Logarithmic Hardy-Littlewood-Sobolev Inequality on Pseudo-Einstein 3-manifolds and the Logarithmic Robin Mass

TL;DR

This paper extends the logarithmic Hardy-Littlewood-Sobolev inequality to three-dimensional pseudo-Einstein CR manifolds, introducing the Robin mass concept and establishing conditions for the inequality's validity through conformal geometry analysis.

Contribution

It introduces the Robin mass in the CR setting and proves an Aubin type result ensuring the existence of minimizers for the total mass, leading to the LHLS inequality.

Findings

Existence of conformal contact structures satisfying LHLS inequality

Introduction of Robin mass as a key geometric invariant

Establishment of a variational principle for total 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 \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.