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

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 $${\int }_{{\mathbb{R}}_{+}^d}{\int }_{{\mathbb{R}}_{+}^d}\frac{\mathrm{\mid }u(x)-u(y){\mathrm{\mid }}^p}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d+sp}}dydx\ge {\mathcal{D}}_{d,s,p}{\int }_{{\mathbb{R}}_{+}^d}\frac{\mathrm{\mid }u(x){\mathrm{\mid }}^p}{x_d^{sp}}dx+C_{d,s,p,\tau }{\int }_{{\mathbb{R}}_{+}^d}\frac{\mathrm{\mid }u(x){\mathrm{\mid }}^p}{x_d^{sp-\tau }{(x_{d-1}^2+x_d^2)}^{\tau \mathrm{/}2}}dx,$$ 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.