A simple proof of logarithmic Sobolev inequality on the Heisenberg groups

TL;DR

This paper presents a straightforward, dimension-independent proof of the logarithmic Sobolev inequality on Heisenberg groups, utilizing measure-preserving transformations of Brownian motion, and corrects previous errors in the literature.

Contribution

It provides a simplified and corrected proof of the logarithmic Sobolev inequality on Heisenberg groups that is independent of dimension.

Findings

Proof is simple and dimension-independent

Corrects previous flaws in the literature

Utilizes measure-preserving transformations of Brownian motion

Abstract

In this note we give a simple, dimension independent, proof of the logarithmic Sobolev inequality on the Heisenberg groups $H_n=\R^{2n+1}$ using the measure preserving transformations of the Brownian motion. We have corrected some serious flaws and simplified the proofs.