Fractional Sobolev inequalities : symmetrization, isoperimetry and interpolation

Martín, Joaquim ; Milman, Mario

Astérisque, no. 366 (2014) , 137 p.

Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez le site de la revue.

MR Zbl

@book{AST_2014__366__R1_0,
     author = {Mart{\'\i}n, Joaquim and Milman, Mario},
     title = {Fractional {Sobolev} inequalities : symmetrization, isoperimetry and interpolation},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {366},
     year = {2014},
     mrnumber = {3308452},
     zbl = {1321.46001},
     language = {en},
     url = {http://archive.numdam.org/item/AST_2014__366__R1_0/}
}

TY  - BOOK
AU  - Martín, Joaquim
AU  - Milman, Mario
TI  - Fractional Sobolev inequalities : symmetrization, isoperimetry and interpolation
T3  - Astérisque
PY  - 2014
IS  - 366
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_2014__366__R1_0/
LA  - en
ID  - AST_2014__366__R1_0
ER  -

%0 Book
%A Martín, Joaquim
%A Milman, Mario
%T Fractional Sobolev inequalities : symmetrization, isoperimetry and interpolation
%S Astérisque
%D 2014
%N 366
%I Société mathématique de France
%U http://archive.numdam.org/item/AST_2014__366__R1_0/
%G en
%F AST_2014__366__R1_0

Martín, Joaquim; Milman, Mario. Fractional Sobolev inequalities : symmetrization, isoperimetry and interpolation. Astérisque, no. 366 (2014), 137 p. http://numdam.org/item/AST_2014__366__R1_0/

Bibliographie
Cité par

[1] D. Aalto - "Weak $L^{\infty}$ and $B M O$ in metric spaces", Boll. Unione Mat. Ital. (9) 5 (2012), no. 2, p. 369-385. | MR | Zbl

[2] D. Aalto, L. Berkovits, O. E. Maasalo & H. Yue - "John-Nirenberg lemmas for a doubling measure", arXiv: 0910.1228. | Zbl

[3] F. J. Almgren, Jr. & E. H. Lieb - "Symmetric decreasing rearrangement is sometimes continuous", J. Amer. Math. Soc. 2 (1989), no. 4, p. 683-773. | MR | Zbl | DOI

[4] L. Ambrosio - "Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces", Adv. Math. 159 (2001), no. 1, p. 51-67. | Zbl | MR

[5] D. Bakry, T. Coulhon, M. Ledoux & L. Saloff-Coste - "Sobolev inequalities in disguise", Indiana Univ. Math. J. 44 (1995), no. 4, p. 1033-1074. | Zbl | DOI

[6] D. Bakry & P. A. Meyer - "Sur les inégalités de Sobolev logarithmiques. II", in Seminar on Probability, XVI, Lecture Notes in Math., vol. 920, Springer, Berlin-New York, 1982, p. 138-145, 146-150. | Zbl | EuDML | Numdam

[7] F. Barthe, P. Cattiaux & C. Roberto - "Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry", Rev. Mat. Iberoam. 22 (2006), no. 3, p. 993-1067. | Zbl | EuDML | DOI

[8] F. Barthe, P. Cattiaux & C. Roberto, "Isoperimetry between exponential and Gaussian", Electron. J. Probab. 12 (2007), p. no. 44, 1212-1237 (electronic). | Zbl | EuDML | DOI

[9] J. Bastero, M. Milman & F. J. Ruiz Blasco - "A note on $L (\infty, q)$ spaces and Sobolev embeddings", Indiana Univ. Math. J. 52 (2003), no. 5, p. 1215-1230. | Zbl | DOI

[10] V. Bayle - "Propriétés de concavité du profil isopérimétrique et applications", Ph.D. Thesis, 2003.

