Revista Matemática Iberoamericana

Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds

Xiang Dong Li

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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 $a\geq 0$).

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Rev. Mat. Iberoamericana, Volume 22, Number 2 (2006), 591-648.

First available in Project Euclid: 26 October 2006

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Primary: 31C12: Potential theory on Riemannian manifolds [See also 53C20; for Hodge theory, see 58A14] 53C20: Global Riemannian geometry, including pinching [See also 31C12, 58B20] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60H30: Applications of stochastic analysis (to PDE, etc.)

Bakry-Emery Ricci curvature diffusion operator Riesz transform ultracontractivity


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

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