A note on the boundedness of Riesz transform for some subelliptic operators

TL;DR

This paper proves the boundedness of the Riesz transform on certain subelliptic manifolds under curvature conditions, extending results to Sasakian manifolds, Carnot groups, and principal bundles with nonnegative Ricci curvature.

Contribution

It establishes the boundedness of the Riesz transform for a broad class of subelliptic operators satisfying a generalized curvature inequality, including new geometric settings.

Findings

Riesz transform bounded in L^p for p>1 under curvature conditions

Applicable to Sasakian manifolds with nonnegative Tanaka-Webster Ricci curvature

Includes Carnot groups of step two and certain principal bundles

Abstract

Let $\M$ be a smooth connected non-compact manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. We show that if $L$ satisfies, with a non negative curvature parameter $\rho_1$, the generalized curvature inequality in \eqref{CD} below, then the Riesz transform is bounded in $L^p (\bM)$ for every $p>1$, that is \[\| \sqrt{\Gamma((-L)^{-1/2}f)}\|_p \le C_p \| f \|_p, \quad f \in C^\infty_0(\bM), \] where $\Gamma$ is the \textit{carr\'e du champ} associated to $L$. Our results apply in particular to all Sasakian manifolds whose horizontal Tanaka-Webster Ricci curvature is nonnegative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.