Heat kernel on manifolds with ends
Annales de l'Institut Fourier, Volume 59 (2009) no. 5, p. 1917-1997

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.

DOI : https://doi.org/10.5802/aif.2480
Classification:  58J65,  31C12,  35K10,  60J60
Keywords: Heat kernel, manifold with ends
@article{AIF_2009__59_5_1917_0,
     author = {Grigor'yan, Alexander and Saloff-Coste, Laurent},
     title = {Heat kernel on manifolds with ends},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {5},
     year = {2009},
     pages = {1917-1997},
     doi = {10.5802/aif.2480},
     mrnumber = {2573194},
     zbl = {pre05641405},
     language = {en},
     url = {http://www.numdam.org/item/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/item/AIF_2009__59_5_1917_0/

[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 2018351 | Zbl 1036.22005

[2] Aronson, D.G. Bounds for the fundamental solution of a parabolic equation, Bull. Amer. Math. Soc., Tome 73 (2003), pp. 890-896 | Article | MR 217444 | Zbl 0153.42002

[3] Auschern, P.; Coulhon, T.; Grigor’Yan, A. Heat kernels and analysis on manifolds, graphs, and metric spaces, AMS, Contemporary Mathematics 338 (2003) | MR 2041910 | Zbl 1029.00030

[4] Barlow, M.T. Diffusions on fractals, Lectures on Probability Theory and Statistics, Ecole d’été de Probabilités de Saint-Flour XXV - 1995, Springer (Lecture Notes Math. 1690) (1998), pp. 1-121 | MR 1668115 | Zbl 0916.60069

[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., Tome 72 (1996), pp. 215-240 | Article | MR 1357093 | Zbl 0853.58098

[6] Cai, M. Ends of Riemannian manifolds with nonnegative Ricci curvature outside a compact set, Bull. Amer. Math. Soc., Tome 24 (1991), pp. 371-377 | Article | MR 1071028 | Zbl 0728.53026

[7] Carlen, E. A.; Kusuoka, S.; Stroock, D. W. Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist., Tome 23 (1987) no. 2, suppl., pp. 245-287 | Numdam | MR 898496 | Zbl 0634.60066

[8] Carron, G.; Coulhon, T.; Hassell, A. Riesz transform and L p -cohomology for manifolds with Euclidean ends, Duke Math. J., Tome 133 (2006) no. 1, pp. 59-93 | Article | MR 2219270 | Zbl 1106.58021

[9] Chavel, I.; Feldman, E.A. Isoperimetric constants, the geometry of ends, and large time heat diffusion in Riemannian manifolds, Proc London Math. Soc., Tome 62 (1991), pp. 427-448 | Article | MR 1085648 | Zbl 0723.58048

[10] Chavel, I.; Feldman, E.A. Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J., Tome 64 (1991) no. 3, pp. 473-499 | Article | MR 1141283 | Zbl 0753.58031

[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., Tome 17 (1982), pp. 15-53 | MR 658471 | Zbl 0493.53035

[12] Cheeger, J.; Yau, S.-T. A lower bound for the heat kernel, J. Diff. Geom., Tome 34 (1981), pp. 465-480 | MR 615626 | Zbl 0481.35003

[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., Tome 103 (1981) no. 5, pp. 1021-1063 | Article | MR 630777 | Zbl 0484.53035

[14] Coulhon, T. Noyau de la chaleur et discrétisation d’une variété riemannienne, Israël J. Math., Tome 80 (1992), pp. 289-300 | Article | MR 1202573 | Zbl 0772.58055

[15] Coulhon, T. Itération de Moser et estimation Gaussienne du noyau de la chaleur, J. Operator Theory, Tome 29 (1993), pp. 157-165 | MR 1277971 | Zbl 0882.47014

[16] Coulhon, T. Dimensions at infinity for Riemannian manifolds, Potential Anal., Tome 4 (1995) no. 5, pp. 335-344 | Article | MR 1354888 | Zbl 0847.53022

[17] Coulhon, T. Ultracontractivity and Nash type inequalities, J. Funct. Anal., Tome 141 (1996) no. 2, pp. 510-539 | Article | MR 1418518 | Zbl 0887.58009

[18] Coulhon, T.; Grigor’Yan, A. On-diagonal lower bounds for heat kernels on non-compact manifolds and Markov chains, Duke Math. J., Tome 89 (1997) no. 1, pp. 133-199 | Article | MR 1458975 | Zbl 0920.58064

