On sharp inequalities on nilpotent Lie groups

Abstract

In this paper, we review some recent works on sharp fractional inequalities on nilpotent Lie groups, studying their sharp constants, extremizers, stability and other improvements. We also prove a new sharp subcritical inequality in Section sub . The conformal symmetry, compactness, spectral estimates and flow methods are discussed. We remark that a positivity-type restriction on the singular exponent is required in cases for groups with high dimensional centres, which brings extra difficulty.