  This paper is mainly devoted to the study of the reversed
Hardy-Littlewood-Sobolev (HLS) inequality on Heisenberg group $\mathbb{H}^n$
and CR sphere $\mathbb{S}^{2n+1}$. First, we establish the roughly reversed HLS
inequality and give a explicitly lower bound for the sharp constant. Then, the
existence of the extremal functions with sharp constant is proved by
subcritical approach and some compactness techniques. Our method is
rearrangement free and can be applied to study the classical HLS inequality and
other similar inequalities.