[19] Coulhon, T.; Saloff-Coste, L. Minorations pour les chaînes de Markov unidimensionnelles, Prob. Theory Relat. Fields, Tome 97 (1993), pp. 423-431 | Article | MR 1245253 | Zbl 0792.60063

[20] Davies, E.B. Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., Tome 109 (1987), pp. 319-334 | Article | MR 882426 | Zbl 0659.35009

[21] Davies, E.B. Gaussian upper bounds for the heat kernel of some second-order operators on Riemannian manifolds, J. Funct. Anal., Tome 80 (1988), pp. 16-32 | Article | MR 960220 | Zbl 0759.58045

[22] Davies, E.B. Heat kernels and spectral theory, Cambridge University Press (1989) | MR 990239 | Zbl 0699.35006

[23] Davies, E.B. Heat kernel bounds, conservation of probability and the Feller property, J. d’Analyse Math., Tome 58 (1992), pp. 99-119 | Article | MR 1226938 | Zbl 0808.58041

[24] Davies, E.B. The state of art for heat kernel bounds on negatively curved manifolds, Bull. London Math. Soc., Tome 25 (1993), pp. 289-292 | Article | MR 1209255 | Zbl 0802.58053

[25] Davies, E.B. Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc., Tome 55 (1997) no. 1, pp. 105-125 | Article | MR 1423289 | Zbl 0879.35064

[26] Davies, E.B.; Pang, M.M.H. Sharp heat kernel bounds for some Laplace operators, Quart. J. Math., Tome 40 (1989), pp. 281-290 | Article | MR 1010819 | Zbl 0701.35004

[27] Davies, E.B.; Simon, B. L p norms of non-critical Schrödinger semigroups, J. Funct. Anal., Tome 102 (1991), pp. 95-115 | Article | MR 1138839 | Zbl 0743.47047

[28] Dodziuk, J. Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J., Tome 32 (1983), pp. 703-716 | Article | MR 711862 | Zbl 0526.58047

[29] Gaffney, M.P. The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math., Tome 12 (1959), pp. 1-11 | Article | MR 102097 | Zbl 0102.09202

[30] Grigor’Yan, A. On the fundamental solution of the heat equation on an arbitrary Riemannian manifold, (in Russian) Mat. Zametki, Tome 41 (1987) no. 3, pp. 687-692 (Engl. transl. Math. Notes 41 (1987), no. 5-6, p. 386-389) | MR 898129 | Zbl 0661.58030

[31] Grigor’Yan, A. The heat equation on non-compact Riemannian manifolds, (in Russian) Matem. Sbornik, Tome 182 (1991) no. 1, pp. 55-87 (Engl. transl. Math. USSR Sb. 72 (1992), no. 1, p. 47-77) | Zbl 0743.58031

[32] Grigor’Yan, A. Heat kernel on a manifold with a local Harnack inequality, Comm. Anal. Geom., Tome 2 (1994) no. 1, pp. 111-138 | MR 1312681 | Zbl 0845.58056

[33] Grigor’Yan, A. Heat kernel upper bounds on a complete non-compact manifold, Revista Matemática Iberoamericana, Tome 10 (1994) no. 2, pp. 395-452 | MR 1286481 | Zbl 0810.58040

[34] Grigor’Yan, A. Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom., Tome 45 (1997) no. 1, pp. 32-52 | MR 1443330 | Zbl 0865.58042

[35] Grigor’Yan, A. Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., Tome 36 (1999), pp. 135-249 | Article | MR 1659871 | Zbl 0927.58019

[36] Grigor’Yan, A. Heat kernels on weighted manifolds and applications, Contemporary Mathematics, Tome 398 (2006), pp. 93-191 | Zbl 1106.58016

[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, Tome 6 (1999) no. 3-4, pp. 307-321 | MR 1713132 | Zbl 0957.58023

[41] Grigor’Yan, A.; Saloff-Coste, L. Dirichlet heat kernel in the exterior of a compact set, Comm. Pure Appl. Math, Tome 55 (2002), pp. 93-133 | Article | MR 1857881 | Zbl 1037.58018

[42] Grigor’Yan, A.; Saloff-Coste, L. Hitting probabilities for Brownian motion on Riemannian manifolds, J. Math. Pures et Appl., Tome 81 (2002), pp. 115-142 | Article | MR 1994606 | Zbl 1042.58022

[43] Grigor’Yan, A.; Saloff-Coste, L. Stability results for Harnack inequalities, Ann. Inst. Fourier, Grenoble, Tome 55 (2005) no. 3, pp. 825-890 | Article | Numdam | MR 2149405 | Zbl 1115.58024

