A Note on the Boundedness of Riesz Transform for Some Subelliptic Operators

Let $\mathbb M$ be a smooth connected noncompact manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying L1=0, and that is symmetric with respect to μ. We show that if L satisfies, with a nonnegative curvature parameter ρ1, the generalized curvature inequality in (2.9), then the Riesz transform is bounded in $L^p(\mathbb M)$ for every p>1, that is $$\parallel\sqrt{\Gamma ((-L)^{-1/2}f)}\parallel_p \leq C_p\parallel f\parallel_p,\quad f\in C^{\infty}_0(\mathbb M),$$ where Γ 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.