Gradient Estimates for the Heat Semigroup on H-Type Groups

By utilizing the Poincaré inequality and representation formulae, it is shown that on the Heisenberg type group, ℍ(2n, m), there exists a constant C > 0 such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ |\nabla e^{t \Delta} f|(g) \leq C e^{t \Delta}(|\nabla f|)(g), \quad \forall g \in \mathbb{H}(2n, m), t > 0, f \in C_o^{\infty}(\mathbb{H}(2n, m)). $$\end{document}