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Multiple sets exponential concentration and higher order eigenvalues. (English) Zbl 1434.35026

Summary: On a generic metric measured space, we introduce a notion of improved concentration of measure that takes into account the parallel enlargement of \(k\) distinct sets. We show that the \(k\)-th eigenvalues of the metric Laplacian gives exponential improved concentration with \(k\) sets. On compact Riemannian manifolds, this allows us to recover estimates on the eigenvalues of the Laplace-Beltrami operator in the spirit of an inequality of F. R. K. Chung et al. [Adv. Math. 117, No. 2, 165–178 (1996; Zbl 0844.53029)].

MSC:

35P15 Estimates of eigenvalues in context of PDEs
60E15 Inequalities; stochastic orderings
26D10 Inequalities involving derivatives and differential and integral operators

Citations:

Zbl 0844.53029
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References:

[1] Aida, S.; Stroock, D., Moment estimates derived from Poincaré and logarithmic Sobolev inequalities, Math. Res. Lett., 1, 1, 75-86 (1994) · Zbl 0862.60064 · doi:10.4310/MRL.1994.v1.n1.a9
[2] Ambrosio, L., Ghezzi, R.: Sobolev and bounded variation functions on metric measure spaces. In: Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds. Vol. II, EMS Ser. Lect. Math., pp. 211-273. Eur. Math. Soc., Zürich (2016) · Zbl 1362.53002
[3] Atkinson, K.; Han, W., Spherical Harmonics and Approximations on the Unit Sphere: An Introduction, Volume 2044 of Lecture notes in Mathematics (2012), Heidelberg: Springer, Heidelberg · Zbl 1254.41015
[4] Bakry, D.; Émery, M., Diffusions hypercontractives, Lecture Notes in Mathematics, 177-206 (1985), Berlin, Heidelberg: Springer Berlin Heidelberg, Berlin, Heidelberg · Zbl 0561.60080
[5] Bakry, D.; Gentil, I.; Ledoux, M., Analysis and geometry of Markov diffusion operators, volume 348 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (2014), Cham: Springer, Cham · Zbl 1376.60002
[6] Bobkov, S.; Ledoux, M., Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution, Probab. Theory Related Fields, 107, 3, 383-400 (1997) · Zbl 0878.60014 · doi:10.1007/s004400050090
[7] Buser, P., A note on the isoperimetric constant, Ann. Sci. É,cole Norm. Sup. (4), 15, 2, 213-230 (1982) · Zbl 0501.53030 · doi:10.24033/asens.1426
[8] Chavel, I., Eigenvalues in Riemannian Geometry, volume 115 of Pure and Applied Mathematics (1984), Orlando: Academic Press, Inc., Orlando · Zbl 0551.53001
[9] Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., 9, 3, 428-517 (1999) · Zbl 0942.58018 · doi:10.1007/s000390050094
[10] Chung, F.R.K., Grigor’yan, A., Yau, S.-T.: Eigenvalues and diameters for manifolds and graphs. In: Tsing Hua Lectures on Geometry & Analysis (Hsinchu, 1990-1991), pp. 79-105. Int. Press, Cambridge (1997) · Zbl 0890.58093
[11] Chung, Frk; Grigor’Yan, A.; Yau, S-T, Upper bounds for eigenvalues of the discrete and continuous Laplace operators, Adv. Math., 117, 2, 165-178 (1996) · Zbl 0844.53029 · doi:10.1006/aima.1996.0006
[12] Friedman, J.; Tillich, J-P, Laplacian eigenvalues and distances between subsets of a manifold, J. Differential Geom., 56, 2, 285-299 (2000) · Zbl 1032.58015 · doi:10.4310/jdg/1090347645
[13] Funano, K., Estimates of Eigenvalues of the Laplacian by a reduced number of subsets, Israel J. Math., 217, 1, 413-433 (2017) · Zbl 1368.53033 · doi:10.1007/s11856-017-1453-7
[14] Funano, K.; Shioya, T., Concentration, Ricci curvature, and Eigenvalues of Laplacian, Geom. Funct. Anal., 23, 3, 888-936 (2013) · Zbl 1277.53038 · doi:10.1007/s00039-013-0215-x
[15] Gozlan, N.; Roberto, C.; Samson, P-M, From dimension free concentration to the P,oincaré inequality, Calc. Var Partial Differential Equations, 52, 3-4, 899-925 (2015) · Zbl 1352.60028 · doi:10.1007/s00526-014-0737-6
[16] Gromov, M.; Milman, Vd, A topological application of the isoperimetric inequality, Amer. J. Math., 105, 4, 843-854 (1983) · Zbl 0522.53039 · doi:10.2307/2374298
[17] Heinonen, J., Lectures on Lipschitz Analysis, volume 100 of Report. University of Jyväskylä Department of Mathematics and Statistics (2005), Jyväskylä: University of Jyväskylä, Jyväskylä · Zbl 1086.30003
[18] Kirszbraun, M., Uber die zusammenziehende und lipschitzsche transformationen, Fundam. Math., 22, 77-108 (1934) · Zbl 0009.03904 · doi:10.4064/fm-22-1-77-108
[19] Ledoux, M., The Concentration of Measure Phenomenon, Volume 89 of Mathematical Surveys and Monographs (2001), Providence: American Mathematical Society, Providence · Zbl 0995.60002
[20] Lichnerowicz, A., Géométrie des groupes de transformations. Travaux et recherches mathématiques, III (1958), Paris: Dunod, Paris · Zbl 0096.16001
[21] Liu, S.: An optimal dimension-free upper bound for eigenvalue ratios. Arxiv e-prints, May (2014)
[22] Mcshane, Ej, Extension of range of functions, Bull. Amer. Math. Soc., 40, 12, 837-842 (1934) · doi:10.1090/S0002-9904-1934-05978-0
[23] Milman, E., Spectral estimates, contractions and hypercontractivity, J. Spectr. Theory, 8, 2, 669-714 (2018) · Zbl 1396.35005 · doi:10.4171/JST/210
[24] Milman, E., On the role of convexity in isoperimetry, spectral gap and concentration, Invent. Math., 177, 1, 1-43 (2009) · Zbl 1181.52008 · doi:10.1007/s00222-009-0175-9
[25] Milman, E., Isoperimetric and concentration inequalities: equivalence under curvature lower bound, Duke Math. J., 154, 2, 207-239 (2010) · Zbl 1205.53038 · doi:10.1215/00127094-2010-038
[26] Schmuckenschläger, M., Martingales, Poincaré type inequalities, and deviation inequalities, J. Funct. Anal., 155, 2, 303-323 (1998) · Zbl 0924.58111 · doi:10.1006/jfan.1997.3218
[27] Wang, F-Y, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170, 1, 219-245 (2000) · Zbl 0946.58010 · doi:10.1006/jfan.1999.3516
[28] Whitney, H., Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36, 1, 63-89 (1934) · doi:10.1090/S0002-9947-1934-1501735-3
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