Let be a complete Riemannian manifold, be a Markovian symmetric diffusion operator with an invariant measure , where , is the Riemannian volume measure on . A fundamental question in harmonic analysis and potential theory asks whether or not the Riesz transform is bounded in for all and for certain . An affirmative answer to this problem has many important applications in elliptic or parabolic PDEs, potential theory, probability theory, the -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 is bounded in for all and provided that generates a ultracontractive Markovian semigroup in the sense that for all , for all for some constants and , and satisfies
for some constants and , where denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature on , i.e.,
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 is bounded in for all and for all (or even for all ).
Cite this article
Xiang-Dong Li, Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds. Rev. Mat. Iberoam. 22 (2006), no. 2, pp. 591–648DOI 10.4171/RMI/467