Functional inequalities, semigroup properties and spectrum estimates

Infinite Dimensional Analysis, Quantum Probability and Related Topics

Volumn 3, Issue 2, 2000, Pages 263-295

  Wang, Feng Yu ^a,b  

^a BEIJING NORMAL UNIVERSITY   (China)

^b BIELEFELD UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034419013     PISSN: 02190257     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0219025700000194     Document Type: Article

References (67)

1. S. Aida, Uniformly positivity improving property, Sobolev inequalities and spectral gaps, J. Funct. Anal. 158 (1998) 152-158.
2. S. Aida, T. Masuda and I. Shigekawa, Logarithmic Sobolev inequalities and exponential integrability, J. Funct. Anal. 126 (1994) 83-101.
3. S. Aida and D. Stroock, Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett. 1 (1994) 75-86.
4. D. Bakry, L'hypercontractivité et Son Utilisation en Théorie des Semigroups, Lecture Notes in Math. Vol. 158 (Springer, 1994), pp. 1-114.
5. D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995) 1033-1074.
6. D. Bakry and M. Emery, Hypercontractivité de semigroups de diffusion, C. R. Acad. Sci. Paris Sér. I 209 (1984) 775-778.
7. D. Bakry, M. Ledoux and Z. Qian, Logarithmic Sobolev inequalities, Poincaré inequalities and heat kernel bounds (1997) preprint.
8. N. Bourbaki, Espaces Vectoriels Topologiques (Livere V), (Hermann, 1955), Chaps. III-IV.
9. E. A. Carlen, S. Kusuoka and D. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Prob. Statist. 23 (1987) 245-287.
10. I. Chavel, Eigenvalues in Riemannian Geometry (Academic Press, 1984).
11. J. Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, in Problems in Analysis, A Symposium in Honor of S. Bochner 195-199 (Princeton Univ. Press, 1970).
12. M. F. Chen, Optimal Markovian coupling and applications, Acta Math. Sin. New Ser. 10 (1994) 260-275.
13. M. F. Chen, Optimal couplings and application to Riemannian geometry, Probability Theory Math. Stat., eds. B. Grigelionis et al. (1994) pp. 121-142.
14. M. F. Chen, Estimation of spectral gap for Markov chains, Acta Math. Sin. New Ser. 12 (1996) 337-360.
15. M. F. Chen, Coupling, spectral gap and related topics (I; II; III), Chinese Sci. Bull. 42 (1997) 1321-1327; 1409-1416; 1497-1505.
16. M. F. Chen, Trilogy of couplings and formulas for lower bound of spectral gap, in Probability Towards 2000 (New York, 1995), Lecture Notes in Stat., Vol. 128 (Springer, 1998), pp. 123-136.
17. M. F. Chen, Nash inequalities for general symmetric forms, Act. Math. Sin. (English Ser.) 15 (1999) 353-370.
18. M. F. Chen, Log-Sobolev inequality for symmetric forms, to appear in Sci. Sin. (A).
19. M. F. Chen, Eigenvalues, inequalities and ergodic theory, to appear in Chinese Sci. Bull.
20. M. F. Chen, Eigenvalues, inequalities and ergodic theory, to appear in Adv. Math.
21. M. F. Chen and F.-Y. Wang, Application of coupling method to the first eigenvalue on manifolds, Sci. Sin. (A) (English Ed.) 37 (1994) 1-14.
22. M. F. Chen and F.-Y. Wang, Estimates of the first eigenvalue of second order elliptic operators, J. Funct. Anal. 131 (1995) 345-363.
23. M. F. Chen and F.-Y. Wang, General formula for lower bound of the first eigenvalue on Riemannian manifolds, Sci. Sin. (A) (English Ed.) 40 (1997) 384-394.
24. M. F. Chen and F.-Y. Wang, Estimation of spectral gap for elliptic operators, Trans. Amer. Math. Soc. 349 (1997) 1239-1267.
25. M. F. Chen and F.-Y. Wang, Estimates of log-Sobolev inequalities: An improvement of Bakry-Emery criterion, J. Funct. Anal. 144 (1997) 287-300.
26. M. F. Chen and F.-Y. Wang, Cheeger's inequalities for general symmetric forms and existence criteria for spectral gap, to appear in Ann. Prob. Results claimed in Chinese Sci. Bull. 43 (1998) 1516-1518.
27. T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal. 141 (1996) 510-539.
28. C. B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École. Norm. Sup. 13 (1980) 419-435.
29. E. B. Davies, Heat Kernels and Spectral Theory (Cambridge Univ. Press, 1989).
30. E. B. Davies and B. Simon, Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians, J. Funct. Anal. 59 (1984) 335-395.
