## Large scale Sobolev inequalities on metric measure spaces and applications.(English)Zbl 1194.53036

Summary: For functions on a metric measure space, we introduce a notion of “gradient at a given scale”. This allows us to define Sobolev inequalities at a given scale. We prove that satisfying a Sobolev inequality at a large enough scale is invariant under large-scale equivalence, a metric-measure version of coarse equivalence. We prove that for a Riemmanian manifold satisfying a local Poincaré inequality, our notion of Sobolev inequalities at large scale is equivalent to its classical version. These notions provide a natural and efficient point of view to study the relations between the large time on-diagonal behavior of random walks and the isoperimetry of the space. Specializing our main result to locally compact groups, we obtain that the $$L^p$$-isoperimetric profile, for every $$1\leq p\leq \infty$$ is invariant under quasi-isometry between amenable unimodular compactly generated locally compact groups. A qualitative application of this new approach is a very general characterization of the existence of a spectral gap on a quasi-transitive measure space $$X$$, providing a natural point of view to understand this phenomenon.

### MSC:

 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 22D05 General properties and structure of locally compact groups 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 60G50 Sums of independent random variables; random walks 20F65 Geometric group theory
Full Text:

### References:

 [1] Brooks, R.: The fundamental group and the spectrum of the Laplacian. Comm. Math. Helv. 56 (1981), 581-598. · Zbl 0495.58029 [2] Chavel, I.: Riemannian geometry: a modern introduction . Cambridge Tracts in Mathematics 108 . Cambridge University Press, Cambridge, 1993. · Zbl 0810.53001 [3] Chavel, I. and Feldman, E.: Modified isoperimetric constants, and large time heat diffusion in Riemannian manifolds. Duke Math. J. 64 (1991), no. 3, 473-499. · Zbl 0753.58031 [4] Coulhon, T.: Dimension à l’infini d’un semi-groupe analytique. Bull. Sci. Math. 114 (1990), 485-500. · Zbl 0738.47032 [5] Coulhon, T.: Espaces de Lipschitz et inégalités de Poincaré. J. Funct. Anal. 136 (1996), 81-113. · Zbl 0859.58009 [6] Coulhon, T.: Random walks and geometry on infinite graphs. In Lecture notes on analysis on metric spaces (Trento, 1999) , 5-36. Appunti Corsi Tenuti Docenti Sc., Scuola Norm. Sup., Pisa, 2000. · Zbl 1063.60063 [7] Coulhon, T.: Heat kernel and isoperimetry on non-compact Riemannian manifolds. In Heat kernels and analysis on manifolds, graphs and metric spaces (Paris, 2002) , 65-99. Contemp. Math. 338 . Amer. Math. Soc., Providence, RI, 2003. · Zbl 1045.58016 [8] Coulhon, T. and Grigor’yan, A.: On-diagonal lower bounds for heat kernel and Markov chains. Duke Math. J. 89 (1997), 133-199. · Zbl 0920.58064 [9] Coulhon, T. and Ledoux, M.: Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: un contre-exemple. Ark. Mat. 32 (1994), 63-77. · Zbl 0826.53035 [10] Coulhon, T. and Saloff-Coste, L.: Variétés riemanniennes isométriques à l’infini. Rev. Mat. Iberoamericana 11 (1995), no. 3, 687-726. · Zbl 0845.58054 [11] Coulhon, T. and Saloff-Coste, L.: Isopérimétrie pour les groupes et les variétés. Rev. Mat. Iberoamericana 9 (1993), no. 2, 293-314. · Zbl 0782.53066 [12] Grigor’yan, A.: Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoamericana 10 (1994), no. 2, 395-452. · Zbl 0810.58040 [13] Gromov, M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53-73. · Zbl 0474.20018 [14] Gromov, M.: Structures métriques pour les variétés riemanniennes. Textes Mathématiques 1 . CEDIC, Paris, 1981. [15] Gromov, M.: Infinite groups as geometric objects. In Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) , 385-392. PWN, Warsaw, 1984. · Zbl 0599.20041 [16] Gromov, M.: Asymptotic invariants of infinite groups. In Geometric group theory, Vol. 2 (Sussex, 1991) , 1-295. London Math. Soc. Lecture Note Ser. 182 . Cambridge Univ. Press, Cambridge, 1993. · Zbl 0841.20039 [17] Heinonen, J.: Lectures on analysis on metric spaces . Universitext. Springer-Verlag, New York, 2001. · Zbl 0985.46008 [18] Hjorth, G.: Classification and orbit equivalence relations . Mathematical Surveys and Monographs 75 . American Mathematical Society, Providence, RI, 2000. · Zbl 0942.03056 [19] Kanai, M.: Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds. J. Math. Soc. Japan 37 (1985), 391-413. · Zbl 0554.53030 [20] Kesten, H.: Symmetric random walks on groups. Trans. Amer. Math. Soc. 92 (1959), 336-354. · Zbl 0092.33503 [21] Pier, J. P.: Amenable locally compact groups . Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1984. · Zbl 0597.43001 [22] Pittet, C.: The isoperimetric profile of homogeneous Riemannian manifolds. J. Differential Geom. 54 (2000), no. 2, 255-302. · Zbl 1035.53069 [23] Pittet, C. and Saloff-Coste, L.: A survey on the relationships between volume growth, isoperimetry, and the behaviour of simple random walk on Cayley graphs, with examples. Unpublished manuscript, 1997. [24] Roe, J.: Lectures on coarse geometry . University Lecture Series 31 . American Mathematical Society, Providence, RI, 2003. · Zbl 1042.53027 [25] Saloff-Coste, L.: Analysis on Riemannian co-compact covers. In Surveys in differential Geometry IX , 351-384. International Press, Sommerville, MA, 2004. · Zbl 1082.31006 [26] Saloff-Coste, L. and Woess, W.: Transition operators on co-compact $$G$$-spaces. Rev. Mat. Iberoamericana 22 (2006), no. 3, 747-799. · Zbl 1116.22007 [27] Salvatori, M.: On the norms of group-invariant transition operators on graphs. J. Theoret. Probab. 5 (1992), 563-576. · Zbl 0751.60068 [28] Semmes, S.: Finding Curves on General Spaces through quantitative topology, with applications for Sobolev and Poincaré inequalities. Selecta Math. (N.S.) 2 (1996), 155-295. · Zbl 0870.54031 [29] Soardi, P. M. and Woess, W.: Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z. 205 (1990), 471-486. · Zbl 0693.43001 [30] Tessera, R.: Asymptotic isoperimetry of balls in metric measure spaces. Publ. Math. 50 (2006), no. 2, 315-348. · Zbl 1116.53028 [31] Tessera, R.: Asymptotic isoperimetry on groups and uniform embeddings into Banach spaces. Preprint, 2006. Available at math.GR/0603138. · Zbl 1274.43009 [32] Tessera, R.: Vanishing of the first reduced cohomology with values in a $$L^p$$-representation. To appear in Ann. Inst. Fourier (Grenoble) . · Zbl 1225.22019 [33] Varopoulos, N. Th.: Analysis on Lie groups. J. Funct. Anal. 76 (1988), no. 2, 346-410. · Zbl 0634.22008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.