The $\Box_b$-heat equation on finite type CR manifolds with comparable Levi form

Abstract

UDC 517.9
The main purpose of this paper is to study the initial-value problems for the heat equations associated with the operator $\Box_b$ on compact CR manifolds of finite type. The critical component of our analysis is the condition called $D^{\epsilon}(q)$ and introduced by K. D. Koenig [Amer. J. Math. -- 2002. -- {\bf 124}. -- P. 129--197]. Actually, it states that the $\min\{q, n-1-q\}$th smallest eigenvalue of the Levi form is comparable with the largest eigenvalue of the Levi form.