LOGARITHMIC HARDY–LITTLEWOOD–SOBOLEV INEQUALITY ON PSEUDO-EINSTEIN 3-MANIFOLDS AND THE LOGARITHMIC ROBIN MASS

Abstract

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 [30] in the Riemannian setting, yet the differential operators that we are dealing with are of very different nature. For this reason, we introduce the notion of Robin mass as the constant term appearing in the expansion of the Green’s function of the $P^{\prime }$-operator. We show that the LHLS inequality appears when we study the variation of the total mass under conformal change. This can be tied to the value of the regularized Zeta function of the operator at $1$ and hence we prove a CR version of the results in [27]. We also exhibit an Aubin-type result guaranteeing the existence of a minimizer for the total mass which yields the classical LHLS inequality.