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



Shu-Cheng Chang

Jingzhi Tie

Chin-Tung Wu


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 $H^n$.


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.

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