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.
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.
Keywords: Heat kernel, manifold with ends
Mot clés : noyau de la chaleur, variétés à bouts
@article{AIF_2009__59_5_1917_0, author = {Grigor{\textquoteright}yan, Alexander and Saloff-Coste, Laurent}, title = {Heat kernel on manifolds with ends}, journal = {Annales de l'Institut Fourier}, pages = {1917--1997}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {59}, number = {5}, year = {2009}, doi = {10.5802/aif.2480}, mrnumber = {2573194}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2480/} }
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%0 Journal Article %A Grigor’yan, Alexander %A Saloff-Coste, Laurent %T Heat kernel on manifolds with ends %J Annales de l'Institut Fourier %D 2009 %P 1917-1997 %V 59 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2480/ %R 10.5802/aif.2480 %G en %F AIF_2009__59_5_1917_0
Grigor’yan, Alexander; Saloff-Coste, Laurent. Heat kernel on manifolds with ends. Annales de l'Institut Fourier, Volume 59 (2009) no. 5, pp. 1917-1997. doi : 10.5802/aif.2480. http://www.numdam.org/articles/10.5802/aif.2480/
[1] 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] Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., Volume 73 (2003), pp. 890-896 | DOI | MR | Zbl
[3] Heat kernels and analysis on manifolds, graphs, and metric spaces, Contemporary Mathematics 338, AMS, 2003 | MR | Zbl
[4] 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] 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] 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] 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] Riesz transform and -cohomology for manifolds with Euclidean ends, Duke Math. J., Volume 133 (2006) no. 1, pp. 59-93 | DOI | MR | Zbl
[9] 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] 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] 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] A lower bound for the heat kernel, J. Diff. Geom., Volume 34 (1981), pp. 465-480 | MR | Zbl
[13] 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] Noyau de la chaleur et discrétisation d’une variété riemannienne, Israël J. Math., Volume 80 (1992), pp. 289-300 | DOI | MR | Zbl
[15] Itération de Moser et estimation Gaussienne du noyau de la chaleur, J. Operator Theory, Volume 29 (1993), pp. 157-165 | MR | Zbl
[16] Dimensions at infinity for Riemannian manifolds, Potential Anal., Volume 4 (1995) no. 5, pp. 335-344 | DOI | MR | Zbl
[17] Ultracontractivity and Nash type inequalities, J. Funct. Anal., Volume 141 (1996) no. 2, pp. 510-539 | DOI | MR | Zbl
[18] 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] Minorations pour les chaînes de Markov unidimensionnelles, Prob. Theory Relat. Fields, Volume 97 (1993), pp. 423-431 | DOI | MR | Zbl
[20] Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., Volume 109 (1987), pp. 319-334 | DOI | MR | Zbl
[21] 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] Heat kernels and spectral theory, Cambridge University Press, 1989 | MR | Zbl
[23] Heat kernel bounds, conservation of probability and the Feller property, J. d’Analyse Math., Volume 58 (1992), pp. 99-119 | DOI | MR | Zbl
[24] 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] Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc., Volume 55 (1997) no. 1, pp. 105-125 | DOI | MR | Zbl
[26] Sharp heat kernel bounds for some Laplace operators, Quart. J. Math., Volume 40 (1989), pp. 281-290 | DOI | MR | Zbl
[27] norms of non-critical Schrödinger semigroups, J. Funct. Anal., Volume 102 (1991), pp. 95-115 | DOI | MR | Zbl
[28] 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] The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math., Volume 12 (1959), pp. 1-11 | DOI | MR | Zbl
[30] 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] 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] Heat kernel on a manifold with a local Harnack inequality, Comm. Anal. Geom., Volume 2 (1994) no. 1, pp. 111-138 | MR | Zbl
[33] Heat kernel upper bounds on a complete non-compact manifold, Revista Matemática Iberoamericana, Volume 10 (1994) no. 2, pp. 395-452 | MR | Zbl
[34] Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom., Volume 45 (1997) no. 1, pp. 32-52 | MR | Zbl
[35] 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] Heat kernels on weighted manifolds and applications, Contemporary Mathematics, Volume 398 (2006), pp. 93-191 | Zbl
[37] Heat kernel on manifolds with parabolic ends (in preparation)
[38] Heat kernel upper bounds on manifolds with ends (in preparation)
[39] Surgery of Faber-Krahn inequalities and applications to heat kernel upper bounds on manifolds with ends (in preparation)
[40] Heat kernel on connected sums of Riemannian manifolds, Math. Research Letters, Volume 6 (1999) no. 3-4, pp. 307-321 | MR | Zbl
[41] Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math, Volume 55 (2002), pp. 93-133 | DOI | MR | Zbl
[42] Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures et Appl., Volume 81 (2002), pp. 115-142 | DOI | MR | Zbl
[43] Stability results for Harnack inequalities, Ann. Inst. Fourier, Grenoble, Volume 55 (2005) no. 3, pp. 825-890 | DOI | Numdam | MR | Zbl
[44] Sobolev Met Poincaré, Memoirs of the AMS 688, 2000 | MR | Zbl
[45] 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] 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] Green’s function, harmonic functions and volume comparison, J. Diff. Geom., Volume 41 (1995), pp. 227-318 | Zbl
[48] On the parabolic kernel of the Schrödinger operator, Acta Math., Volume 156 (1986) no. 3-4, pp. 153-201 | DOI | MR | Zbl
[49] 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] Global heat kernel bounds via desingularizing wieghts, J. Funct. Anal., Volume 212 (2004), pp. 373-398 | DOI | MR | Zbl
[51] Weighted Sobolev inequalities and Ricci flat manifolds (to appear in GAFA)
[52] 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] 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] The Laplacian on a Riemannian manifold, London Mathematical Society Student Texts 31, Cambridge University Press, 1997 | MR | Zbl
[55] A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices, Volume 2 (1992), pp. 27-38 | DOI | MR | Zbl
[56] Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., Volume 36 (1992), pp. 417-450 | MR | Zbl
[57] Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, Volume 4 (1995), pp. 429-467 | DOI | MR | Zbl
[58] Aspects of Sobolev inequalities, LMS Lecture Notes Series 289, Cambridge University Press, 2002 | MR | Zbl
[59] Spaces of harmonic functions, J. London Math. Soc. (2) (2000) no. 3, pp. 789-806 | DOI | MR | Zbl
[60] Brownian motion and random walks on manifolds, Ann. Inst. Fourier, Volume 34 (1984), pp. 243-269 | DOI | Numdam | MR | Zbl
[61] Hardy-Littlewood theory for semigroups, J. Funct. Anal., Volume 63 (1985) no. 2, pp. 240-260 | DOI | MR | Zbl
[62] Isoperimetric inequalities and Markov chains, J. Funct. Anal., Volume 63 (1985) no. 2, pp. 215-239 | DOI | MR | Zbl
[63] Random walks and Brownian motion on manifolds, Sympos. Math., Volume 29 (1986), pp. 97-109 | MR | Zbl
[64] 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] Global lower bound for the heat kernel of , Proceedings of the Amer. Math. Soc., Volume 129 (2000), pp. 1105-1112 | DOI | MR | Zbl
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