Precise estimates for the subelliptic heat kernel on H-type groups. (English) Zbl 1178.35096
Author’s abstract: We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups \(G\) of \(H\)-type. Specifically, we show that there exist positive constants \(C_1\), \(C_2\) and a polynomial correction function \(Q_t\) on \(G\) such that
\[ C_1 Q_t e^{-\frac{d^2}{4t}} \leq p_t \leq C_2 Q_t e^{-\frac{d^2}{4t}} \]
where \(p_t\) is the heat kernel, and d the Carnot-Carathéodory distance on \(G\). We also obtain similar bounds on the norm of its subelliptic gradient \(|\nabla p_t|\). Along the way, we record explicit formulas for the distance function \(d\) and the sub-Riemannian geodesics of \(H\)-type groups.
\[ C_1 Q_t e^{-\frac{d^2}{4t}} \leq p_t \leq C_2 Q_t e^{-\frac{d^2}{4t}} \]
where \(p_t\) is the heat kernel, and d the Carnot-Carathéodory distance on \(G\). We also obtain similar bounds on the norm of its subelliptic gradient \(|\nabla p_t|\). Along the way, we record explicit formulas for the distance function \(d\) and the sub-Riemannian geodesics of \(H\)-type groups.
Reviewer: Marco Biroli (Milano)
MSC:
| 35B45 | A priori estimates in context of PDEs |
| 35H20 | Subelliptic equations |
| 53C17 | Sub-Riemannian geometry |
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