Towards a Unified Theory of Sobolev Inequalities

  • Joaquim Martín
  • Mario Milman
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 108)


We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated with a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.



We are grateful to E. Milman for a number of useful comments that helped improve the presentation.

The author J. Martín was Partially supported in part by Grants MTM2010-14946, MTM-2010-16232.

The author M. Milman was partially supported by a grant from the Simons Foundation (#207929 to Mario Milman).


  1. 1.
    Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 314. Springer, Berlin (1999)Google Scholar
  2. 2.
    Allen, G.D.: Locally Continuous Operators II. Indiana Univ. Math. J. 38, 711–743 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Almgren, F., Lieb, E.: Symmetric Decreasing Rearrangement is sometimes continuous. J. Am. Math. Soc. 2, 683–773 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Alvino, A., Trombetti, G., Lions, P.L.: On optimization problems with prescribed rearrangements. Nonlinear Anal. 13, 185–220 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bakry, D., Coulhon, T., Ledoux, M., Saloff-Coste, L.: Sobolev inequalities in disguise. Indiana Univ. Math. J. 44, 1033–1074 (1995)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Barthe, F.: Levels of concentration between exponential and Gaussian. Ann. Fac. Sci. Toulouse Math. 10, 393–404 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Barthe, F.: Log-concave and spherical models in isoperimetry. Geom. Funct. Anal. 12, 32–55 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    F. Barthe, P. Cattiaux and C. Roberto, Isoperimetry between exponential and Gaussian, Orlicz hyper-contractivity and isoperimetry, Rev. Mat. Iber. 22 (2006), 993–1067.CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Barthe, F., Cattiaux, P., Roberto, C.: Isoperimetry between exponential and Gaussian. Electron. J. Probab. 12, 1212–1237 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bastero, J., Milman, M., Ruiz, F.: On the connection between weighted norm inequalities, commutators and real interpolation. preprint, Sem A. Galdeano (1996)Google Scholar
  11. 11.
    Bastero, J., Milman, M., Ruiz, F.: A note on L(, q) spaces and Sobolev embeddings. Indiana Math. J. 52, 1215–1230 (2003)CrossRefzbMATHGoogle Scholar
  12. 12.
    Bayle, V.: Propriétés de concavité du profil isopérimétrique et applications. Ph.D. thesis, Institut Joseph Fourier (2004)Google Scholar
  13. 13.
    Beckner, W., Persson, M.: On sharp Sobolev embedding and the logarithmic Sobolev inequality. Bull. Lond. Math. Soc. 30, 80–84 (1998)CrossRefGoogle Scholar
  14. 14.
    Bennett, C., DeVore, R., Sharpley, R.: Weak-L and BMO. Ann. Math. 113, 601–611 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)zbMATHGoogle Scholar
  16. 16.
    Bergh, J., Löfström, J.: Interpolation Spaces: An Introduction. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  17. 17.
    Bobkov, S.G.: Isoperimetric and analytic inequalities for log-concave probability measures. Ann. Probab. 27, 1903–1921 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Bobkov, S.G., Houdré, C.: Some connections between isoperimetric and Sobolev-type inequalities. Mem. Am. Math. Soc. 129 (1997).Google Scholar
  19. 19.
    Bobkov, S.G., Zegarlinski, B.: Entropy bounds and isoperimetry. Mem. Am. Math. Soc. 176 (2005)Google Scholar
  20. 20.
    Bobkov, S.G., Zegarlinski, B.: Distributions with slow tails and ergodicity of Markov semigroups in infinite dimensions. In: Laptev, A. (ed.) Around the Research of Vladimir Maz’ya I: Function Spaces, pp 13–79. Springer, New York (2010)CrossRefGoogle Scholar
  21. 21.
    Borell, C.: The Ehrhard inequality. C. R. Math. Acad. Sci. Paris 337, 663–666 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Borell, C.: The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Brezis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convolution inequalities. Commun. Partial Differ. Equ. 5, 773–789 (1980)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Calderón, A.P.: Spaces between L 1and L and the theorem of Marcinkiewicz. Stud. Math. 26, 273–299 (1966)zbMATHGoogle Scholar
  25. 25.
