On gradient estimates of the heat semigroups on step-two Carnot groups

TL;DR

This paper establishes gradient estimates for heat semigroups on certain step-two Carnot groups, providing new conditions and extending results to high order gradients and Riemannian cases.

Contribution

It introduces a sufficient condition for quasi Bakry-Émery curvature on step-two Carnot groups and applies it to derive gradient estimates for the heat semigroup on the free group with three generators.

Findings

Gradient estimate for heat semigroup on N_{3,2}

High order gradient estimates derived

Riemannian counterparts also established

Abstract

In this work, we give a sufficient condition for a step-two Carnot group to satisfy the quasi Bakry-\'Emery curvature condition. As an application, we establish the gradient estimate for the heat semigroup on the free step-two Carnot group with three generators $N_{3,2}$. Moreover, high order gradient estimates and the Riemannian counterparts are also deduced under an extra condition.