Sharp weighted fractional Hardy inequalities

Abstract

We investigate the weighted fractional order Hardy inequality $${\int }_{\Omega }{\int }_{\Omega }\frac{|f(x)-f(y){|}^p}{|x-y{|}^{d+sp}}\text{dist}(x,\partial \Omega {)}^{-\alpha }\text{dist}(y,\partial \Omega {)}^{-\beta }dydx\ge C{\int }_{\Omega }\frac{|f(x){|}^p}{\text{dist}(x,\partial \Omega {)}^{sp+\alpha +\beta }}dx,$$ 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.

14 pages, 1 figure

References in corpus (1)

• Fractional Sobolev spaces with power weights

Cited by in corpus (6)

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• Asymptotics of weighted Gagliardo seminorms
• Weighted fractional Hardy inequalities with singularity on any flat submanifold
• Fractional Hardy inequality with singularity on submanifold
• Fractional boundary Hardy inequality for the critical cases
• Robust nonlocal trace and extension theorems