On the H.-Q. Li inequality on step-two Carnot groups

Abstract

In this note we show that the gradient estimate of the heat semigroup, or more precisely the H.-Q. Li inequality, is preserved under tensorization, some suitable group epimorphism, and central sum. We also establish the Riemannian counterpart of the H.-Q. Li inequality. As a byproduct, we provide a simpler proof of the fact that the constant in H.-Q. Li inequality is strictly larger than $1$.