Gaussian Estimates for the Heat Kernel of the Weighted Laplacian and Fractal Measures

Setti, Alberto G.

Canadian journal of mathematics, Tome 44 (1992) no. 5, pp. 1061-1078

Résumé

Let 0 < wbe a smooth function on a complete Riemannian manifold Mn, and define L = — Δ — ▽ (log w) and Rw =Ric - w -1 Hess w.In this paper we show that if Rw ≥ —nK, (K ≥0), then the positive solutions of (L + ∂/∂t)u —0 satisfy a gradient estimate of the same form as that obtained by Li and Yau ([LY]) when Lis the Laplacian. This is used to obtain a parabolic Harnack inequality, which in turn, yields upper and lower Gaussian estimates for the heat kernel of L.The results obtained are applied to study the LPmapping properties of t→ e-tL μfor measures μ which are α-dimensional in a sense that generalises the local uniform α-dimensionality introduced by R. S. Strichartz ([St2], [St3]).

Détail

DOI : 10.4153/CJM-1992-065-4

Mots-clés : Primary: 58G11, 58C35, secondary: 35K05, 28A75

Bibliographie

[Bkl] Bakry, D., Étude des transformations de Riesz dans les variétés riemannienne à courbure de Ricci minorée. In: Séminaire de Probabilités XXI, Lecture Notes in Math. 1247 Springer-Verlag, Berlin-Heidelberg, 1987, 137–172. Google Scholar

[Bk2] Bakry, D., Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complete, C.R. Acad. Se. Paris 303(1986), 23–26. Google Scholar

[BE] Bakry, D. and Emery, M., Diffusions hypercontractives. In: Séminaire de Probabilités XIX, Lecture Notes in Math. 1123, Springer-Verlag, Berlin-Heidelberg, 1985, 179–206. Google Scholar

[Ca] Calabi, E., An extension of E. Hopf's maximum principle with an application to Riemannian geometry, Duke Math. J. 25 (1958), 45–56. Google Scholar

[CG] Cheeger, J. and Gromoll, D., The splitting theorem for manifolds with nonne gative Ricci curvature, J. Diff. Geom. 6(1971), 119–128. Google Scholar

[CGT] Cheeger, J., Gromoll, D., Gromov, M. and Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds J. Diff. Geom. 17(1982), 15–53. Google Scholar

[CY] Cheng, S.Y. and Yau, S.T., Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math. 28(1975), 333–354. Google Scholar

[Dl] Davies, E.B., Heat Kernels and Spectral Theory, Cambridge University Press, 1989. Google Scholar

[D2] Davies, E.B., Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math. 109(1987), 319–334. Google Scholar

[D3] Davies, E.B., Heat kernel bounds for second order elliptic operators on Riemannian manifolds, Amer. J. Math. 109(1987), 545–570. Google Scholar

[D4] Davies, E.B., Gaussian upper bounds for the heat kernel of some second order operators on Riemannian manifolds, J. Funct. Anal. 80(1988), 16–32. Google Scholar

[D5] Davies, E.B., Heat kernel bounds, conservation of probability and the Feller property, preprint. Google Scholar

[DS] Deuschel, J.D. and Stroock, D., Hypercontractivity and spectral gap of symmetric diffusion with applications to the stochastic Ising model, J. Funct. Anal. 92(1990), 30–48. Google Scholar

[F] Fukushima, M., Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980. Google Scholar

[Ka] Kannai, Y., Off diagonal short time asymptoticsfor fundamental solutions of diffusion equations, Comm. in P.D.E. 2(1977),781–830. Google Scholar

[La] Lau, K.S., Fractal measures and mean p-variations, preprint, 1990. Google Scholar

[Le] Lebedev, N.N., Special Functions and Their Applications, Dover, New York, 1972. Google Scholar

[LY] Li, P. and Yau, S.T., On the parabolic kernel of the Schrôdinger operator, Acta Math. 156(1986), 153–201. Google Scholar

[MkS] McKean, H.P. and Singer, I.M., Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. 1(1967), 43–69. Google Scholar

[RS] Reed, M. and Simon, B., Methods of Modem Mathematical Physics. II, Academic Press, New York, 1975. Google Scholar

[S] Setti, A., Eigenvalue estimates for the weighted Laplacian on a Riemannian manifold, preprint, 1990. Google Scholar

[Stl] Strichartz, R.S., Analysis of the Laplacian on the complete Riemannian manifold, J. Funct Anal.. 52(1983), 48–79. Google Scholar

[St2] Strichartz, R.S., Spectral asymptotics of fractal measures, J. Funct. Anal. 89(1990), 154–187. Google Scholar

[St3] Strichartz, R.S., Spectral asymptotics of fractal measures on Riemannian manifolds, J. Funct. Anal., to appear. Google Scholar

[V] Varopoulos, N.T., Small time Gaussian estimates of heat diffusion kernels. Part I: The semigroup technique, Bull. Sc. Math.. 113(1989), 253–277. Google Scholar

[Y] Yau, S.T., Some function theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J.. 25(1976), 659–670. Google Scholar