## Log-Sobolev inequalities: different roles of Ric and Hess.(English)Zbl 1187.60061

From the author’s abstract: Let $$P_t$$ be the diffusion semigroup generated by $$L:=\Delta +\nabla\Delta$$ on a complete connected Riemannian manifold with $$\mathrm{Ric}\geq -(\sigma^2\rho_0^2+c)$$ for some constants $$\sigma, c>0$$ and $$\rho_0$$ the Riemannian distance to a fixed point. It is shown that $$P_t$$ is hypercontractive, or the log–Sobolev inequality holds for the associated Dirichlet form, provided $$-\mathrm{Hess}_V\geq \delta$$ holds outside of a compact set for some constant $$\delta>(1+\sqrt 2)\sigma\sqrt{d-1}$$. This indicates that $$\mathrm{Ric}$$ and $$-\mathrm{Hess}_V$$ play quite different roles for the log–Sobolev inequality to hold.

### MSC:

 60J60 Diffusion processes 58J32 Boundary value problems on manifolds
Full Text:

### References:

 [1] Aida, S. (1998). Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158 152-185. · Zbl 0914.47041 [2] Aida, S., Masuda, T. and Shigekawa, I. (1994). Logarithmic Sobolev inequalities and exponential integrability. J. Funct. Anal. 126 83-101. · Zbl 0846.46020 [3] Arnaudon, M., Thalmaier, A. and Wang, F.-Y. (2006). Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130 223-233. · Zbl 1089.58024 [4] Bakry, D. and Émery, M. (1984). Hypercontractivité de semi-groupes de diffusion. C. R. Acad. Sci. Paris Sér. I Math. 299 775-778. · Zbl 0563.60068 [5] Bogachev, V. I., Röckner, M. and Wang, F.-Y. (2001). Elliptic equations for invariant measures on finite and infinite dimensional manifolds. J. Math. Pures Appl. 80 177-221. · Zbl 0996.58023 [6] Bismut, J.-M. (1984). Large Deviations and the Malliavin Calculus. Progress in Mathematics 45 . Birkhäuser Boston, Boston, MA. · Zbl 0537.35003 [7] Chen, M.-F. and Wang, F.-Y. (1994). Application of coupling method to the first eigenvalue on manifold. Sci. China Ser. A 37 1-14. · Zbl 0799.53044 [8] Cranston, M. (1991). Gradient estimates on manifolds using coupling. J. Funct. Anal. 99 110-124. · Zbl 0770.58038 [9] Chen, M.-F. and Wang, F.-Y. (1997). Estimates of logarithmic Sobolev constant: An improvement of Bakry-Emery criterion. J. Funct. Anal. 144 287-300. · Zbl 0872.58066 [10] Davies, E. B. and Simon, B. (1984). Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59 335-395. · Zbl 0568.47034 [11] Elworthy, K. D. and Li, X.-M. (1994). Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125 252-286. · Zbl 0813.60049 [12] Émery, M. (1989). Stochastic Calculus in Manifolds . Springer, Berlin. · Zbl 0697.60060 [13] Greene, R. E. and Wu, H. (1979). Function Theory on Manifolds Which Possess a Pole. Lecture Notes in Math. 699 . Springer, Berlin. · Zbl 0414.53043 [14] Gross, L. (1975). Logarithmic Sobolev inequalities. Amer. J. Math. 97 1061-1083. JSTOR: · Zbl 0318.46049 [15] Hsu, E. P. (1997). Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds. Comm. Math. Phys. 189 9-16. · Zbl 0892.58083 [16] Kendall, W. S. (1987). The radial part of Brownian motion on a manifold: A semimartingale property. Ann. Probab. 15 1491-1500. · Zbl 0647.60086 [17] Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. In Séminaire de Probabilités , XXXIII. Lecture Notes in Math. 1709 120-216. Springer, Berlin. · Zbl 0957.60016 [18] Röckner, M. and Wang, F.-Y. (2003). Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds. Forum Math. 15 893-921. · Zbl 1062.47044 [19] Wang, F.-Y. (1997). Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 417-424. · Zbl 0887.35012 [20] Wang, F.-Y. (2001). Logarithmic Sobolev inequalities: Conditions and counterexamples. J. Operator Theory 46 183-197. · Zbl 0993.58019 [21] Wang, F.-Y. (2005). Functional Inequalities , Markov Properties , and Spectral Theory . Science Press, Beijing.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.