Gradient estimates for the subelliptic heat kernel on H-type groups

Abstract

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type: $$|\nabla P_t f| \le K P_t(|\nabla f|)$$ where $P_t$ is the heat semigroup corresponding to the sublaplacian on $G$, $\nabla$ is the subelliptic gradient, and $K$ is a constant. This extends a result of H.-Q. Li for the Heisenberg group. The proof is based on pointwise heat kernel estimates, and follows an approach used by Bakry, Baudoin, Bonnefont, and Chafa\"i.Comment: 23 pages; updated with peer-review revision

Duality on gradient estimates and Wasserstein controls

“…Our work is strongly motivated by recent development on gradient estimates on a Lie group endowed with a sub-Riemannian structure [5,8,12,13,22,27]. To explain a consequence of our duality, we deal with the 3-dimensional Heisenberg group here.…”

Analysis on Extended Heisenberg Group

“…with ∇ ∇ ≡ (X, Y ) and some constant C t ∈ (0, ∞); (see [12] for q > 1 and more general groups via stochastic methods, and [23], [3], for Heisenberg group, and [14], [20] for general H-type groups with q = 1).…”

Gradient Estimates for the Heat Semigroup on H-Type Groups

“…32, we should remark that there is another idea, i.e. by Lemma 3.1 of Eldredge [12], to calculate this determinant. However, we will follow the method in Hu and Li (submitted for publication), we believe, in which our method is more effective, to give the details about the valuation of the determinant.…”

“…The second authors thanks Dominique Bakry for his invitation. The authors are grateful to Fabrice Baudoin who kindly inform us the reference [12], Thierry Coulhon for interesting comments, Xiang-Dong Li, Qing-Xue Wang and Jian-Gang Ying for reading the manuscript. The authors would also like to thank the referees for their suggestions and their careful reading of manuscript.…”

“…This method is very efficient to compute the determinant of the same kind of matrices. Note that the methods in [5] are also available to prove Theorem 1.1, see [12]. Just as in [1,5], we could get some interesting applications of the inequality in Theorem 1.1.…”