Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds

Abstract

Let $(M,g)$ be a complete Riemannian manifold, $L=\Delta -\nabla \varphi \cdot \nabla$ be a Markovian symmetric diffusion operator with an invariant measure $d\mu (x)=e^{-\varphi (x)}d\nu (x)$, where $\varphi \in C^2(M)$, $\nu$ 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 $a\ge 0$. 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 $p\ge 2$ provided that $L$ generates a ultracontractive Markovian semigroup $P_t=e^{tL}$ in the sense that $P_{t}1=1$ for all $t\ge 0$, $\Vert P_t{\Vert }_{1,\infty }<C{t}^{-n/2}$ for all $t\in (0,1]$ for some constants $C>0$ and $n>1$, and satisfies

for some constants $c\ge 0$ and $ϵ>0$, where $K(x)$ denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature $Ric(L)=Ric+{\nabla }^2\varphi$ 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 $p\ge 2$ and for all $a>0$ (or even for all $a\ge 0$).