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

2011/05/03 by F. Baudoin, Baudoin, F., N. Garofalo +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #math.AP #math.DG #math.FA

paper · pdf · doi:10.48550/arxiv.1105.0467

arxiv created 2011/05/03 · arxiv updated 2011/05/04

Abstract

Let \M be a smooth connected non-compact manifold endowed with a smooth measure μ and a smooth locally subelliptic diffusion operator L satisfying L1=0, and which is symmetric with respect to μ. We show that if L satisfies, with a non negative curvature parameter ρ₁, the generalized curvature inequality in \eqrefCD below, then the Riesz transform is bounded in L^p (\bM) for every p>1, that is ‖ √Γ((-L)^-1/2f)‖_p ≤ C_p ‖ f ‖_p, f ∈ C^∞₀(\bM), 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.

Related

• Sobolev, Poincare and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality
• Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries
• Riesz transforms on non-compact manifolds
• Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality
• Riesz transform on manifolds and heat kernel regularity