31. J. D. Deuschel and D. Stroock, Large Deviations, Pure and Appl. Math. Ser., Vol. 137 (Academic Press, 1989).
32. B. Driver and Y. Z. Hu, On heat kernel logarithmic Sobolev inequalities, in Stochastic Analysis and Applications, Powys, 1995 (World Scientific, 1996), pp. 189-200.
33. D. Elworthy and X. M. Li, Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1994) 252-286.
34. F. Z. Gong and F.-Y. Wang, Functional inequalities for uniformly integrable semigroups and applications, SFB-343 preprint 99-80, Bielefeld.
35. L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1976) 1061-1083.
36. L. Gross, Logarithmic Sobolev Inequalities and Contractivity Properties of Semigroups, Lecture Notes in Math., Vol. 1563 (Springer, 1993), pp. 54-88.
37. L. Gross, Hypercontractivity over complex manifolds, Acta Math. 182 (1999) 159-206.
38. L. Gross and O. Rothaus, Herbst inequalities for supercontractive semigroups, J. Math. Kyoto Univ. 38 (1998) 295-318.
39. E. P. Hsu, Logarithmic Sobolev inequalities on path spaces over Riemannian manifolds, Comm. Math. Phys. 189 (1997) 9-16.
40. H. König, Eigenvalues Distribution of Compact Operators (Birkhäuser, 1986).
41. 2 spectrum for Markov chain and Markov processes: A generalization of Cheeger's inequality, Trans. Amer. Math. Soc. 309 (1988) 301-413.
42. M. Ledoux, Concentration of Measures and Logarithmic Sobolev Inequali-ties, Lecture Notes in Math., to appear.
43. M. Ledoux, A simple proof of an inequality by P. Buser, Proc. Amer. Math. Soc. 121 (1994) 951-958.
44. A. Lichnerowicz, Geometrie des Groupes des Transformationes (Dunod, 1958).
45. Z. M. Ma and M. Röckner, Introduction to Theory of (Non-symmetric) Dirichlet Forms (Springer-Verlag, 1992).
46. M. Röckner and F.-Y. Wang, Ultracontractivity and supercontractivity of Markov semigroups, SFB-343 preprint 99-88, Bielefeld.
47. M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis (Academic Press, 1980).
48. p norms, Proc. Amer. Math. Soc. 126 (1998) 2309-2314.
49. M. Tomisaki, Comparison theorems on Dirichlet norms and their applications, Forum. Math. 2 (1990) 277-295.
50. F.-Y. Wang, Application of coupling method to the Neumann eigenvalue problem, Prob. Theory Related Fields 98 (1994) 299-306.
51. F.-Y. Wang, Spectral gap for diffusion processes on noncompact manifolds, Chinese Sci. Bull. 40 (1995) 1145-1149.
52. F.-Y. Wang, Logarithmic Sobolev inequalities for diffusion processes with application to path spaces, Chin. J. Appl. Prob. Stat. 12 (1996) 255-264.
53. F.-Y. Wang, On estimation of logarithmic Sobolev constant and gradient estimates of heat semigroups, Prob. Theory Related Fields 108 (1997) 87-101.
54. F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Prob. Theory Related Fields 109 (1997) 417-424.
55. F.-Y. Wang, Existence of spectral gap for elliptic operators, Arkiv. Mat. 37 (1999) 395-407.
56. F.-Y. Wang, General formulae for the lower bound of the first two Dirichlet eigenvalues, Taiwanese J. Math. 3 (1999) 235-242.
57. F.-Y. Wang, Harnack inequalities for log-Sobolev functions and estimates of log-Sobolev constant, Ann. Prob. 27 (1999) 653-663.
58. F.-Y. Wang, Sobolev type inequalities for general symmetric forms, to appear in Proc. Amer. Math. Soc.
59. F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal. 170 (2000) 219-245.
60. F.-Y. Wang, On Estimation of the Dirichlet spectral gap, to appear in Archiv. Math.
61. F.-Y. Wang, Logarithmic Sobolev inequalities: Conditions and counterexamples, J. Operator Theory, to appear.
62. F.-Y. Wang, Estimates of semigroups and eigenvalues using functional inequalities, 1999 preprint.
63. L. M. Wu, Large deviations for Markov processes under superboundedness, C. R. Acad. Sci. Paris Série. I 324 (1995) 777-782.
64. L. M. Wu, Uniformly integrable operators and large deviations for Markov processes, to appear in J. Funct. Anal.
65. S. T. Yau, Isoperimetric constant and the first eigenvalue of a compact manifold, Ann. Ec. Norm. Super. Paris 8 (1987) 487-507.
66. K. Yoshida, Functional Analysis (Springer-Verlag, 1974), 4th ed.
67. J. Q. Zhong and H. C. Yang, Estimates of the first eigenvalue of a compact Riemannian manifolds, Sci. Sin. 27 (1984) 1251-1265.