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 $$\int_{\mathbb{R}^{{d}<em>{+}}\int</em>{\mathbb{R}^{{d}<em>{+}}\frac{|u(x)-u(y)|^{p}{|x-y|^{{d+sp}}dy\,dx\ge\mathcal{D}}}</em>{d,s,p}\int_{\mathbb{R}^{{d}<em>{+}}\frac{|u(x)|^{p}{x_d^{sp}}dx+C}</em>{d,s,p,τ}\int_{\mathbb{R}^{{d}<em>{+}}\frac{|u(x)|^p}{x</em>{d}^{{sp-τ}\left(x_{d-1}^{2+x_d^{2\right)^{τ/2}}dx,}}}}}}}$$ where $\mathcal{D}<em>{d,s,p}$ stands for the sharp constant in the fractional Hardy inequality on a half-space $\mathbb{R}^{d}</em>{+}$. We also obtain a similar result in the setting of Sobolev--Bregman forms.