p}}{{\rm dist}(x,\partial \Omega )^{sp+\alpha +\beta }}\,dx,$$ 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 a weighted version of the fractional Hardy–Sobolev–Maz’ya inequality. The proofs are based on general Hardy inequalities and the nonlinear ground state representation, established by Frank and Seiringer.
