Mathematics
LOGARITHMIC HARDY–LITTLEWOOD–SOBOLEV INEQUALITY ON PSEUDO-EINSTEIN 3-MANIFOLDS AND THE LOGARITHMIC ROBIN MASS
Document Type
Article
Abstract
Given a three-dimensional pseudo-Einstein CR manifold (M, T1,0M, θ), we study the existence of a contact structure conformal to θ 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 P0-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. © 2023 Universitat Autonoma de Barcelona. All rights reserved.
Publication Title
Publicacions Matematiques
Publication Date
2023
Volume
67
Issue
2
First Page
515
Last Page
540
ISSN
0214-1493
DOI
10.5565/PUBLMAT6722302
Keywords
logarithmic Hardy–Littlewood–Sobolev inequality, P0-operator, pseudo-Einstein CR manifolds, spectral Zeta function
Repository Citation
Maalaoui, Ali, "LOGARITHMIC HARDY–LITTLEWOOD–SOBOLEV INEQUALITY ON PSEUDO-EINSTEIN 3-MANIFOLDS AND THE LOGARITHMIC ROBIN MASS" (2023). Mathematics. 9.
https://commons.clarku.edu/mathematics/9