[44] Hajłasz, P.; Koskela, P. Sobolev Met Poincaré, Memoirs of the AMS 688 (2000) | MR 1683160 | Zbl 0954.46022

[45] Kasue, A. Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature I., Geometry and Analysis on Manifolds (Katata/Kyoto, 1987), Springer (Lecture Notes Math. 1339) (1988), pp. 158-181 | MR 961480 | Zbl 0685.31004

[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 561406 | Zbl 0416.35033

[47] Li, P.; Tam, L.F. Green’s function, harmonic functions and volume comparison, J. Diff. Geom., Tome 41 (1995), pp. 227-318 | Zbl 0827.53033

[48] Li, P.; Yau, S.-T. On the parabolic kernel of the Schrödinger operator, Acta Math., Tome 156 (1986) no. 3-4, pp. 153-201 | Article | MR 834612 | Zbl 0611.58045

[49] Liskevich, V.; Sobol, Z. Estimates of integral kernels for semigroups associated with second-order elliptic operators with singular coefficients, Potential Analysis, Tome 18 (2003), pp. 359-390 | Article | MR 1953267 | Zbl 1023.35041

[50] Milman, P.D.; Semenov, Yu.A. Global heat kernel bounds via desingularizing wieghts, J. Funct. Anal., Tome 212 (2004), pp. 373-398 | Article | MR 2064932 | Zbl 1057.47043

[51] Minerbe, V. Weighted Sobolev inequalities and Ricci flat manifolds (to appear in GAFA) | Zbl pre05528908

[52] Molchanov, S.A. Diffusion processes and Riemannian geometry, (in Russian) Uspekhi Matem. Nauk, Tome 30 (1975) no. 1, pp. 3-59 (Engl. transl. Russian Math. Surveys 30 (1975), no 1, p. 1-63) | MR 413289 | Zbl 0315.53026

[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, Tome 39 (1984) no. 3, pp. 101-156 (Engl. transl. Russian Math. Surveys 39 (1984), no 3, p. 119-178) | MR 747792 | Zbl 0582.35052

[54] Rosenberg, S. The Laplacian on a Riemannian manifold, Cambridge University Press, London Mathematical Society Student Texts 31 (1997) | MR 1462892 | Zbl 0868.58074

[55] Saloff-Coste, L. A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices, Tome 2 (1992), pp. 27-38 | Article | MR 1150597 | Zbl 0769.58054

[56] Saloff-Coste, L. Uniformly elliptic operators on Riemannian manifolds, J. Diff. Geom., Tome 36 (1992), pp. 417-450 | MR 1180389 | Zbl 0735.58032

[57] Saloff-Coste, L. Parabolic Harnack inequality for divergence form second order differential operators, Potential Analysis, Tome 4 (1995), pp. 429-467 | Article | MR 1354894 | Zbl 0840.31006

[58] Saloff-Coste, L. Aspects of Sobolev inequalities, Cambridge University Press, LMS Lecture Notes Series 289 (2002) | MR 1872526 | Zbl 0991.35002

[59] Sung, C.-J.; Tam, L.-F.; Wang, J. Spaces of harmonic functions, J. London Math. Soc. (2) (2000) no. 3, pp. 789-806 | Article | MR 1766105 | Zbl 0963.31004

[60] Varopoulos, N.Th. Brownian motion and random walks on manifolds, Ann. Inst. Fourier, Tome 34 (1984), pp. 243-269 | Article | Numdam | MR 746500 | Zbl 0523.60071

[61] Varopoulos, N.Th. Hardy-Littlewood theory for semigroups, J. Funct. Anal., Tome 63 (1985) no. 2, pp. 240-260 | Article | MR 803094 | Zbl 0608.47047

[62] Varopoulos, N.Th. Isoperimetric inequalities and Markov chains, J. Funct. Anal., Tome 63 (1985) no. 2, pp. 215-239 | Article | MR 803093 | Zbl 0573.60059

[63] Varopoulos, N.Th. Random walks and Brownian motion on manifolds, Sympos. Math., Tome 29 (1986), pp. 97-109 | MR 951181 | Zbl 0651.60013

[64] Varopoulos, N.Th. Small time Gaussian estimates of heat diffusion kernel. I. The semigroup technique, Bull. Sci. Math.(2), Tome 113 (1989) no. 3, pp. 253-277 | MR 1016211 | Zbl 0703.58052

[65] Zhang, Q. S. Global lower bound for the heat kernel of -Δ+c |x|2, Proceedings of the Amer. Math. Soc., Tome 129 (2000), pp. 1105-1112 | Article | MR 1814148 | Zbl 0964.35024