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

Reversed Hardy-Littlewood-Sobolev inequality on Heisenberg group $\mathbb{H}^n$ and CR sphere $\mathbb{S}^{2n+1}$