JournalsrmiVol. 22, No. 2pp. 591–648

Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds

  • Xiang-Dong Li

    Chinese Academy of Sciences, Beijing, China
Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds cover
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Abstract

Let (M,g)(M, g) be a complete Riemannian manifold, L=ΔϕL=\Delta -\nabla \phi \cdot \nabla be a Markovian symmetric diffusion operator with an invariant measure dμ(x)=eϕ(x)dν(x)d\mu(x)=e^{-\phi(x)}d\nu(x), where ϕC2(M)\phi\in C^2(M), ν\nu is the Riemannian volume measure on (M,g)(M, g). A fundamental question in harmonic analysis and potential theory asks whether or not the Riesz transform Ra(L)=(aL)1/2R_a(L)=\nabla(a-L)^{-1/2} is bounded in Lp(μ)L^p(\mu) for all 1<p<1<p<\infty and for certain a0a\geq 0. An affirmative answer to this problem has many important applications in elliptic or parabolic PDEs, potential theory, probability theory, the LpL^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 Ra(L)=(aL)1/2R_a(L)=\nabla(a-L)^{-1/2} is bounded in Lp(μ)L^p(\mu) for all a>0a>0 and p2p\geq 2 provided that LL generates a ultracontractive Markovian semigroup Pt=etLP_t=e^{tL} in the sense that Pt1=1P_t 1=1 for all t0t\geq 0, Pt1,<Ctn/2\|P_t\|_{1, \infty} < Ct^{-n/2} for all t(0,1]t\in (0, 1] for some constants C>0C>0 and n>1n > 1, and satisfies

(K+c)Ln2+ϵ(M,μ)(K+c)^{-}\in L^{{n\over 2}+\epsilon}(M, \mu)

for some constants c0c\geq 0 and ϵ>0\epsilon>0, where K(x)K(x) denotes the lowest eigenvalue of the Bakry-Emery Ricci curvature Ric(L)=Ric+2ϕRic(L)=Ric+\nabla^2\phi on TxMT_x M, i.e.,

K(x)=inf{Ric(L)(v,v):vTxM,v=1}, xM.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 Ra(L)R_a(L) is bounded in Lp(μ)L^p(\mu) for all p2p\geq 2 and for all a>0a>0 (or even for all a0a\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