Sharp trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere

TL;DR

This paper proves sharp Sobolev trace inequalities for fractional sublaplacians on the Heisenberg group and CR sphere, extending Euclidean results to these non-Euclidean geometries.

Contribution

It extends Euclidean conformally invariant fractional Sobolev inequalities to the Heisenberg group and CR sphere, including sharp trace Beckner-Onofri inequalities.

Findings

Established sharp Sobolev trace inequalities on the Heisenberg group and CR sphere.

Extended Euclidean results to non-Euclidean settings.

Derived trace Beckner-Onofri inequalities on the CR sphere.

Abstract

We establish sharp Sobolev trace inequalities for conformally invariant fractional powers of the sublaplacian on the Heisenberg group and the CR sphere, extending the corresponding Euclidean results of Einav-Loss, Beckner, and Bez-Machihara-Sugimoto to these non-Euclidean settings. In the limiting case, sharp trace Beckner-Onofri inequalities are also established on the CR sphere. The proofs are based on a duality argument due to Bez-Machihara-Sugimoto, together with the Frank-Lieb sharp form of the Hardy-Littlewood-Sobolev inequalities on the Heisenberg group and the CR sphere. The same approach also yields trace Beckner-Onofri inequalities on the standard sphere.