Sobolev type inequalities for general symmetric forms
HTML articles powered by AMS MathViewer
- by Feng-Yu Wang PDF
- Proc. Amer. Math. Soc. 128 (2000), 3675-3682 Request permission
Abstract:
A general version of the Sobolev type inequality, including both the classical Sobolev inequality and the logarithmic Sobolev one, is studied for general symmetric forms by using isoperimetric constants. Some necessary and sufficient conditions are presented as results. The main results are illustrated by two examples of birth-death processes.References
- Shigeki Aida, Takao Masuda, and Ichir\B{o} Shigekawa, Logarithmic Sobolev inequalities and exponential integrability, J. Funct. Anal. 126 (1994), no. 1, 83–101. MR 1305064, DOI 10.1006/jfan.1994.1142
- Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
- Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Papers dedicated to Salomon Bochner, 1969) Princeton Univ. Press, Princeton, N. J., 1970, pp. 195–199. MR 0402831
- Mu Fa Chen, From Markov chains to nonequilibrium particle systems, World Scientific Publishing Co., Inc., River Edge, NJ, 1992. MR 1168209, DOI 10.1142/1389
- Chen, M. F. and Wang, F. Y., Cheeger’s inequalities for general symmetric forms and existence criteria for spectral gap, MSRI Preprint, 1998-024, to appear in Ann. Probab.
- Fan R. K. Chung, Spectral graph theory, CBMS Regional Conference Series in Mathematics, vol. 92, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1997. MR 1421568
- Leonard Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), no. 4, 1061–1083. MR 420249, DOI 10.2307/2373688
- Gregory F. Lawler and Alan D. Sokal, Bounds on the $L^2$ spectrum for Markov chains and Markov processes: a generalization of Cheeger’s inequality, Trans. Amer. Math. Soc. 309 (1988), no. 2, 557–580. MR 930082, DOI 10.1090/S0002-9947-1988-0930082-9
- Ledoux, M., Concentration of measure and logarithmic Sobolev inequalities, 1997 preprint.
- Laurent Saloff-Coste, Lectures on finite Markov chains, Lectures on probability theory and statistics (Saint-Flour, 1996) Lecture Notes in Math., vol. 1665, Springer, Berlin, 1997, pp. 301–413. MR 1490046, DOI 10.1007/BFb0092621
Additional Information
- Feng-Yu Wang
- Affiliation: Department of Mathematics, Beijing Normal University, Beijing 100875, People’s Republic of China
- Address at time of publication: Universität Bielefeld, Fakultät für Mathematik, Postfach 100131, D-33501 Bielefeld, Germany
- Email: wangfy@bnu.edu.cn, fwang@mathematik.uni-bielefeld.de
- Received by editor(s): February 27, 1998
- Received by editor(s) in revised form: September 21, 1998, and February 10, 1999
- Published electronically: June 7, 2000
- Additional Notes: The author’s research was supported in part by the Alexander von Humboldt Foundation, NSFC(19631060), the Fok Ying-Tung Educational Foundation and the Research Foundation for Returned Overseas Chinese Scholars.
- Communicated by: Stanley Sawyer
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 3675-3682
- MSC (1991): Primary 60J25, 47A75
- DOI: https://doi.org/10.1090/S0002-9939-00-05471-X
- MathSciNet review: 1691009