Geometric asymptotics and the logarithmic Sobolev inequality

Forum Mathematicum

Volumn 11, Issue 1, 1999, Pages 105-137

  Beckner, William ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0033249421     PISSN: 09337741     EISSN: None     Source Type: Journal    
DOI: 10.1515/form.11.1.105     Document Type: Article

References (50)

1. Aubin, T.: Problemes isoperimetriques et espaces de Sobolev. J. Differential Geom. 11 (1976), 573-598
2. Aubin, T.: Non-linear analysis on manifolds, Monge-Ampére equations. Springer-Verlag, New York 1982
3. Bakry, D. and Ledoux, M.: Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator. Invent, math. 123 (1996), 259-281
4. Beckner, W.: Inequalities in Fourier analysis. Ann. Math. 102 (1975), 159-182
5. n. Proc. Nat. Acad. Sci. 89 (1992), 4816-4819
6. Beckner, W.: A generalized Poincaré inequality for Gaussian measures. Proc. Amer. Math. Soc. 105 (1989), 397-400
7. Beckner, W.: Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality. Ann. Math. 138 (1993), 213-242
8. Beckner, W.: Geometric inequalities in Fourier analysis. Essays on Fourier Analysis in Honor of Elias M. Stein, Princeton University Press, 1995, pp. 36-68
9. Beckner, W.: Pitt's inequality and the uncertainty principle. Proc. Amer. Math. Soc. 123 (1995), 1897-1905
10. Beckner, W.: Logarithmic Sobolev inequalities and the existence of singular integrals. Forum Math. 9 (1997), 303-323
11. Beckner, W.: Sharp inequalities and geometric manifolds. J. Fourier Anal. Appl. 3 (1997), 825-836
12. Beckner, W.: Geometric proof of Nash's inequality. Int. Math. Res. Notices (1998), 67-72
13. Beckner, W.: On the Hardy-Littlewood-Sobolev inequality (in preparation)
14. Beckner, W.: Pitt's inequality on the Heisenberg group (in preparation)
15. Beckner, W. and Pearson, M.: On sharp Sobolev embedding and the logarithmic Sobolev inequality. Bull. London Math. Soc. 30 (1998), 80-84
16. Brothers, J. E. and Ziemer, W. P.: Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384 (1988), 153-179
17. Carlen, E. A. and Loss, M.: Sharp constant in Nash's inequality. Int. Math. Res. Not. (1993), 213-215
18. Davies, E. B.: Heat kernels and spectral theory. Cambridge University Press, 1989
19. du Plessis, N.: Some theorems about the Riesz fractional integral. Trans. Amer. Math. Soc. 80 (1955), 124-134
20. Erdélyi, A. (editor): Bateman Manuscript Project, Higher transcendental functions. McGraw-Hill, 1953
21. Escobar, J. F.: Sharp constant in a Sobolev trace inequality. Indiana Math. J. 37 (1988), 687-698
22. Folland, G. B. and Sitaran, A.: The uncertainty principle. J. Fourier Anal. Appl. 3 (1997), 207-238
23. b complex and analysis on the Heisenberg group. Comm. Pure. Appl. Math. 27 (1974), 429-522
24. Folland, G. B. and Stein, E. M.: Hardy spaces on homogeneous groups. Princeton University Press, 1982
25. Gromov, M.: Isoperimetric inequalities in Riemannian manifolds. Lecture Notes in Mathematics, 1200. Springer-Verlag, 1986, pp. 114-129
26. Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), 1061-1083
27. Hardy, G. H., Littlewood, J. E. and Pólya, G.: Inequalities. Cambridge University Press, 1952
28. Jerison, D. and Lee, J. M.: A subelliptic nonlinear eigenvalue problem on CR manifolds. Contemporary Math. 27 (1984), 57-63
29. Jerison, D. and Lee, J. M.: Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem. J. Amer. Math. Soc. 1 (1988), 1-13
30. Jost, R. (editor): Local quantum theory. Academic Press, 1969
31. 2 (R). Addison-Wesley, 1975
32. Ledoux, M.: Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes. Comptes rendus 306 (1988), 79-82
33. Li, Y. Y. and Zhu, M.: Sharp Sobolev trace inequalities on Riemannian manifolds with boundaries. Comm. Pure Appl. Math. 50 (1997), 449-487
34. Lieb, E.H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. 118 (1983), 349-374
35. Lieb, E.H.: The stability of matter: from atoms to stars. Springer-Verlag, 1997
36. McKean, H. P.: Geometry of differential space. Ann. Probability 1 (1973), 197-206
37. Nagy, B. Sz.: Über Integralungleichungen zwischen einer Funktion und ihrer Ableitung. Acta Sci. Math. (Szeged) 10 (1941), 64-74
38. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958), 931-954
39. Peierls, R. E.: The laws of nature. Charles Scribner's Sons, 1956
40. Shannon, C. E. and Weaver, W.: The mathematical theory of communication. University of Illinois Press, 1948
41. Sinai, Y.G.: Probability theory. Springer-Verlag, 1992
42. Stam, A. J.: Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control 2 (1959), 101-112
43. Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton University Press, 1970
44. Stein, E. M.: Topics in harmonic analysis. Princeton University Press, 1970
45. Stroock, D.W.: Probability theory. Cambridge University Press, 1993
46. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110 (1976), 353-372
47. Watson, G. N.: A treatise on the theory of Bessel functions. Cambridge University Press, 1944
48. Weil, A.: L'intégration dans les groupes topologiques et ses applications. Hermann, 1965
49. Weissler, F. B.: Logarithmic Sobolev inequalities for the heat-diffusion semi-group. Trans. Amer. Math. Soc. 237 (1978), 255-269
50. Whittaker, E. T. and Watson, G. N.: A course of modern analysis. Cambridge University Press, 1927