Sharp reversed Hardy--Littlewood--Sobolev inequality on the half space ${\mathbb{R}}_{+}^n$

This is the second in our series of papers concerning some reversed Hardy–Littlewood–Sobolev inequalities. In the present work, we establish the following sharp reversed Hardy–Littlewood–Sobolev inequality on the half space ${\mathbb{R}}_{+}^n$ $${\int }_{{\mathbb{R}}_{+}^n}{\int }_{\mathrm{\partial }{\mathbb{R}}_{+}^n}f(x)\mathrm{\mid }x-y{\mathrm{\mid }}^{\lambda }g(y)dxdy⩾{\mathcal{C}}_{n,p,r}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(\mathrm{\partial }{\mathbb{R}}_{+}^n)}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^r({\mathbb{R}}_{+}^n)}$$ for any nonnegative functions $f\in L^p(\mathrm{\partial }{\mathbb{R}}_{+}^n)$, $g\in L^r({\mathbb{R}}_{+}^n)$, and $p,r\in (0,1)$,$\lambda > 0$suchthat$(1-1/n)1/p + 1/r -(\lambda-1) /n =2$. Some estimates for ${\mathcal{C}}_{n,p,r}$ as well as the existence of extrema functions for this inequality are also considered. New ideas are also introduced in this paper.