Heat Kernel Bounds on Metric Measure Spaces and Some Applications

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)