    Cianchi, A., Pick, L.: Optimal Gaussian Sobolev embeddings. J. Funct. Anal. 256, 3588–3642 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Coulhon, T.: Espaces de Lipschitz et inégalités de Poincaré. J. Funct. Anal. 136, 81–113 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Coulhon, T.: Dimensions at infinity for Riemannian manifolds. Potential Anal. 4, 335–344 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Coulhon, T.: Heat kernel and isoperimetry on non-compact Riemmanian manifolds. Contemp. Math. 338, 65–99 (2003)CrossRefMathSciNetGoogle Scholar
  29. 29.
    Cwikel, M., Jawerth, B., Milman, M.: A note on extrapolation of inequalities, preprint (2010)Google Scholar
  30. 30.
    Cwikel, M., Pustylnik, E.: Sobolev type embeddings in the limiting case. J. Fourier Anal. Appl. 4, 433–446 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Ditzian, Z., Totik, V.: Moduli of Smoothness. Springer, New York (1987)CrossRefzbMATHGoogle Scholar
  32. 32.
    Ditzian, Z., Ivanov, K.G.: Strong converse inequalities. J. D’Analise Math. 61, 61–111 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Ditzian, Z., Lubinsky, D.S.: Jackson and smoothness theorems for Freud weights in L p (0 < p < ). Constr. Approx. 13, 99–152 (1997)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces and Embeddings. Springer, Berlin (2004)zbMATHGoogle Scholar
  35. 35.
    Ehrhard, A.: Symétrisation dans l’espace de Gauss. Math. Scand. 53, 281–301 (1983)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Ehrhard, A.: Sur l’inégalité de Sobolev logarithmique de Gross. In: Séminaire de Probabilités XVII. Lecture Notes in Mathematics, vol. 1059, pp. 194–196. Springer, Heidelberg (1984)Google Scholar
  37. 37.
    Ehrhard, A.: Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes. Ann. Sci. Ecole. Norm. Sup. 17, 317–332 (1984)zbMATHMathSciNetGoogle Scholar
  38. 38.
    Fiorenza, A.: Duality and reflexivity in grand Lebesgue spaces. Collect. Math. 51, 131–148 (2000)zbMATHMathSciNetGoogle Scholar
  39. 39.
    Fiorenza, A., Karadzhov, G.E.: Grand and Small Lebesgue Spaces and their analogs. Z. Anal. Anwendungen 23, 657–681 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Fiorenza, A., Krbec, M., Schmeisser, H.J.: An improvement of dimension-free Sobolev imbeddings in r.i. spaces, Journal of Functional Analysis 267, 243–261 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Fontana, L., Morpurgo, C.: Optimal limiting embeddings for Δ-reduced Sobolev spaces in L 1. Ann. de l’Inst. Henri Poincaré (C) Non Linear Analysis, 31, 217–230 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Gallot, S.: Inégalités isopérimétriques et analytiques sur les variétés Riemanniennes. Astérisque 163–164, 31–91 (1988)Google Scholar
  43. 43.
    Garsia, A.M.: Combinatorial inequalities and smoothness of functions. Bull. Am. Math. Soc. 82, 157–170 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    Garsia, A.M.: A remarkable inequality and the uniform convergence of Fourier series. Indiana Univ. Math. J. 25, 85–102 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Garsia, A., Rodemich, E.: Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier 24, 67–116 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Griebel, M.: Sparse grids and related approximation schemes for higher dimensional problems. In: Pardo, L., Pinkus, A., Suli, E., Todd, M. (eds.) Foundations of Computational Mathematics (FoCM05), Santander, pp. 106–161. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  47. 47.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)CrossRefGoogle Scholar
  48. 48.
    Hajlasz, P.: Sobolev inequalities, truncation method, and John domains. In: Papers in Analysis, Rep. Univ. Jyväskylä Dep. Math. Stat. 83, Univ. Jyväskylä, Jyväskylä, pp 109–126 (2001)Google Scholar
  49. 49.
    Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Am. Math. Soc. 145, 101 pages (2000)Google Scholar
  50. 50.
    Hansson, K.: Imbedding theorems of Sobolev type in potential theory. Math Scand 45, 77–102 (1979)zbMATHMathSciNetGoogle Scholar
  51. 51.
