LAUSR

The purpose of this paper is five-fold. First, we employ the harmonic analysis techniques to establish the following Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel on the upper half space $$\int_{\mathbb{R}_ + ^n} {{{\int_{\partial \mathbb{R}_ + ^n} {f\left( \xi \right)P\left( {x,\xi,\alpha } \right)g\left( x \right)d\xi dx \leq {C_{n,\alpha,p,q'}}\parallel g\parallel } }_{Lq'\left( {\mathbb{R}_ + ^n} \right)}}} \parallel f{\parallel _{Lq'\left( {\partial \mathbb{R}_ + ^n} \right),}}$$∫R+n∫∂R+nf(ξ)P(x,ξ,α)g(x)dξdx≤Cn,α,p,q′∥g∥Lq′(R+n)∥f∥Lq′(∂R+n), where $$f\, \in \,{L^p}\left( {\partial \mathbb{R}_ + ^n} \right),\,g\, \in \,{L^{q'}}\left( {\mathbb{R}_ + ^n}\right)\,\text{and}\;\,p,\,q'\, \in \,\left( {1 + \infty } \right),\,2\, \leq {\alpha