[11] C. Bennett, R. A. Devore & R. Sharpley - "Weak $L^{\infty}$ and $B M O$ ", Ann. of Math. (2) 113 (1981), no. 3, p. 601-611. | Zbl

[12] C. Bennett & K. Rudnick - "On Lorentz-Zygmund spaces", Dissertationes Math. (Rozprawy Mat.) 175 (1980), p. 67. | Zbl

[13] C. Bennett & R. Sharpley - Interpolation of operators, Pure Appl. Math., vol. 129, Academic Press, Inc., Boston, MA, 1988. | Zbl

[14] J. Bergh & J. Löfström - Interpolation spaces. An introduction, Grundlehren Math. Wiss., vol. 223, Springer-Verlag, Berlin-New York, 1976. | Zbl

[15] G. Besson - "From isoperimetric inequalities to heat kernels via symmetrisation", in Surveys in differential geometry. Vol. IX (A. Grigor'yan & S. Yau, eds.), Surv. Differ. Geom., IX, Int. Press, Somerville, MA, 2004, p. 27-51. | Zbl | DOI

[16] S. G. Bobkov - "Extremal properties of half-spaces for log-concave distributions", Ann. Probab. 24 (1996), no. 1, p. 35-48. | Zbl | DOI

[17] S. G. Bobkov & F. Götze - "Exponential integrability and transportation cost related to logarithmic Sobolev inequalities", J. Fund. Anal. 163 (1999), no. 1, p. 1-28. | Zbl | DOI

[18] S. G. Bobkov & C. Houdré - "Some connections between isoperimetric and Sobolev-type inequalities", Mem. Amer. Math. Soc. 129 (1997), no. 616, p. viii+111. | Zbl

[19] C. Borell - "Intrinsic bounds for some real-valued stationary random functions", in Probability in Banach spaces, V (Medford, Mass., 1984), Lecture Notes in Math., vol. 1153, Springer, Berlin, 1985, p. 72-95. | Zbl | DOI

[20] P. S. Bourdon, J. H. Shapiro & W. T. Sledd - "Fourier series, mean Lipschitz spaces, and bounded mean oscillation", in Analysis at Urbana, Vol. I (Urbana, IL, 1986-1987), London Math. Soc. Lecture Note Ser., vol. 137, Cambridge Univ. Press, Cambridge, 1989, p. 81-110. | Zbl

[21] D. W. Boyd - "Indices of function spaces and their relationship to interpolation", Canad. J. Math. 21 (1969), p. 1245-1254. | Zbl | DOI

[22] P. L. Butzer & H. Behrens - Semigroups of operators and approximation, Grundlehren Math. Wiss., vol. 145, Springer-Verlag New York Inc., New York, 1967. | Zbl

[23] C. P. Calderón - "Lacunary differentiability of functions in $𝐑^{n}$ ", J. Approx. Theory 40 (1984), no. 2, p. 148-154. | Zbl | DOI

[24] J. Cheeger - "Differentiability of Lipschitz functions on metric measure spaces", Geom. Funct. Anal. 9 (1999), no. 3. p. 428-517. | Zbl | DOI

[25] A. Cianchi - "Continuity properties of functions from Orlicz-Sobolev spaces and embedding theorems", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 23 (1996), no. 3, p. 575-608. | Zbl | EuDML | Numdam

[26] R. R. Coifman & G. Weiss - Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., vol. 242, Springer-Verlag, Berlin-New York, 1971. | Zbl

[27] T. Coulhon - "Espaces de Lipschitz et inégalités de Poincaré", J. Funct. Anal. 136 (1996), no. 1, p. 81-113. | Zbl | DOI

[28] T. Coulhon, "Heat kernel and isoperimetry on non-compact Riemannian manifolds", in Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, p. 65-99. | Zbl | DOI

[29] M. Cwikel, B. Jawerth & M. Milman - "A note on extrapolation of inequalities".

[30] M. Cwikel, Y. Sagher & P. Shvartsman - "A geometrical/combinatorical question with implications for the John-Nirenberg inequality for $B M O$ functions", in Marcinkiewicz centenary volume, Banach Center Publ., vol. 95, Polish Acad. Sci. Inst. Math., Warsaw, 2011, p. 45-53. | Zbl

[31] P. De Land - "Moduli of continuity for exponential Lipschitz classes", Trans. Amer. Math. Soc. 229 (1977), p. 175-189. | Zbl | DOI

[32] Z. Ditzian & K. G. Ivanov - "Strong converse inequalities", J. Anal. Math. 61 (1993), p. 61-111. | Zbl | DOI

[33] Z. Ditzian & D. S. Lubinsky - "Jackson and smoothness theorems for Freud weights in $L_{p} (0 \leq p \leq \infty)$ ", Constr. Approx. 13 (1997), no. 1, p. 99-152. | Zbl

