Fractional Hardy--Maz'ya inequality on a half-space

Abstract

The main purpose of this article is to provide a fractional counterpart of the well-known Maz'ya inequality on the half-space, that is $${\int }_{{\mathbb{R}}_{+}^d}{\int }_{{\mathbb{R}}_{+}^d}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{d+sp}}dydx\ge {\mathcal{D}}_{d,s,p}{\int }_{{\mathbb{R}}_{+}^d}\frac{|u(x){|}^p}{x_d^{sp}}dx+C_{d,s,p,\tau }{\int }_{{\mathbb{R}}_{+}^d}\frac{|u(x){|}^p}{x_d^{sp-\tau }{\left(x_{d-1}^2+x_d^2\right)}^{\tau /2}}dx,$$ where ${\mathcal{D}}_{d,s,p}$ stands for the sharp constant in the fractional Hardy inequality on a half-space ${\mathbb{R}}_{+}^d$. We also obtain a similar result in the setting of Sobolev--Bregman forms.

23 pages