    Hedberg, L.I.: On Maz’ya’s work in potential theory and the theory of function spaces. In: The Maz’ya Anniversary Collection, vol. 1, pp. 7–16, Rostock, 1998. Operator Theory: Advances and Applications, vol. 109. Birkhäuser, Basel (1999)Google Scholar
  52. 52.
    Heinonen, J.: Lectures on Analysis on metric spaces. Lecture Notes. University of Michigan (1996)Google Scholar
  53. 53.
    Houdre, C., Ledoux, M., Milman, E., Milman, M.: Concentration, functional inequalities and isoperimetry. Contemp. Math. 545 (2011)Google Scholar
  54. 54.
  55. 55.
    Iwaniec, T., Sbordone, C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Rational Mech. Anal. 119, 129–143 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  56. 56.
    Jawerth, B., Milman, M.: Extrapolation theory with applications. Mem. Am. Math. Soc. 89, 440 (1991)MathSciNetGoogle Scholar
  57. 57.
    Jawerth, B., Milman, M.: Interpolation of weak type spaces. Math. Z. 201, 509–520 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    Johnen, H., Scherer, K.: On the equivalence of the K-functional and moduli of continuity and some applications. In: Constructive Theory of Functions of Several Variables. Lecture Notes in Mathematics, vol. 571, pp. 119–140. Springer, Berlin (1977)Google Scholar
  59. 59.
    Karadzhov, G.E., Mehmood, Q.: Optimal regularity properties of the generalized Sobolev spaces. J. Funct. Spaces Appl. 761648, p 10 (2013)MathSciNetGoogle Scholar
  60. 60.
    Karadzhov, G.E., Milman, M.: Extrapolation theory: new results and applications. J. Approx. Theory 133, 38–99 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  61. 61.
    Kesavan, S.: Symmetrization and Applications. World Scientific, Hackensack (2006)zbMATHGoogle Scholar
  62. 62.
    Kolyada, V.I. (1989) Rearrangements of functions and embedding theorems. Uspekhi Mat. Nauk 44, 61–95 (1989); transl. Russ. Math. Surv. 44, 73–117 (1989)Google Scholar
  63. 63.
    Krbec, M., Schmeisser, H.J.: On dimension-free Sobolev imbeddings I. J. Math. Anal. Appl. 387, 114–125 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  64. 64.
    Krbec, M., Schmeisser, H.J.: On dimension-free Sobolev imbeddings II. Rev. Mat. Complutense 25, 247–265 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  65. 65.
    Ledoux, M.: Isoperimetry and gaussian analysis. Ecole d’Eté de Probabilités de Saint-Flour 1994. Springer Lecture Notes, vol. 1648, pp 165–294. Springer, Heidelberg (1996)Google Scholar
  66. 66.
    Ledoux, M.: Isopérimétrie et inégalitées de Sobolev logarithmiques gaussiennes. C. R. Acad. Sci. Paris Ser. I Math. 306, 79–92 (1988)zbMATHMathSciNetGoogle Scholar
  67. 67.
    Ledoux, M.: The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89. American Mathematical Society, Providence (2001)Google Scholar
  68. 68.
    Ledoux, M.: From concentration to isoperimetry: Semigroup proofs. Contemp. Math. 545, 155–166 (2011)CrossRefMathSciNetGoogle Scholar
  69. 69.
    Leoni, G.: A First Course in Sobolev Spaces. Graduate Studies in Mathematics, vol. 105. American Mathematical Society, New York (2009)Google Scholar
  70. 70.
    Malý, J., Pick, L.: An elementary proof of Sharp Sobolev embeddings. Proc. Am. Math. Soc. 130, 555–563 (2002)CrossRefzbMATHGoogle Scholar
  71. 71.
    Martín, J., Milman, M.: Symmetrization inequalities and Sobolev embeddings. Proc. Am. Math. Soc. 134, 2335–2347 (2006)CrossRefzbMATHGoogle Scholar
  72. 72.
    Martín, J., Milman, M.: Higher-order symmetrization inequalities and applications. J. Math. Anal. Appl. 330, 91–113 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  73. 73.
