# Boundedness of the Riesz Transforms for some Subelliptic Operators

## Nicola Garofalo

### (Purdue University)

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 di usion operator $L$ satisfying $L_1 = 0$, and which is symmetric with respect to $\mu$. We show that if L satis es, 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 /carré 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.