# 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, $L=Δ−∇ϕ⋅∇$ be a Markovian symmetric diffusion operator with an invariant measure $dμ(x)=e_{−ϕ(x)}dν(x)$, where $ϕ∈C_{2}(M)$, $ν$ is the Riemannian volume measure on $(M,g)$. A fundamental question in harmonic analysis and potential theory asks whether or not the Riesz transform $R_{a}(L)=∇(a−L)_{−1/2}$ is bounded in $L_{p}(μ)$ for all $1<p<∞$ and for certain $a≥0$. 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 $R_{a}(L)=∇(a−L)_{−1/2}$ is bounded in $L_{p}(μ)$ for all $a>0$ and $p≥2$ provided that $L$ generates a ultracontractive Markovian semigroup $P_{t}=e_{tL}$ in the sense that $P_{t}1=1$ for all $t≥0$, $∥P_{t}∥_{1,∞}<Ct_{−n/2}$ for all $t∈(0,1]$ for some constants $C>0$ and $n>1$, and satisfies

for some constants $c≥0$ and $ϵ>0$, where $K(x)$ denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature $Ric(L)=Ric+∇_{2}ϕ$ 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 $L_{p}(μ)$ for all $p≥2$ and for all $a>0$ (or even for all $a≥0$).

## 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