    Martin, J., Milman, M.: A note on Sobolev inequalities and limits of Lorentz spaces. Contemp. Math. 445, 237–245 (2007)CrossRefMathSciNetGoogle Scholar
  74. 74.
    Martin, J., Milman, M.: Isoperimetry and Symmetrization for Logarithmic Sobolev inequalities. J. Funct. Anal. 256, 149–178 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  75. 75.
    Martin, J., Milman, M.: Addendum to Isoperimetry and Symmetrization for Logarithmic Sobolev inequalities. arXiv:0901.1839Google Scholar
  76. 76.
    Martin, J., Milman, M.: Isoperimetry and Symmetrization for Sobolev spaces on metric spaces. Comptes Rendus Math. 347, 627–630 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  77. 77.
    Martín, J., Milman, M.: Isoperimetric Hardy type and Poincaré inequalities on metric spaces. In: Laptev, A. (ed.) Around the Research of Vladimir Maz’ya I. Function Spaces. International Mathematical Series, vol. 11, pp. 285–298. Springer, New York (2010)Google Scholar
  78. 78.
    Martin, J., Milman, M.: Pointwise symmetrization inequalities for Sobolev functions and applications. Adv. Math. 225, 121–199 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  79. 79.
    Martin, J., Milman, M.: Sobolev inequalities, rearrangements, isoperimetry and interpolation spaces. Contemp. Math. 545, 167–193 (2011)CrossRefMathSciNetGoogle Scholar
  80. 80.
    Martin, J., Milman, M.: Fractional Sobolev inequalities: symmetrization, isoperimetry and interpolation, to appear in Astérisque (arXiv:1205.1584)Google Scholar
  81. 81.
    Martin, J., Milman, M.: A note on Coulhon type inequalities, to appear in Proc. Am. Math. Soc. (arXiv:1206.1584)Google Scholar
  82. 82.
    Martin, J., Milman, M.: Integral isoperimetric transference and dimensionless Sobolev inequalities, to appear in Revista Matemática Complutense (arXiv:1309.1980)Google Scholar
  83. 83.
    Martin, J., Milman, M.: A note on iterated Sobolev inequalities involving the isoperimetric profile, preprintGoogle Scholar
  84. 84.
    Martin, J., Milman, M.: On the Calderón-Maz’ya-Rubio de Francia extrapolation principle, preprint (2013)Google Scholar
  85. 85.
    Martin, J., Milman, M.: Symmetrization methods in the theory of Sobolev inequalities. Lecture Notes, in preparationGoogle Scholar
  86. 86.
    Martin, J., Milman, M., Pustylnik, E.: Sobolev inequalities: symmetrization and self improvement via truncation. J. Funct. Anal. 252, 677–695 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  87. 87.
    Mastylo, M.: The modulus of smoothness in metric spaces and related problems. Potential Anal. 35, 301–328 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  88. 88.
    Maz’ya, V.G.: 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)Google Scholar
  89. 89.
    Maz’ya, V.G.: The p-conductivity and theorems on imbedding certain functional spaces into a C-space (Russian), Dokl. Akad. Nauk SSSR 140, 299–302 (1961) (English translation: in Soviet Math. Dokl. 3 (1962)Google Scholar
  90. 90.
    Milman, E.: Concentration and isoperimetry are equivalent assuming curvature lower bound. C. R. Math. Acad. Sci. Paris 347, 73–76 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  91. 91.
    Milman, E.: On the role of convexity in isoperimetry, spectral-gap and concentration. Invent. Math. 177, 1–43 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  92. 92.
    Milman, E.: On the role of convexity in functional and isoperimetric inequalities. Proc. Lond. Math. Soc. Proc. 999, 32–66 (2009)CrossRefMathSciNetGoogle Scholar
  93. 93.
    Milman, E.: Isoperimetric and concentration inequalities - equivalence under curvature lower bound. Duke Math. J. 154, 207–239 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  94. 94.
    Milman, E.: A converse to the Maz’ya inequality for capacities under curvature lower bound. In: Laptev, A. (ed.) Around the Research of Vladimir Maz’ya I: Function Spaces, pp. 321–348. Springer, New York (2010)CrossRefGoogle Scholar
  95. 95.
