Logarithmic Sobolev, Hardy and Poincar\'e inequalities on the Heisenberg group

TL;DR

This paper establishes fundamental inequalities such as Sobolev, Hardy, Poincaré, and logarithmic inequalities on the Heisenberg group, providing explicit constants and extending classical results to this non-commutative setting.

Contribution

It introduces explicit constants for key inequalities on the Heisenberg group, including fractional Sobolev and Gross inequalities, and extends some results to infinite-dimensional settings.

Findings

Sharp constants for first-order Sobolev inequalities

Extension of Gross inequality to the Heisenberg group

Generalized Poincaré inequality with explicit constants

Abstract

In this paper we first prove a number of important inequalities with explicit constants in the setting of the Heisenberg group. This includes the fractional and integer Sobolev, Gagliardo-Nirenberg, (weighted) Hardy-Sobolev, Nash inequalities, and their logarithmic versions. In the case of the first order Sobolev inequality, our constant recovers the sharp constant of Jerison and Lee. Remarkably, we also establish the analogue of the Gross inequality with a semi-probability measure on the Heisenberg group that allows -- as it happens in the Euclidean setting -- an extension to infinite dimensions, and particularly can be regarded as an inequality on the infinite dimensional $\mathbb{H}^{\infty}$. Finally, we prove the so-called generalised Poincar\'e inequality on the Heisenberg group both with respect to the aforementioned semi-probability measure and the Haar measure, also with explicit constants.