[34] Z. Ditzian & V. Totik - Moduli of smoothness, Springer Ser. Comput. Math., vol. 9, Springer-Verlag, New York, 1987. | Zbl

[35] G. F. D. Duff - "Differences, derivatives, and decreasing rearrangements", Canad. J. Math. 19 (1967), p. 1153-1178. | Zbl | DOI

[36] A. Ehrhard - "Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes", Ann. Sci. École Norm. Sup. (4) 17 (1984), no. 2, p. 317-332. | Zbl | EuDML | Numdam | DOI

[37] H. Federer & W. H. Fleming - "Normal and integral currents", Ann. of Math. (2) 72 (1960), p. 458-520. | Zbl | DOI

[38] C. Fefferman & E. M. Stein - " $H^{p}$ spaces of several variables", Acta Math. 129 (1972), no. 3-4, p. 137-193. | Zbl | DOI

[39] G. F. Feissner - "Hypercontractive semigroups and Sobolev's inequality", Trans. Amer. Math. Soc. 210 (1975), p. 51-62. | Zbl

[40] A. M. Garsia & E. Rodemich - "Monotonicity of certain functionals under rearrangement", Ann. Inst. Fourier (Grenoble) 24 (1974), no. 2, p. vi, 67-116. | Zbl | EuDML | Numdam | DOI

[41] A. M. Garsia, E. Rodemich & H. Rumsey, Jr. - "A real variable lemma and the continuity of paths of some Gaussian processes", Indiana Univ. Math. J. 20 (1970/1971), p. 565-578. | Zbl | DOI

[42] A. M. Garsia - "On the smoothness of functions satisfying certain integral inequalities", in Functional Analysis (Proc. Sympos., Monterey, Calif., 1969), Academic Press, New York, 1969, p. 127-162. | Zbl

[43] A. M. Garsia, "Combinatorial inequalities and smoothness of functions", Bull. Amer. Math. Soc. 82 (1976), no. 2, p. 157-170. | Zbl | DOI

[44] A. M. Garsia, "A remarkable inequality and the uniform convergence of Fourier series", Indiana Univ. Math. J. 25 (1976), no. 1, p. 85-102. | Zbl | DOI

[45] A. E. Gatto & W. O. Urbina - "On Gaussian Lipschitz spaces and the bound-edness of fractional integrals", arxiv.org/abs/0911.3962 [math.CA], 2010.

[46] S. Geiss & A. Toivola - "On fractional smoothness and $L_{p}$ -approximation on the Wiener space", preprint, Innsbruck, 2012.

[47] A. Gogatishvili, P. Koskela & N. Shanmugalingam - "Interpolation properties of Besov spaces defined on metric spaces", Math. Nachr. 283 (2010), no. 2, p. 215-231. | Zbl | DOI

[48] P. Hajłasz & P. Koskela - "Sobolev met Poincaré", Mem. Amer. Math. Soc. 145 (2000), no. 688, p. x+101. | Zbl

[49] D. D. Haroske - Envelopes and sharp embeddings of function spaces, Chapman & Hall, 2007. | Zbl

[50] T. Holmstedt - "Equivalence of two methods of interpolation", Math. Scand. 18 (1966), p. 45-52. | Zbl | EuDML | DOI

[51] B. Jawerth & M. Milman - "Interpolation of weak type spaces", Math. Z. 201 (1989), no. 4, p. 509-519. | Zbl | EuDML | DOI

[52] B. Jawerth & A. Torchinsky - "Local sharp maximal functions", J. Approx. Theory 43 (1985), no. 3, p. 231-270. | Zbl | DOI

[53] F. John - "Quasi-isometric mappings", in Seminari 1962/63 Anal. Alg. Geom. e Topol, vol. 2, Ist. Naz. Alta Mat., Ediz. Cremonese, Rome, 1965, p. 462-473. | Zbl

[54] H. Johnen & K. Scherer - "On the equivalence of the $K$ -functional and moduli of continuity and some applications", in Constructive theory of functions of several variables (Proc. Conf, Math. Res. Inst., Oberwolfach, 1976), Lecture Notes in Math., vol. 571, Springer, Berlin, 1977, p. 119-140. | Zbl | DOI

