Asian Journal of Mathematics

Volume 14 (2010)

Number 1

Subgradient Estimate and Liouville-type Theorem for the $CR$ Heat Equation on Heisenberg Groups

Pages: 41 – 72

DOI: https://dx.doi.org/10.4310/AJM.2010.v14.n1.a4

Authors

Shu-Cheng Chang (Department of Mathematics, National Taiwan University, Taipei, Taiwan)

Jingzhi Tie

Chin-Tung Wu

Abstract

In this paper, we first get a subgradient estimate of the $CR$ heat equation on a closed pseudohermitian $(2n + 1)$-manifold. Secondly, by deriving the $CR$ version of sub-Laplacian comparison theorem on an $(2n + 1)$-dimensional Heisenberg group $H^n$, we are able to establish a subgradient estimate and then the Liouville-type theorem for the $CR$ heat equation on $H^n$.

Keywords

Subgradient estimate, Liouville-type theorem, heat kernel, pseudohermitian manifold, Heisenberg group, $CR$-pluriharmonic, $CR$-Paneitz operator, sub-Laplacian, Li-Yau Harnack inequality

2010 Mathematics Subject Classification

Primary 32V05, 32V20. Secondary 53C56.

Published 1 January 2010