A note on Coulhon type inequalities

Martín, Joaquim; Milman, Mario

doi:10.1090/S0002-9939-2014-12133-2

A note on Coulhon type inequalities
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by Joaquim Martín and Mario Milman

Proc. Amer. Math. Soc. 142 (2014), 4221-4237

DOI: https://doi.org/10.1090/S0002-9939-2014-12133-2

Published electronically: August 6, 2014

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Abstract:

T. Coulhon introduced an interesting reformulation of the usual Sobolev inequalities. We characterize Coulhon type inequalities in terms of rearrangement inequalities.

References

Nadine Badr, Gagliardo-Nirenberg inequalities on manifolds, J. Math. Anal. Appl. 349 (2009), no. 2, 493–502. MR 2456207, DOI 10.1016/j.jmaa.2008.09.013
D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), no. 4, 1033–1074. MR 1386760, DOI 10.1512/iumj.1995.44.2019
F. Barthe, P. Cattiaux, and C. Roberto, Isoperimetry between exponential and Gaussian, Electron. J. Probab. 12 (2007), no. 44, 1212–1237. MR 2346509, DOI 10.1214/EJP.v12-441
Jesús Bastero, Mario Milman, and Francisco J. Ruiz Blasco, A note on $L(\infty ,q)$ spaces and Sobolev embeddings, Indiana Univ. Math. J. 52 (2003), no. 5, 1215–1230. MR 2010324, DOI 10.1512/iumj.2003.52.2364
Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
Serguei G. Bobkov and Christian Houdré, Some connections between isoperimetric and Sobolev-type inequalities, Mem. Amer. Math. Soc. 129 (1997), no. 616, viii+111. MR 1396954, DOI 10.1090/memo/0616
Thierry Coulhon, Espaces de Lipschitz et inégalités de Poincaré, J. Funct. Anal. 136 (1996), no. 1, 81–113 (French, with English summary). MR 1375154, DOI 10.1006/jfan.1996.0022
Thierry Coulhon, Dimensions at infinity for Riemannian manifolds, Potential Anal. 4 (1995), no. 4, 335–344. Potential theory and degenerate partial differential operators (Parma). MR 1354888, DOI 10.1007/BF01053451
Thierry Coulhon, Heat kernel and isoperimetry on non-compact Riemannian manifolds, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 65–99. MR 2039952, DOI 10.1090/conm/338/06071
M. Cwikel and M. Milman, Sobolev inequalities in disguise and real interpolation, unpublished manuscript, 1996.
A. M. Garsia and E. Rodemich, Monotonicity of certain functionals under rearrangement, Ann. Inst. Fourier (Grenoble) 24 (1974), no. 2, vi, 67–116 (English, with French summary). MR 414802
Alexander Grigor′yan, Isoperimetric inequalities and capacities on Riemannian manifolds, The Maz′ya anniversary collection, Vol. 1 (Rostock, 1998) Oper. Theory Adv. Appl., vol. 109, Birkhäuser, Basel, 1999, pp. 139–153. MR 1747869
Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
Björn Jawerth and Mario Milman, Extrapolation theory with applications, Mem. Amer. Math. Soc. 89 (1991), no. 440, iv+82. MR 1046185, DOI 10.1090/memo/0440
Jan Kališ and Mario Milman, Symmetrization and sharp Sobolev inequalities in metric spaces, Rev. Mat. Complut. 22 (2009), no. 2, 499–515. MR 2553946, DOI 10.5209/rev_{r}ema.2009.v22.n2.16292
Michel Ledoux, Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes, C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 2, 79–82 (French, with English summary). MR 929114
Giovanni Leoni, A first course in Sobolev spaces, Graduate Studies in Mathematics, vol. 105, American Mathematical Society, Providence, RI, 2009. MR 2527916, DOI 10.1090/gsm/105
Joaquim Martín and Mario Milman, Pointwise symmetrization inequalities for Sobolev functions and applications, Adv. Math. 225 (2010), no. 1, 121–199. MR 2669351, DOI 10.1016/j.aim.2010.02.022
Joaquim Martín and Mario Milman, Sobolev inequalities, rearrangements, isoperimetry and interpolation spaces, Concentration, functional inequalities and isoperimetry, Contemp. Math., vol. 545, Amer. Math. Soc., Providence, RI, 2011, pp. 167–193. MR 2858472, DOI 10.1090/conm/545/10771
Joaquim Martín and Mario Milman, Isoperimetry and symmetrization for logarithmic Sobolev inequalities, J. Funct. Anal. 256 (2009), no. 1, 149–178. MR 2475420, DOI 10.1016/j.jfa.2008.09.001
J. Martin and M. Milman, Fractional Sobolev inequalities: Symmetrization, isoperimetry and interpolation, preprint, arXiv:1205.1231.
Joaquim Martin, Mario Milman, and Evgeniy Pustylnik, Sobolev inequalities: symmetrization and self-improvement via truncation, J. Funct. Anal. 252 (2007), no. 2, 677–695. MR 2360932, DOI 10.1016/j.jfa.2007.05.017
Vladimir Maz’ya, Sobolev spaces with applications to elliptic partial differential equations, Second, revised and augmented edition, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 342, Springer, Heidelberg, 2011. MR 2777530, DOI 10.1007/978-3-642-15564-2
S. Poornima, An embedding theorem for the Sobolev space $W^{1,1}$, Bull. Sci. Math. (2) 107 (1983), no. 3, 253–259 (English, with French summary). MR 719267
Jie Xiao, The $p$-Faber-Krahn inequality noted, Around the research of Vladimir Maz’ya. I, Int. Math. Ser. (N. Y.), vol. 11, Springer, New York, 2010, pp. 373–390. MR 2723828, DOI 10.1007/978-1-4419-1341-8_{1}7