  In this paper, we establish the following Stein-Weiss inequality with the
fractional Poisson kernel (see Theorem 1.1): \begin{equation}\label{int1}
\int_{\mathbb{R}^n_{+}}\int_{\partial\mathbb{R}^n_{+}}|\xi|^{-\alpha}f(\xi)P(x,\xi,\gamma)g(x)|x|^{-\beta}d\xi
dx\leq
C_{n,\alpha,\beta,p,q'}\|g\|_{L^{q'}(\mathbb{R}^n_{+})}\|f\|_{L^p(\partial
\mathbb{R}^{n}_{+})}, \end{equation} where
$P(x,\xi,\gamma)=\frac{x_n}{(|x'-\xi|^2+x_n^2)^{\frac{n+2-\gamma}{2}}}$, $2\le
\gamma<n$, $f\in L^{p}(\partial\mathbb{R}^n_{+})$, $g\in
L^{q'}(\mathbb{R}^n_{+})$ and $p,\ q'\in (1,\infty)$ and satisfy
$\frac{n-1}{n}\frac{1}{p}+\frac{1}{q'}+\frac{\alpha+\beta+2-\gamma}{n}=1$. Then
we prove that there exist extremals for the Stein-Weiss inequality (0.1) and
the extremals must be radially decreasing about the origin (see Theorem 1.5).
We also provide the regularity and asymptotic estimates of positive solutions
to the integral systems which are the Euler-Lagrange equations of the extremals
to the Stein-Weiss inequality (0.1) with the fractional Poisson kernel (see
Theorems 1.7 and 1.8). Our result is inspired by the work of Hang, Wang and Yan
[29] where the Hardy-Littlewood-Sobolev type inequality was first
establishedmwhen $\gamma=2$ and $\alpha=\beta=0$ (see (1.5)). The proof of the
Stein-Weiss inequality (0.1) with the fractional Poisson kernel in this paper
uses our recent work on the Hardy-Littlewood-Sobolev inequality with the
fractional Poisson kernel [18] and the present paper is a further study in this
direction.
