Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds. (English) Zbl 1119.53022

Li, Xiang Dong

Rev. Mat. Iberoam. 22, No. 2, 591-648 (2006).

Summary: 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 \[ (K+c)^{-}\in L^{{n\over 2}+\varepsilon}(M, \mu) \] for some constants \(c\geq 0\) and \(\varepsilon>0\), 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 \(a\geq 0\)).

MSC:

53C20	Global Riemannian geometry, including pinching
31C12	Potential theory on Riemannian manifolds and other spaces
58J65	Diffusion processes and stochastic analysis on manifolds
60H30	Applications of stochastic analysis (to PDEs, etc.)

Keywords:

Bakry-Emery Ricci curvature; diffusion operator; Riesz transform; ultracontractivity

References:

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