Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group

In this paper, we study two types of weighted Hardy–Littlewood–Sobolev (HLS) inequalities, also known as Stein–Weiss inequalities, on the Heisenberg group. More precisely, we prove the |u||u| weighted HLS inequality in Theorem 1.1 and the |z||z| weighted HLS inequality in Theorem 1.5 (where we have denoted u=(z,t)u=(z,t) as points on the Heisenberg group). Then we provide regularity estimates of positive solutions to integral systems which are Euler–Lagrange equations of the possible extremals to the Stein–Weiss inequalities. Asymptotic behavior is also established for integral systems associated to the |u||u| weighted HLS inequalities around the origin. By these a priori estimates, we describe asymptotically the possible optimizers for sharp versions of these inequalities.