# Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds

### Xiang-Dong Li

Chinese Academy of Sciences, Beijing, China

## 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

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.,

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 $a\geq 0$).

## Cite this article

Xiang-Dong Li, Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds. Rev. Mat. Iberoam. 22 (2006), no. 2, pp. 591–648

DOI 10.4171/RMI/467