Estimates for powers of sub-Laplacian on the non-isotropic Heisenberg group

Assume that\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $${\mathcal{L}}_\alpha = - \frac{1}{2}\sum\nolimits_{j = 1}^n {(Z_j \bar Z_j + \bar Z_j Z_j ) + i\alpha T} $$ \end{document}is the sub-Laplacian on the nonisotropic Heisenberg groupHn;Zj,Zjfor j = 1, 2, …,n andTare the basis of the Lie algebra hn.We apply the Laguerre calculus to obtain the explicit kernel for the fundamental solution of the powers of Lαand the heat kernel exp{−sLα}.Estimates for this kernel in various function spaces can be deduced easily.