Stein-Weiss inequalities with the fractional Poisson kernel

In this paper, we establish the following Stein-Weiss inequality with the fractional Poisson kernel: (*) integral(R+n) integral(delta R+n) vertical bar xi vertical bar(-alpha) f(xi) P(x, xi, gamma) g(x) vertical bar x vertical bar(-beta) d xi dx <= C-n,C-alpha,C-beta,C-gamma,C-p,C-q' parallel to g parallel to(Lq'(R+n)) parallel to f parallel to L-p(partial derivative R-+(n)), where P(x, xi, gamma) = x(n)/vertical bar x' -xi vertical bar(2) +x(n)(2))((n+2-gamma)/2), 2 <= gamma < n, f is an element of L-p (partial derivative R-n(+)), g is an element of L-q' (R-n(+)), and p, q'is an element of (1,infinity) and satisfy (n-1)/(np)+1/q' + (alpha+ beta+2-gamma)/n = 1. Then we prove that there exist extremals for the Stein-Weiss inequality (*), and that the extremals must be radially decreasing about the origin. 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 (*) with the fractional Poisson kernel. Our result is inspired by the work of Hang, Wang and Yan, where the Hardy-Littlewood-Sobolev type inequality was first established when gamma = 2 and alpha = beta = 0. The proof of the Stein-Weiss inequality (*) with the fractional Poisson kernel in this paper uses recent work on the Hardy-Littlewood-Sobolev inequality with the fractional Poisson kernel by Chen, Lu and Tao, and the present paper is a further study in this direction.