Isoperimetry and symmetrization for Sobolev spaces on metric spaces

Abstract

Using isoperimetry we obtain new symmetrization inequalities that allow us to provide a unified framework to study Sobolev inequalities in metric spaces. The applications include concentration inequalities, Poincaré inequalities, as well as metric versions of the Pólya-Szegö and Faber-Krahn principles. To cite this article: J. Martín, M. Milman, C. R. Acad. Sci. Paris, Ser. I 347 (2009). Résumé Isopérimétrie et symetrisation dans des espaces de Sobolev sur les espaces métriques. En utilisant l'isopérimétrie nous obtenons des nouvelles inégalités de symetrisation qui nous permettent de fournir un cadre unifié pour étudier des inégalités de Sobolev dans des espaces métriques. Les applications incluent des inégalités de concentration, inégalités de Poincaré, et des versions métriques des principes de Pólya-Szegö et de Faber-Krahn. Pour citer cet article : J. Martín, M. Milman, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

“…Finally, let us state that our main focus in this paper was to develop our methods and illustrate their reach, but without trying to state the results in their most general form. We refer the reader to [27] for a general theory of isoperimetry and symmetrization in the metric setting.…”

Pointwise symmetrization inequalities for Sobolev functions and applications

“…In our recent work (cf. [90], [86], [87]) we have developed new symmetrization inequalities that address all these issues and can be applied to provide a unified treatment of sharp Sobolev-Poincaré inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations. Our inequalities combine three basic features, each of which may have been considered before but, apparently, not all of them simultaneously; namely our inequalities are (i) pointwise rearrangement inequalities, (ii) incorporate in their formulation the isoperimetric profile and (iii) are formulated in terms of oscillations.…”

“…Theorem 1), the isoperimetric inequality can be reformulated on metric probability spaces (Ω, d, µ), (cf. [87], and also [16], [70], [90], [86], for Euclidean or Gaussian versions, see also [41] for a somewhat different perspective) as follows 2 (1.4)…”

Towards a Unified Theory of Sobolev Inequalities

“…When working on the Gaussian inequalities, we realized early on that, with a suitable definition of modulus of the gradient 14 |∇f | , and having at hand an associated co-area formula 15 , we could indeed prove (10) in the general setting of metric measure spaces (this was informally first announced in [73] and more formally in [75]). Fortunately, all the tools that we need to implement this insight had already been developed by Bobkov-Houdre [18].…”