Heat kernel on manifolds with ends

Grigor’yan, Alexander; Saloff-Coste, Laurent

Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 1917-1997.

Nous obtenons des bornes inférieures et supérieures du noyau de la chaleur sur des variétés riemanniennes non-paraboliques à bouts, sous l’hypothèse que sur chaque bout, séparément, une estimation de type Li-Yau est vérifiée.

We prove two-sided estimates of heat kernels on non-parabolic Riemannian manifolds with ends, assuming that the heat kernel on each end separately satisfies the Li-Yau estimate.

Classification : 58J65, 31C12, 35K10, 60J60
Keywords: Heat kernel, manifold with ends
Mot clés : noyau de la chaleur, variétés à bouts

[1] Anker, J-Ph.; Ostellari, P. The heat kernel on noncompact symmetric spaces, in : Lie groups and symmetric spaces, Amer. Math. Soc. Transl. Ser. 2, 210 (2003), pp. 27-46 | MR | Zbl

[2] Aronson, D.G. Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., Volume 73 (2003), pp. 890-896 | DOI | MR | Zbl

[3] Auschern, P.; Coulhon, T.; Grigor’yan, A. Heat kernels and analysis on manifolds, graphs, and metric spaces, Contemporary Mathematics 338, AMS, 2003 | MR | Zbl

[4] Barlow, M.T. Diffusions on fractals, Lectures on Probability Theory and Statistics, Ecole d’été de Probabilités de Saint-Flour XXV - 1995 (Lecture Notes Math. 1690), Springer, 1998, pp. 1-121 | MR | Zbl

[5] Benjamini, I.; Chavel, I.; Feldman, E.A. Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash, Proceedings of London Math. Soc., Volume 72 (1996), pp. 215-240 | DOI | MR | Zbl

[6] Cai, M. Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc., Volume 24 (1991), pp. 371-377 | DOI | MR | Zbl

[7] Carlen, E. A.; Kusuoka, S.; Stroock, D. W. Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist., Volume 23 (1987) no. 2, suppl., pp. 245-287 | Numdam | MR | Zbl

[8] Carron, G.; Coulhon, T.; Hassell, A. Riesz transform and $L^p$-cohomology for manifolds with Euclidean ends, Duke Math. J., Volume 133 (2006) no. 1, pp. 59-93 | DOI | MR | Zbl

[9] Chavel, I.; Feldman, E.A. Isoperimetric constants, the geometry of ends, and large time heat diffusion in Riemannian manifolds, Proc London Math. Soc., Volume 62 (1991), pp. 427-448 | DOI | MR | Zbl

[10] Chavel, I.; Feldman, E.A. Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J., Volume 64 (1991) no. 3, pp. 473-499 | DOI | MR | Zbl

[11] Cheeger, J.; Gromov, M.; Taylor, M. Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom., Volume 17 (1982), pp. 15-53 | MR | Zbl

[12] Cheeger, J.; Yau, S.-T. A lower bound for the heat kernel, J. Diff. Geom., Volume 34 (1981), pp. 465-480 | MR | Zbl

[13] Cheng, S.Y.; Li, P.; Yau, S.-T. On the upper estimate of the heat kernel of a complete Riemannian manifold, Amer. J. Math., Volume 103 (1981) no. 5, pp. 1021-1063 | DOI | MR | Zbl

[14] Coulhon, T. Noyau de la chaleur et discrétisation d’une variété riemannienne, Israël J. Math., Volume 80 (1992), pp. 289-300 | DOI | MR | Zbl

[15] Coulhon, T. Itération de Moser et estimation Gaussienne du noyau de la chaleur, J. Operator Theory, Volume 29 (1993), pp. 157-165 | MR | Zbl

[16] Coulhon, T. Dimensions at infinity for Riemannian manifolds, Potential Anal., Volume 4 (1995) no. 5, pp. 335-344 | DOI | MR | Zbl

[17] Coulhon, T. Ultracontractivity and Nash type inequalities, J. Funct. Anal., Volume 141 (1996) no. 2, pp. 510-539 | DOI | MR | Zbl

[18] Coulhon, T.; Grigor’yan, A. On-diagonal lower bounds for heat kernels on non-compact manifolds and Markov chains, Duke Math. J., Volume 89 (1997) no. 1, pp. 133-199 | DOI | MR | Zbl

