Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds

Abstract

Let (M, g) be a complete Riemannian manifold, L=\Delta -\nabla \phi \cdot \nabla be a Markovian symmetric diffusion operator with an invariant measure d\mu(x)=e^{-\phi(x)}d\nu(x) , where \phi\in C^2(M) , \nu is the Riemannian volume measure on (M, g) . A fundamental question in harmonic analysis and potential theory asks whether or not the Riesz transform R_a(L)=\nabla(a-L)^{-1/2} is bounded in L^p(\mu) for all 1<p<\infty and for certain a\geq 0 . An affirmative answer to this problem has many important applications in elliptic or parabolic PDEs, potential theory, probability theory, the L^p -Hodge decomposition theory and in the study of Navier-Stokes equations and boundary value problems. Using some new interplays between harmonic analysis, differential geometry and probability theory, we prove that the Riesz transform R_a(L)=\nabla(a-L)^{-1/2} is bounded in L^p(\mu) for all a>0 and p\geq 2 provided that L generates a ultracontractive Markovian semigroup P_t=e^{tL} in the sense that P_t 1=1 for all t\geq 0 , \|P_t\|_{1, \infty} < Ct^{-n/2} for all t\in (0, 1] for some constants C>0 and n > 1 , and satisfies (K+c)^{-}\in L^{{n\over 2}+\epsilon}(M, \mu) for some constants c\geq 0 and \epsilon>0 , where K(x) denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature Ric(L)=Ric+\nabla^2\phi on T_x M , i.e., K(x)=\inf\limits\{Ric(L)(v, v): v\in T_x M, \|v\|=1\}, \quad\forall\ x\in M. Examples of diffusion operators on complete non-compact Riemannian manifolds with unbounded negative Ricci curvature or Bakry-Emery Ricci curvature are given for which the Riesz transform R_a(L) is bounded in L^p(\mu) for all p\geq 2 and for all a>0 (or even for all a\geq 0 ).