[55] V. I. Kolyada - "Estimates for rearrangements and embedding theorems", Mat. Sb. (N.S.) 136 (1988), no. 1, p. 3-23, in Russian; English transl.: Math. USSR-Sb. 55 (1989), p. 1-21. | Zbl

[56] M. Krbec & H.-J. Schmeisser - "On dimension-free Sobolev imbeddings I", J. Math. Anal. Appl. 387 (2012), no. 1, p. 114-125. | Zbl | DOI

[57] M. Krbec & H.-J. Schmeisser, "On dimension-free Sobolev imbeddings II", Rev. Mat. Complut. 25 (2012), no. 1, p. 247-265. | Zbl | DOI

[58] S. G. Kreĭn, Y. Ī. Petunīn & E. M. Semënov - Interpolation of linear operators, Transl. Math. Monogr., vol. 54, Amer. Math. Soc., Providence, R.I., 1982. | Zbl

[59] M. Ledoux - "Isopérimétrie et inégalités de Sobolev logarithmiques gaussiennes", C. R. Acad. Sci. Paris Sér. I Math. 306 (1988), no. 2, p. 79-82. | Zbl

[60] G. Leoni - A first course in Sobolev spaces, Grad. Stud. Math., vol. 105, Amer. Math. Soc., Providence, RI, 2009. | Zbl

[61] A. K. Lerner - "On pointwise estimates for maximal and singular integral operators", Studia Math. 138 (2000), no. 3, p. 285-291. | Zbl | EuDML | DOI

[62] G. G. Lorentz - "On the theory of spaces $Λ$ ", Pacific J. Math. 1 (1951), p. 411-429. | Zbl | DOI

[63] J. Martín - "Symmetrization inequalities in the fractional case and Besov embeddings", J. Math. Anal. Appl. 344 (2008), no. 1, p. 99-123. | Zbl | DOI

[64] J. Martín & M. Milman - "Modes of convergence: interpolation methods. I", J. Approx. Theory 111 (2001), no. 1, p. 91-127. | Zbl | DOI

[65] J. Martín & M. Milman, "Symmetrization inequalities and Sobolev embeddings", Proc. Amer. Math. Soc. 134 (2006), no. 8, p. 2335-2347. | Zbl | DOI

[66] J. Martín & M. Milman, "A note on Sobolev inequalities and limits of Lorentz spaces", in Interpolation theory and applications, Contemp. Math., vol. 445, Amer. Math. Soc., Providence, RI, 2007, p. 237-245. | Zbl | DOI

[67] J. Martín & M. Milman, "Sharp Gagliardo-Nirenberg inequalities via symmetrization", Math. Res. Lett. 14 (2007), no. 1, p. 49-62. | Zbl | DOI

[68] J. Martín & M. Milman, "Self improving Sobolev-Poincaré inequalities, truncation and symmetrization", Potential Anal. 29 (2008), no. 4, p. 391-408. | Zbl | DOI

[69] J. Martín & M. Milman, "Isoperimetry and symmetrization for logarithmic Sobolev inequalities", J. Funct. Anal. 256 (2009), no. 1, p. 149-178. | Zbl | DOI

[70] J. Martín & M. Milman, "Pointwise symmetrization inequalities for Sobolev functions and applications", Adv. Math. 225 (2010), no. 1, p. 121-199. | Zbl

[71] J. Martín & M. Milman, "Sobolev inequalities, rearrangements, isoperimetry and interpolation spaces", in Concentration, functional inequalities and isoperimetry, Contemp. Math., vol. 545, Amer. Math. Soc., Providence, RI, 2011, p. 167-193. | Zbl | DOI

[72] J. Martín, M. Milman & E. Pustylnik - "Sobolev inequalities: symmetrization and self-improvement via truncation", J. Funct. Anal. 252 (2007), no. 2, p. 677-695. | Zbl | DOI

[73] M. Mastyło - "The modulus of smoothness in metric spaces and related problems", Potential Anal. 35 (2011), no. 4, p. 301-328. | Zbl | DOI

[74] V. G. Maz'Ja - "Classes of domains and imbedding theorems for function spaces", Soviet Math. Dokl. 1 (1960), p. 882-885. | Zbl