    Milman, E.: Isoperimetric bounds on convex manifolds. Contemp. Math 545, 195–208 (2011)CrossRefMathSciNetGoogle Scholar
  96. 96.
    Milman, M.: Local operators vs Lorentz-Marcinkiewicz spaces. Interpolation Spaces and Related Topics (Haifa, 1990). Israel Mathematical Conference Proceedings, vol. 5, pp. 151–157 (1992)MathSciNetGoogle Scholar
  97. 97.
    Milman, M., Pustylnik, E.: On sharp higher order Sobolev embeddings. Commun. Contemp. Math. 6, 495–511 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  98. 98.
    Oklander, E.: Interpolacion, espacios de Lorentz y teorema de Marcinkiewicz. In: Cursos y Seminarios 20, Univ. Buenos Aires (1965) [See also Oklander, E.: On interpolation of Banach spaces, Thesis, Univ. Chicago (1963)]Google Scholar
  99. 99.
    O’Neil, R.: Convolution operators and L(p,q) spaces. Duke Math. J. 30, 129–142 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  100. 100.
    Pérez Lázaro, F.J.: A note on extreme cases of Sobolev embeddings. J. Math. Anal. Appl. 320, 973–982 (2006)CrossRefMathSciNetGoogle Scholar
  101. 101.
    Pick, L., Kufner, A., John, O., Fucik, S.: Function Spaces, vol. 1. Walter de Gruyter & Co, Berlin (2012)CrossRefGoogle Scholar
  102. 102.
    Pisier, G.: Factorization of operators through L p∞ or L p1 and non-commutative generalizations. Math. Ann. 276, 105–136 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  103. 103.
    Pustylnik, E.: On compactness of Sobolev embeddings in rearrangement-invariant spaces. Forum Math. 18, 839–852 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  104. 104.
    Rakotoson, J.M.: Réarrangement relatif. Un instrument d’estimations dans les problèmes aux limites. Mathematics & Applications, vol. 64. Springer, Berlin (2008)Google Scholar
  105. 105.
    Ros, A.: The isoperimetric problem. In: Global Theory of Minimal Surfaces. Clay Mathematics Proceedings, vol. 2, pp. 175–209. American Mathematical Society, Providence (2005)Google Scholar
  106. 106.
    Rota, G.C.: Ten Lessons I wish I had been Taught.
  107. 107.
    Saloff-Coste, L.: Aspects of Sobolev Inequalities. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  108. 108.
    Slavíková, L.: Compactness of higher order Sobolev embeddings, Master Thesis, Charles University (2012)Google Scholar
  109. 109.
    Sudakov, V.N., Tsirelson, B.S.: Extremal properties of half-spaces for spherically invariant measures. J. Soviet. Math. 9, 918 (1978); translated from Zap. Nauch. Sem. L.O.M.I. 41, 1424 (1974)Google Scholar
  110. 110.
    Talenti, G.: Inequalities in rearrangement-invariant function spaces. In: Nonlinear Analysis, Function Spaces and Applications, vol. 5, pp. 177–230. Prometheus, Prague (1995) (for a comprehensive bibliography)Google Scholar
  111. 111.
    Tartar, L.: Imbedding theorems of Sobolev spaces into Lorentz spaces. Boll. Unione Mat. Ital. Sez B Artic. Ric. Mat. 1(8), 479–500 (1998)zbMATHMathSciNetGoogle Scholar
  112. 112.
    Trudinger, N.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17 473–483 (1967)zbMATHMathSciNetGoogle Scholar
  113. 113.
    Triebel, H.: Tractable embeddings of Besov spaces into Zygmund spaces. Function spaces IX, vol. 92, pp. 361–377. Banach Center, Polish Acad. Sci. Inst. Math., Warsaw (2011)Google Scholar
  114. 114.
    Triebel, H.: Tractable embeddings, preprint, University of Jena (2012)Google Scholar
  115. 115.
    Xiao, J., Zhai, Z.: Fractional Sobolev, Moser-Trudinger, Morrey-Sobolev inequalities under Lorentz norms. J. Math. Sci. 166, 357–376 (2010)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversitat Autònoma de BarcelonaBarcelonaSpain
  2. 2.Department of MathematicsFlorida Atlantic UniversityBoca RatonUSA

Personalised recommendations