Heat kernel upper bounds under the generalized curvature(-dimension) inequality

TL;DR

This paper establishes upper bounds for heat kernels on sub-Riemannian manifolds using generalized curvature-dimension inequalities, extending previous results to cases with negative curvature parameters.

Contribution

It introduces new heat kernel upper bounds under generalized curvature inequalities, combining methods from Baudoin-Garofalo and Wang's Harnack inequality.

Findings

Heat kernel upper bounds derived under generalized curvature-dimension inequality.

Extension of bounds to cases with negative curvature parameters.

Application of combined analytical techniques for sharper estimates.

Abstract

In the sub-Riemannian manifolds, on the one hand, following Baudoin-Garofalo \cite{BaudoinGarofalo}, the upper bound for heat kernels associated to a class of locally subelliptic operators are given under the generalized curvature-dimension inequality with a negative curvature parameter; on the other hand, the argument combining Grigor'yan's integrated maximum principle with Wang's dimension-free Harnack inequality is also shown to derive the upper bound for the heat kernel under the generalized curvature inequality.