The Annals of Probability

Harnack Inequalities for Log-Sobolev Functions and Estimates of Log-Sobolev Constants

Feng-Yu Wang

Full-text: Open access


By using the maximum principle and analysis of heat semigroups, Harnack inequalities are studied for log-Sobolev functions. From this, some lower bound estimates of the log-Sobolev constant are presented by using the spectral gap inequality and the coupling method. The resulting inequalities either recover or improve the corresponding ones proved by Chung and Yau. Especially, Harnack inequalities and estimates of log-Sobolev constants can be dimension-free.

Article information

Ann. Probab. Volume 27, Number 2 (1999), 653-663.

First available in Project Euclid: 29 May 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58G32 60J60: Diffusion processes [See also 58J65]

Harnack inequality log-Sobolev function log-Sobolev constant coupling method


Wang, Feng-Yu. Harnack Inequalities for Log-Sobolev Functions and Estimates of Log-Sobolev Constants. Ann. Probab. 27 (1999), no. 2, 653--663. doi:10.1214/aop/1022677381.

Export citation


  • [1] Bakry, D. and Emery, M. (1984). Hypercontractivit´e de semigroups de diffusion. C. R. Acad. Sci. Paris S´er. I 209 775-778.
  • [2] Chen, M. F. and Wang, F. Y. (1994). Application of coupling method to the first eigenvalue on manifolds. Sci. Sin. A 37 1-14.
  • [3] Chen, M. F. and Wang, F. Y. (1997). General formula for lower bound of the first eigenvalue on Riemannian manifolds. Sci. Sin. A 40 384-394.
  • [4] 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.
  • [5] Chung, F. R. K. and Yau, S. T. (1996). Logarithmic Harnack inequalities. Math. Res. Lett. 3 793-812.
  • [6] Cranston, M. (1991). Gradient estimates on manifolds using coupling. J. Funct. Anal. 99 110-124.
  • [7] Deuschel, J. D. and Stroock, D. W. (1990). Hypercontractivity and spectral gap of symmetric diffusions with applications to the stochastic Ising models. J. Funct. Anal. 92 30-48.
  • [8] Kendall, W. S. (1986). Nonnegative Ricci curvature and the Brownian coupling property. Stochastics 19 111-129.
  • [9] Lu, Y. G. (1994). An estimate on non-zero eigenvalues of Laplacian in non-linear version. Stochastic Anal. Appl. 15 547-554.
  • [10] Rothaus, O. S. (1981). Logarithmic Sobolev inequalities and the spectrum of Schr¨odinger operator. J. Funct. Anal. 42 110-120.
  • [11] Wang, F. Y. (1997). On estimation of logarithmic Sobolev constant and gradient estimates of heat semigroups. Probab. Theory Related Fields 108 87-101.
  • [12] Wang, F. Y. (1997). Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Related Fields 109 417-424.
  • [13] Wang, F. Y. (1994). Application of coupling method to the Neumann eigenvalue problem. Probab. Theory Related Fields 98 299-306.