[19] Coulhon, T.; Saloff-Coste, L. Minorations pour les chaînes de Markov unidimensionnelles, Prob. Theory Relat. Fields, Volume 97 (1993), pp. 423-431 | DOI | MR | Zbl

[20] Davies, E.B. Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., Volume 109 (1987), pp. 319-334 | DOI | MR | Zbl

[21] Davies, E.B. Gaussian upper bounds for the heat kernel of some second-order operators on Riemannian manifolds, J. Funct. Anal., Volume 80 (1988), pp. 16-32 | DOI | MR | Zbl

[22] Davies, E.B. Heat kernels and spectral theory, Cambridge University Press, 1989 | MR | Zbl

[23] Davies, E.B. Heat kernel bounds, conservation of probability and the Feller property, J. d’Analyse Math., Volume 58 (1992), pp. 99-119 | DOI | MR | Zbl

[24] Davies, E.B. The state of art for heat kernel bounds on negatively curved manifolds, Bull. London Math. Soc., Volume 25 (1993), pp. 289-292 | DOI | MR | Zbl

[25] Davies, E.B. Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc., Volume 55 (1997) no. 1, pp. 105-125 | DOI | MR | Zbl

[26] Davies, E.B.; Pang, M.M.H. Sharp heat kernel bounds for some Laplace operators, Quart. J. Math., Volume 40 (1989), pp. 281-290 | DOI | MR | Zbl

[27] Davies, E.B.; Simon, B. $L^p$ norms of non-critical Schrödinger semigroups, J. Funct. Anal., Volume 102 (1991), pp. 95-115 | DOI | MR | Zbl

[28] Dodziuk, J. Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., Volume 32 (1983), pp. 703-716 | DOI | MR | Zbl

[29] Gaffney, M.P. The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math., Volume 12 (1959), pp. 1-11 | DOI | MR | Zbl

[30] Grigor’yan, A. On the fundamental solution of the heat equation on an arbitrary Riemannian manifold, (in Russian) Mat. Zametki, Volume 41 (1987) no. 3, pp. 687-692 Engl. transl. Math. Notes 41 (1987), no. 5-6, p. 386-389 | MR | Zbl

[31] Grigor’yan, A. The heat equation on non-compact Riemannian manifolds, (in Russian) Matem. Sbornik, Volume 182 (1991) no. 1, pp. 55-87 Engl. transl. Math. USSR Sb. 72 (1992), no. 1, p. 47-77 | Zbl

[32] Grigor’yan, A. Heat kernel on a manifold with a local Harnack inequality, Comm. Anal. Geom., Volume 2 (1994) no. 1, pp. 111-138 | MR | Zbl

[33] Grigor’yan, A. Heat kernel upper bounds on a complete non-compact manifold, Revista Matemática Iberoamericana, Volume 10 (1994) no. 2, pp. 395-452 | MR | Zbl

[34] Grigor’yan, A. Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom., Volume 45 (1997) no. 1, pp. 32-52 | MR | Zbl

[35] Grigor’yan, A. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., Volume 36 (1999), pp. 135-249 | DOI | MR | Zbl

[36] Grigor’yan, A. Heat kernels on weighted manifolds and applications, Contemporary Mathematics, Volume 398 (2006), pp. 93-191 | Zbl

[37] Grigor’yan, A.; Saloff-Coste, L. Heat kernel on manifolds with parabolic ends (in preparation)

[38] Grigor’yan, A.; Saloff-Coste, L. Heat kernel upper bounds on manifolds with ends (in preparation)

[39] Grigor’yan, A.; Saloff-Coste, L. Surgery of Faber-Krahn inequalities and applications to heat kernel upper bounds on manifolds with ends (in preparation)

[40] Grigor’yan, A.; Saloff-Coste, L. Heat kernel on connected sums of Riemannian manifolds, Math. Research Letters, Volume 6 (1999) no. 3-4, pp. 307-321 | MR | Zbl

[41] Grigor’yan, A.; Saloff-Coste, L. Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math, Volume 55 (2002), pp. 93-133 | DOI | MR | Zbl

[42] Grigor’yan, A.; Saloff-Coste, L. Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures et Appl., Volume 81 (2002), pp. 115-142 | DOI | MR | Zbl

[43] Grigor’yan, A.; Saloff-Coste, L. Stability results for Harnack inequalities, Ann. Inst. Fourier, Grenoble, Volume 55 (2005) no. 3, pp. 825-890 | DOI | Numdam | MR | Zbl

