Abstract
We show that certain functional inequalities, e.g. Nash-type and Poincaré-type inequalities, for infinitesimal generators of C 0 semigroups are preserved under subordination in the sense of Bochner. Our result improves earlier results by Bendikov and Maheux (Trans Am Math Soc 359:3085–3097, 2007, Theorem 1.3) for fractional powers, and it also holds for non-symmetric settings. As an application, we will derive hypercontractivity, supercontractivity and ultracontractivity of subordinate semigroups.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bendikov A., Coulhon T., Saloff-Coste L.: Ultracontractivity and embedding into L ∞. Math. Ann. 337, 817–853 (2007)
Bendikov A., Maheux P.: Nash-type inequalities for fractional powers of non-negative self-adjoint operators. Trans. Am. Math. Soc. 359, 3085–3097 (2007)
Beckenbach E.F., Bellman R.: Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete (Neue Folge), vol. 40. Springer, Berlin (1961)
Berg C., Boyadzhiev Kh., de Laubenfels R.: Generation of generators of holomorphic semigroups. J. Aust. Math. Soc. (Ser. A) 55, 246–269 (1993)
Bihari I.: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations. Acta Math. Hungarica 7, 81–94 (1956)
Chen M.-F.: Eigenvalues, Inequalities and Ergodic Theory. Springer, London (2005)
Coulhon T.: Ultracontractivity and Nash-type inequalities. J. Func. Anal. 141, 510–539 (1996)
Gentil, I., Maheux, P.: Nash-type inequalities, super-Poincaré inequalities for subordinated semigroups. Preprint. arXiv: 1105. 3095v1
Ôkura H.: Recurrence and transience criteria for subordinated symmetric Markov processes. Forum Math. 14, 121–146 (2002)
Röckner M., Wang F.-Y.: Weak Poincaré inequalities and L 2-convergence rates of Markov semigroups. J. Func. Anal. 185, 564–603 (2001)
Schilling R.L.: A note on invariant sets. Prob. Math. Stat. 24, 47–66 (2004)
Schilling R.L., Song R.M., Vondraček Z.: Bernstein Functions—Theory and Applications. In: Studies in Mathematics, vol. 37. de Gruyter, Berlin (2010)
Walter W.: Differential and Integral Inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55. Springer, Berlin (1970)
Wang F.-Y.: Functional inequalities for empty essential spectrum. J. Funct. Anal. 170, 219–425 (2000)
Wang F.-Y.: Functional inequalities, semigroup properties and spectrum estimates. Infinite Dimens. Anal. Quant. Prob. Rel. Topics 3, 263–295 (2000)
Wang F.-Y.: Functional inequalities and spectrum estimates: the infinite measure case. J. Funct. Anal. 194, 288–310 (2002)
Wang F.-Y.: Functional Inequalities, Markov Semigroups and Spectral Theory. Science Press, Beijing (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
Note added in Proof
After we have finished this paper, Patrick Maheux informed us that he and Ivan Gentil have, independently, obtained similar results in their (at that point still forthcoming) preprint [8]; although our findings partially overlap, the methods used here and in [8] are essentially different.
Rights and permissions
About this article
Cite this article
Schilling, R.L., Wang, J. Functional inequalities and subordination: stability of Nash and Poincaré inequalities. Math. Z. 272, 921–936 (2012). https://doi.org/10.1007/s00209-011-0964-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0964-x
Keywords
- Subordination
- Bernstein function
- Nash-type inequality
- Super-Poincaré inequality
- Weak Poincaré inequality