Abstract
In this last chapter of Part II, we make full use of the Γ-calculus technique to establish important functional inequalities. We have already shown the Poincaré–Lichnerowicz inequality in the previous chapter. In this chapter we further obtain the logarithmic Sobolev inequality, the Beckner inequality, and the Sobolev inequality. We will closely follow the arguments in the linear (Riemannian) case and generalize them to our nonlinear (Finsler) setting. All the estimates will be of the same forms as the linear case, except for the range of the exponent p adopted in the Sobolev inequality in the non-reversible case. We remark that, in contrast with the gradient estimates in Chaps. 14 and 15, linearized heat semigroups will play only a subsidiary role.
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References
Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Springer, Cham (2014)
Bakry, D., Gentil, I., Scheffer, G.: Sharp Beckner-type inequalities for Cauchy and spherical distributions. Studia Math. 251, 219–245 (2020)
Cavalletti, F., Mondino, A.: Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds. Geom. Topol. 21, 603–645 (2017)
Chow, B., Chu, S.-C., Glickenstein, D., Guenther, C., Isenberg, J., Ivey, T., Knopf, D., Lu, P., Luo, F., Ni, L.: The Ricci Flow: Techniques and Applications. Part I. Geometric Aspects. American Mathematical Society, Providence, RI (2007)
Chow, B., Lu, P., Ni, L.: Hamilton’s Ricci Flow. American Mathematical Society, Providence, RI; Science Press Beijing, New York (2006)
Duoandikoetxea, J.: Fourier Analysis. American Mathematical Society, Providence, RI (2001)
Gentil, I., Zugmeyer, S.: A family of Beckner inequalities under various curvature-dimension conditions. Bernoulli 27, 751–771 (2021)
Hebey, E.: Sobolev Spaces on Riemannian Manifolds. Lecture Notes in Mathematics, vol. 1635. Springer, Berlin (1996)
Hebey, E.: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI (1999)
Ledoux, M.: The Concentration of Measure Phenomenon. American Mathematical Society, Providence, RI (2001)
Mai, C.H.: On Riemannian manifolds with positive weighted Ricci curvature of negative effective dimension. Kyushu J. Math. 73, 205–218 (2019)
Milman, E.: Beyond traditional curvature-dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension. Trans. Am. Math. Soc. 369, 3605–3637 (2017)
Nguyen, V.H.: Φ-entropy inequalities and asymmetric covariance estimates for convex measures. Bernoulli 25, 3090–3108 (2019)
Ohta, S.: Some functional inequalities on non-reversible Finsler manifolds. Proc. Indian Acad. Sci. Math. Sci. 127, 833–855 (2017)
Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. Preprint (2002). Available at arXiv:math/0211159
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Ohta, Si. (2021). Functional Inequalities. In: Comparison Finsler Geometry. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-80650-7_16
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DOI: https://doi.org/10.1007/978-3-030-80650-7_16
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