Abstract
Let (X,d,μ) be a R C D ∗(K,N) space with \(K\in \mathbb {R}\) and N∈[1,∞). We derive the upper and lower bounds of the heat kernel on (X,d,μ) by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the L p boundedness of (local) Riesz transforms.
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Ambrosio, L., Gigli, N., Mondino, A., Rajala, T.: Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure. Trans. Amer. Math. Soc. 367, 4661–4701 (2015)
Ambrosio, L., Gigli, N., Savaré, G.: Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Mathematics ETH Zürich. Basel, Birkhäuser (2008)
Ambrosio, L., Gigli, N., Savaré, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163, 1405–1490 (2014)
Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195, 289–391 (2014)
Ambrosio, L., Gigli, N., Savaré, G.: Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43, 339–404 (2015)
Ambrosio, L., Gigli, N., Savaré, G.: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. 29, 969–996 (2013)
Ambrosio, L., Mondino, A., Savaré, G.: On the Bakry-Émery condition, the gradient estimates and the Local-to-Global property of R C D ∗(K,N) metric measure spaces, to appear in Journal of Geometric Analysis. doi:10.1007/s12220-014-9537-7(arXiv:1309.4664)
Ambrosio, L., Mondino, A., Savaré, G.: Nonlinear diffusion equations and curvature conditions in metric measure spaces. pp. 1–108. arXiv:1509.07273 (2015)
Auscher, P., Coulhon, T., Duong, X.T., Hofmann, S.: Riesz transform on manifolds and heat kernel regularity. Ann. Sci. École Norm. Sup. (4) 37, 911–957 (2004)
Baudoin, F., Garofalo, N.: A note on the boundedness of Riesz transform for some subelliptic operators. Int. Math. Res. Not. 2013(2), 398–421 (2012)
Bernicot, F., Coulhon, T., Frey, D.: Gaussian heat kernel bounds through elliptic Moser iteration. arXiv:1407.3906
Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)
Coulhon, T., Sikora, A.: Gaussian heat kernel bounds via Phragmén-Lindelöf theorem. Proc. London Math. Soc. 3(96), 507–544 (2008)
Coulhon, T., Duong, X.T.: Riesz transforms for 1≤p≤2. Trans. Amer. Math. Soc. 351, 1151–1169 (1999)
Davies, E.B.: Non-Gaussian aspects of heat kernel behaviour. J. London Math. Soc. (2) 55, 105–125 (1997)
Duong, X.T., Robinson, D.W.: Semigroup kernels, Poisson bounds and holomorphic functional calculus. J. Funct. Anal. 142(1), 89–128 (1996)
Erbar, M., Kuwada, K., Sturm, K.T.: On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent. Math. 201, 993–1071 (2015)
Garofalo, N., Mondino, A.: Li-Yau and Harnack type inequalities in R C D ∗(K,N) metric measure spaces. Nonlinear Anal. 95, 721–734 (2014)
Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Amer. Math. Soc., vol. 236 (2015). no. 1113
Gong, F.-Z., Wang, F.-Y.: Heat kernel estimates with application to compactness of manifolds. Quart. J. Math. 52, 171–180 (2001)
Hajłasz, P., Koskela, P.: Sobolev meets Poincaré. C. R. Acad. Sci. Paris Sér. I Math. 320(10), 1211–1215 (1995)
Jiang, R.: The Li-Yau inequality and heat kernels on metric measure spaces. J. Math. Pures Appl. (9) 104, 29–57 (2015)
Li, H.Q.: Dimension-Free Harnack inequalities on R C D(K,∞) spaces, to appear in J. Theor. Probab.
Li, P.: Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Ann. of Math. (2) 124, 1–21 (1986)
Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)
Lierl, J., Saloff-Coste, L.: Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms. arXiv:1205.6493
Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. (2) 169, 903–991 (2009)
Mondino, A., Naber, A.: Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds I. arXiv:1405.2222
Rajala, T.: Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc. Var. 44, 477–494 (2012)
Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 243–279 (2000)
Sikora, A.: Riesz transform, Gaussian bounds and the method of wave equation. Math. Z. 247, 643–662 (2004)
Sturm, K.T.: Heat kernel bounds on manifolds. Math. Ann. 292, 149–162 (1992)
Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)
Sturm, K.T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32(2), 275–312 (1995)
Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75(3), 273–297 (1996)
Sturm, K.T.: On the geometry of metric measure spaces I. Acta Math. 196, 65–131 (2006)
Sturm, K.T.: On the geometry of metric measure spaces II. Acta Math. 196, 133–177 (2006)
Wang, F.-Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields 109, 417–424 (1997)
Xu, G.Y.: Large time behavior of the heat kernel. J. Differ. Geom. 98, 467–528 (2014)
Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1978)
Zhang, H.C., Zhu, X.-P.: On a new definition of Ricci curvature on Alexandrov spaces. Acta Math. Sci. Ser. B Engl. Ed. 30, 1949–1974 (2010)
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R.J. is partially supported by NSFC (No. 11301029); H.L. is partially supported by NSFCs (No. 11401403 and No. 11371099) and the ARC grant (DP130101302); H.Z. is partially supported by NSFC (No. 11201492) and by Guangdong Natural Science Foundation (No. S2012040007550).
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Jiang, R., Li, H. & Zhang, H. Heat Kernel Bounds on Metric Measure Spaces and Some Applications. Potential Anal 44, 601–627 (2016). https://doi.org/10.1007/s11118-015-9521-2
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DOI: https://doi.org/10.1007/s11118-015-9521-2