Logarithmic Hardy-Littlewood-Sobolev inequality on pseudo-Einstein 3-manifolds and the logarithmic Robin mass

Given a three dimensional pseudo-Einstein CR manifold $(M,T^{1,0}M,\theta)$, we study the existence of a contact structure conformal to $\theta$ for which the logarithmic Hardy-Littlewood-Sobolev (LHLS) inequality holds. Our approach closely follows \cite{Ok1} in the Riemannian setting. For this purpose, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green's function of the $P'$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. Then we exhibit an Aubin type result guaranteeing the existence of a minimizer for the total mass which yields the classical LHLS inequality.