# Boundedness of the Riesz Transforms for some Subelliptic Operators

### (Purdue University)

**Event:** ERC Workshop on Geometric Analysis on sub-Riemannian and Metric Spaces

**Date:** Oct 11, 2011,
**time:** 08:30

**Place:** Centro De Giorgi, Scuola Normale Superiore

**Abstract.** In this lecture I will describe some recent joint work with Fabrice Baudoin. Let \(\mathbb{M}\) be a smooth connected non-compact manifold endowed with a smooth measure \(\mu\) and a smooth

locally subelliptic diusion operator \(L\) satisfying \(L_1 = 0\), and which is symmetric with respect to \(\mu\). We show that if L satises, with a non negative curvature parameter \(\rho_1\) , a generalization of the curvature-dimension inequality from Riemannian geometry, then the Riesz transform is

bounded in \(L^p(\mathbb{M})\) for every \(p>1\), that is \[\Vert\sqrt{\Gamma{((-L)^{-\frac{1}{2}}f)}}\Vert_p \leq C_p \Vert f \Vert_p, \ \ f\in C_0^\infty(\mathbb{M}),\] where \(\Gamma\) is the /carrĂ© 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.