On F-Sobolev and Orlicz-Sobolev inequalities

Abstract

Let F ∈ C([0,∞)) be a positive increasing function such that Φ(s):= |s|F(|s|) is a Young function. In general, the F-Sobolev inequality and the Φ-Orlicz-Sobolev inequality are not equivalent. In this paper, a growth condition on F is presented for these two inequalities to be equivalent. The main result generalizes the corresponding known one for F(s) = log^δ(1 + s) (δ > 0). As an application, some criteria are presented for the F-Sobolev inequality to hold.

Keywords

Orlicz-Sobolev inequality / F-Sobolev inequality / super Poincaré inequality

References

[1] Bakry D., Ledoux M., Wang F. -Y. Perturbations of functional inequalities using growth conditions. J Math Pures Appl, 2007, 87: 394.407

[2] Bobkov S. G., Götze F. Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J Funct Anal, 1999, 163: 1-28.

[3] Chafai D. Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities. J Math Koyoto Univ, 2004, 44: 325-363.

[4] Chen M. -F. Variational formulas of Poincaré-type inequalities in Banach spaces of functions on the line. Acta Math Sin (Eng Ser), 2002, 18: 417.436

[5] Chen M. -F. Variational formulas of Poincaré-type inequalities for birth-death processes. Acta Math Sin (Eng Ser), 2003, 19: 625-664.

[6] Chen M. -F. Eigenvalues, Inequalities, and Ergodic Theory, 2005, Berlin: Springer.

[7] Chen M. -F. Capacity criteria for Poincaré-type inequalities. Pot Anal, 2005, 23: 303-322.

[8] Coulhon T., Grigor’yan A., Alexander L. D. On isoperimetric profiles of product spaces. Comm Anal Geom, 2003, 11(1): 85-120.

[9] Davies E. B. Heat Kernels and Spectral Theory, 1989, Cambridge: Cambridge Univ Press.

[10] Fukushima M., Uemura T. On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets. Pot Anal, 2003, 18: 59-77.

[11] Gong F. -Z., Wang F. -Y. Functional inequalities for uniformly integrable semigroups and application to essential spectrums. Forum Math, 2002, 14: 293-313.

[12] Gross L. Logarithmic Sobolev inequalities. Amer J Math, 1976, 97: 1061.1083

[13] Latała R., Oleszkiewicz K. Between Sobolev and Poincaré. Lecture Notes in Math, 2000, 1745: 147-168.

[14] Rao M. M., Ren Z. D. Application of Orlicz Spaces, 2002, New York: Marcel Dekker.

[15] Roberto C., Zegarlinski B. Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups. J Funct Anal, 2007, 243: 28-66.

[16] Wang F. Latała-Oleszkiewicz Inequalities. Ph D Thesis, 2003, Beijing: Beijing Normal University.

[17] Wang F. -Y. Functional inequalities for empty essential spectrum. J Funct Anal, 2000, 170: 219-245.

[18] Wang F. -Y. Functional inequalities, semigroup properties and spectrum estimates. Infin Dimens Anal Quant Probab Relat Topics, 2000, 3: 263-295.

[19] Wang F. -Y. A generalization of Poincaré and log-Sobolev inequalities. Pot Anal, 2005, 22: 1-15.