[75] V. G. Maz'Ya - Sobolev spaces with applications to elliptic partial differential equations, augmented ed., Grundlehren Math. Wiss., vol. 342, Springer, Heidelberg, 2011. | Zbl | DOI

[76] P.-A. Meyer - "Interpolation entre espaces d'Orlicz", in Seminar on Probability, XVI, Lecture Notes in Math., vol. 920, Springer, Berlin-New York, 1982, p. 153-158. | Numdam | EuDML | Zbl

[77] E. Milman - "On the role of convexity in isoperimetry, spectral gap and concentration", Invent. Math. 177 (2009), no. 1, p. 1-43. | Zbl | DOI

[78] M. Milman & E. Pustylnik - "On sharp higher order Sobolev embeddings", Commun. Contemp. Math. 6 (2004), no. 3, p. 495-511. | Zbl | DOI

[79] M. Miranda, Jr. - "Functions of bounded variation on "good" metric spaces", J. Math. Pures Appl. (9) 82 (2003), no. 8, p. 975-1004. | Zbl | DOI

[80] C. Mouhot, E. Russ & Y. Sire - "Fractional Poincaré inequalities for general measures", http://arxiv.org/abs/0911.4563. | Zbl | DOI

[81] T. H. Park - "Sobolev type inequalities", Ph.D. Thesis, Univ. San Diego, 1974, (under A. Garsia).

[82] J. Peetre - "A theory of interpolation of normed spaces", Lecture Notes, Brasilia, 1963. | Zbl

[83] J. Peetre, New thoughts on Besov spaces, Duke Univ. Math. Ser., vol. 1, Mathematics Department, Duke University, Durham, N.C., 1976. | Zbl

[84] G. Pisier - "Conditions d'entropie assurant la continuité de certains processus et applications à l'analyse harmonique", in Seminar on Functional Analysis, 1979- 1980 (French), École Polytech., Palaiseau, 1980, p. Exp. No. 13-14, 43. | Zbl | EuDML | Numdam

[85] J. M. Rakotoson & B. Simon - "Relative rearrangement on a finite measure space. Application to the regularity of weighted monotone rearrangement. I", Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.) 91 (1997), no. 1, p. 17-31. | Zbl | EuDML

[86] Y. Sagher & P. Shvartsman - "Rearrangement-function inequalities and interpolation theory", J. Approx. Theory 119 (2002), no. 2, p. 214-251. | Zbl | DOI

[87] S. Semmes - Some novel types of fractal geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2001. | Zbl

[88] R. Sharpley - "Spaces $Λ_{α} (X)$ and interpolation", J. Funct. Anal. 11 (1972), p. 479-513. | Zbl | DOI

[89] E. M. Stein - Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. | Zbl

[90] E. M. Stein, "Editor's note: the differentiability of functions in $𝐑^{n}$ ", Ann. of Math. (2) 113 (1981), no. 2, p. 383-385. | Zbl

[91] J.-O. Strömberg - "Bounded mean oscillation with Orlicz norms and duality of Hardy spaces", Indiana Univ. Math. J. 28 (1979), no. 3, p. 511-544. | Zbl | DOI

[92] R. Tessera - "Large scale Sobolev inequalities on metric measure spaces and applications", Rev. Mat. Iberoam. 24 (2008), no. 3, p. 825-864. | Zbl | EuDML | DOI

[93] A. Torchinsky - Real-variable methods in harmonic analysis, Pure Appl. Math., vol. 123, Academic Press, Inc., Orlando, FL, 1986. | Zbl

[94] H. Triebel - Interpolation theory, function spaces, differential operators, North-Holland Math. Libr. vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. | Zbl

[95] H. Triebel, "Tractable embeddings of Besov spaces into Zygmund spaces", in Function spaces IX, Banach Center Publ., vol. 92, Polish Acad. Sci. Inst. Math., Warsaw, 2011, p. 361-377. | Zbl

[96] J. Xiao & Z. Zhai - "Fractional Sobolev, Moser-Trudinger Morrey-Sobolev inequalities under Lorentz norms", J. Math. Sci. (N. Y.) 166 (2010), no. 3, p. 357-376. | Zbl | DOI

[97] W. P. Ziemer - Weakly differentiable functions, Grad. Texts in Math., vol. 120, Springer-Verlag, New York, 1989. | Zbl | DOI