Abstract
This paper is essentially a survey, with however some variations or new points of view on already published results. The main theme is the study of the relationship between various Sobolev type inequalities on manifolds. In the first part, we introduce a scale of dimensions at infinity adapted to manifolds of polynomial growth, in which we recast the results of [C2], [BCLS], and [CL]. In the second one, we show how Poincaré inequalities allow at the same time to go down in the scale and to refine it into a scale of global inequalities ([CS2], [S], [C2]). In the third part, we take up the more general situation where the volume growth of the manifold is not necessarily governed by a power function.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
BAKRY D., COULHON T., LEDOUX M., SALOFF-COSTE L., Sobolev inequalities in disguise, preprint.
CARRON G., Inégalités isopérimétriques de Faber-Krahn et conséquences, à paraître dans Table ronde de géométrie riemannienne en l’honneur de Marcel Berger, A. L. Besse éd., Astérisque.
CARRON G., Inégalités isopérimétriques sur les variétés riemanniennes, thesis, University of Grenoble, 1994.
CHAVEL I., FELDMAN E., Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds, Duke Math. J., 64, 1991.
COULHON T., Noyau de la chaleur et discrétisation d’une variété riemannienne, Israël J. Math., 80, 1992, 289–300.
COULHON T., Espaces de Lipschitz et inégalités de Poincaré, to appear in J. Funct. Anal..
COULHON T., Ultracontractivity and Nash type inequalities, preprint.
COULHON T., LEDOUX M., Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: un contre-exemple, Ark. Mat., 32, 1994, 63–77.
COULHON T., SALOFF-COSTE L., Puissances d’un opérateur régularisant, Ann. Inst. H. Poincaré, proba. et stat., vol. 26, n° 3, 1990, pp.419–436.
COULHON T., SALOFF-COSTE L., Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamer., 9, 2, 1993, 293–314.
COULHON T., SALOFF-COSTE L., Minorations pour les chaînes de Markov unidimensionnelles, Prob. Th. Rel. F., 97, 1993, pp.423–431.
COULHON T., SALOFF-COSTE L., Variétés riemanniennes isométriques à l’infini, preprint.
GRIGOR’YAN A., Heat kernel upper bounds on a complete non-compact manifolds, to appear, Rev. Mat. Iberoamer..
KANAI M., Analytic inequalities, and rough isometries between non-compact Riemannian manifolds, in Curvature and Topology of Riemannian Manifolds, Springer L. N. n° 1201, 1986, 122–137.
PITTET C., Folner sequences on polycyclic groups, to appear in Rev. Mat. Iberoamer..
SALOFF-COSTE L., On global Sobolev inequalities, Forum Mat., 6, 1994, 271–286.
VAROPOULOS N., Isoperimetric inequalities and Markov chains, J. Funct. Anal., vol. 63, n° 2, 1985, pp.215–239.
VAROPOULOS N., Hardy-Littlewood theory for semigroups, J. Funct. Anal., vol. 63, 2, 1985, 240–260.
VAROPOULOS N., Small time Gaussian estimates of heat diffusion kernels. Part I: the semigroup technique, Bull. Sc. Math., 113, 1989, 253–277.
YAMASAKI M., Parabolic and hyperbolic infinite networks, Hiroshima Math. J., 7, 1977, pp.135–146.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1995 Springer Science+Business Media Dordrecht
About this chapter
Cite this chapter
Coulhon, T. (1995). Dimensions at Infinity for Riemannian Manifolds. In: Biroli, M. (eds) Potential Theory and Degenerate Partial Differential Operators. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-0085-4_3
Download citation
DOI: https://doi.org/10.1007/978-94-011-0085-4_3
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-4042-6
Online ISBN: 978-94-011-0085-4
eBook Packages: Springer Book Archive