[44] Hajłasz, P.; Koskela, P. Sobolev Met Poincaré, Memoirs of the AMS 688, 2000 | MR | Zbl

[45] Kasue, A. Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature I., Geometry and Analysis on Manifolds (Katata/Kyoto, 1987) (Lecture Notes Math. 1339), Springer, 1988, pp. 158-181 | MR | Zbl

[46] Kuz’menko, Yu.T.; Molchanov, S.A. Counterexamples to Liouville-type theorems, (in Russian) Vestnik Moskov. Univ. Ser. I Mat. Mekh. (1979) no. 3, pp. 39-43 Engl. transl. Moscow Univ. Math. Bull. 34 (1979), p. 35-39 | MR | Zbl

[47] Li, P.; Tam, L.F. Green’s function, harmonic functions and volume comparison, J. Diff. Geom., Volume 41 (1995), pp. 227-318 | Zbl

[48] Li, P.; Yau, S.-T. On the parabolic kernel of the Schrödinger operator, Acta Math., Volume 156 (1986) no. 3-4, pp. 153-201 | DOI | MR | Zbl

[49] Liskevich, V.; Sobol, Z. Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients, Potential Analysis, Volume 18 (2003), pp. 359-390 | DOI | MR | Zbl

[50] Milman, P.D.; Semenov, Yu.A. Global heat kernel bounds via desingularizing wieghts, J. Funct. Anal., Volume 212 (2004), pp. 373-398 | DOI | MR | Zbl

[51] Minerbe, V. Weighted Sobolev inequalities and Ricci flat manifolds (to appear in GAFA)

[52] Molchanov, S.A. Diffusion processes and Riemannian geometry, (in Russian) Uspekhi Matem. Nauk, Volume 30 (1975) no. 1, pp. 3-59 Engl. transl. Russian Math. Surveys 30 (1975), no 1, p. 1-63 | MR | Zbl

[53] Porper, F.O.; Eidel’man, S.D. Two-side estimates of fundamental solutions of second-order parabolic equations and some applications, (in Russian) Uspekhi Matem. Nauk, Volume 39 (1984) no. 3, pp. 101-156 Engl. transl. Russian Math. Surveys 39 (1984), no 3, p. 119-178 | MR | Zbl

[54] Rosenberg, S. The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts 31, Cambridge University Press, 1997 | MR | Zbl

[55] Saloff-Coste, L. A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices, Volume 2 (1992), pp. 27-38 | DOI | MR | Zbl

[56] Saloff-Coste, L. Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., Volume 36 (1992), pp. 417-450 | MR | Zbl

[57] Saloff-Coste, L. Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, Volume 4 (1995), pp. 429-467 | DOI | MR | Zbl

[58] Saloff-Coste, L. Aspects of Sobolev inequalities, LMS Lecture Notes Series 289, Cambridge University Press, 2002 | MR | Zbl

[59] Sung, C.-J.; Tam, L.-F.; Wang, J. Spaces of harmonic functions, J. London Math. Soc. (2) (2000) no. 3, pp. 789-806 | DOI | MR | Zbl

[60] Varopoulos, N.Th. Brownian motion and random walks on manifolds, Ann. Inst. Fourier, Volume 34 (1984), pp. 243-269 | DOI | Numdam | MR | Zbl

[61] Varopoulos, N.Th. Hardy-Littlewood theory for semigroups, J. Funct. Anal., Volume 63 (1985) no. 2, pp. 240-260 | DOI | MR | Zbl

[62] Varopoulos, N.Th. Isoperimetric inequalities and Markov chains, J. Funct. Anal., Volume 63 (1985) no. 2, pp. 215-239 | DOI | MR | Zbl

[63] Varopoulos, N.Th. Random walks and Brownian motion on manifolds, Sympos. Math., Volume 29 (1986), pp. 97-109 | MR | Zbl

[64] Varopoulos, N.Th. Small time Gaussian estimates of heat diffusion kernel. I. The semigroup technique, Bull. Sci. Math.(2), Volume 113 (1989) no. 3, pp. 253-277 | MR | Zbl

[65] Zhang, Q. S. Global lower bound for the heat kernel of $-\Delta +\frac{c}{\left|x\right|2}$, Proceedings of the Amer. Math. Soc., Volume 129 (2000), pp. 1105-1112 | DOI | MR | Zbl