Optimality of Function Spaces in Sobolev Embeddings

  • Luboš Pick
Part of the International Mathematical Series book series (IMAT, volume 8)


Abstract We study the optimality of function spaces that appear in Sobolev embeddings. We focus on rearrangement-invariant Banach function spaces. We apply methods of interpolation theory.

It is a great honor for me to contribute to this volume dedicated to the centenary of S.L. Sobolev, one of the greatest analysts of the XXth century. The paper concerns a topic belonging to an area bearing the name, called traditionally Sobolev inequalities or Sobolev embeddings. The focus will be on the sharpness or optimality of function spaces appearing in these embeddings. The results presented in this paper were established in recent years. Most of them were obtained in collaboration with Ron Kerman and Andrea Cianchi.


Function Space Sobolev Inequality Lebesgue Space Orlicz Space Domain Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, Amsterdam (2003)MATHGoogle Scholar
  2. 2.
    Bastero, J., Milman, M., Ruiz, F.: A note in L(∞, q) spaces and Sobolev embeddings. Indiana Univ. Math. J. 52, 1215–1230 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bennett, C., DeVore, R., Sharpley, R.: Weak L and BMO. Ann. Math. 113, 601– 611 (1981)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bennett, C., Rudnick, K.: On Lorentz-Zygmund spaces. Dissert. Math. 175, 1–72 (1980)MathSciNetGoogle Scholar
  5. 5.
    Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)MATHGoogle Scholar
  6. 6.
    Brézis, H., Wainger, S.: A note on limiting cases of Sobolev embeddings and convo lution inequalities. Commun. Partial Differ. Equ. 5, 773–789 (1980)MATHCrossRefGoogle Scholar
  7. 7.
    Brudnyi, Yu.A.: Rational approximation and imbedding theorems (Russian). Dokl. Akad. Nauk SSSR 247, 269–272 (1979); English transl.: Sov. Math., Dokl. 20, 681– 684 (1979)MathSciNetGoogle Scholar
  8. 8.
    Brudnyi, Yu.A., Kruglyak, N.Ya.: Interpolation Functors and Interpolation Spaces, Vol. 1. North-Holland (1991)Google Scholar
  9. 9.
    Carro, M., Gogatishvili, A., Martín, J., Pick, L.: Functional properties of rearran gement invariant spaces defined in terms of oscillations. J. Funct. Anal. 229, no. 2, 375–404 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Carro, M., Gogatishvili, A., Martín, J., Pick, L.: Weighted inequalities involving two Hardy operators with applications to embeddings of function spaces. J. Operator Theory 59, no. 2, 101–124 (2008)Google Scholar
  11. 11.
    Cianchi, A.: A sharp embedding theorem for Orlicz-Sobolev spaces. Indiana Univ. Math. J. 45, 39–65 (1996)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Cianchi, A.: Symmetrization and second order Sobolev inequalities. Ann. Mat. Pura Appl. 183, 45–77 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cianchi, A.: Optimal Orlicz-Sobolev embeddings. Rev. Mat. Iberoam. 20, 427–474 (2004)MATHMathSciNetGoogle Scholar
  14. 14.
    Cianchi, A., Kerman, R., Opic, B., Pick, L.: A sharp rearrangement inequality for fractional maximal operator. Stud. Math. 138, 277–284 (2000)MATHMathSciNetGoogle Scholar
  15. 15.
    Cianchi, A., Kerman, R., Pick, L.: Boundary trace inequalities and rearrangements. J. d'Analyse Math. [To appear]Google Scholar
  16. 16.
    Cianchi, A., Pick, L.: Optimal Gaussian Sobolev embeddings. Manuscript (2008)Google Scholar
  17. 17.
    Cwikel, M., Kamińska, A., Maligranda, L., Pick, L.: Are generalized Lorentz “spaces” really spaces? Proc. Am. Math. Soc. 132, 3615–3625 (2004)MATHCrossRefGoogle Scholar
  18. 18.
    Cwikel, M., Pustylnik, E.: Weak type interpolation near “endpoint” spaces. J. Funct. Anal. 171, 235–277 (1999)CrossRefMathSciNetGoogle Scholar
  19. 19.
    DeVore, R.A., Scherer, K.: Interpolation of linear operators on Sobolev spaces. Ann. Math. 109, 583–599 (1979)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Edmunds, D.E., Kerman, R., Pick, L.: Optimal Sobolev imbeddings involving rearrangement-invariant quasinorms. J. Funct. Anal. 170, 307–355 (2000)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Evans, W.D., Opic, B., Pick, L.: Interpolation of operators on scales of generalized Lorentz-Zygmund spaces. Math. Nachr. 182, 127–181 (1996)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Gogatishvili, A., Opic, B., Pick, L.: Weighted inequalities for Hardy type operators involving suprema. Collect. Math. 57, no. 3, 227–255 (2006)MATHMathSciNetGoogle Scholar
  23. 23.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97, 1061–1083 (1975)CrossRefGoogle Scholar
  24. 24.
    Hempel, J.A., Morris, G.R., Trudinger, N.S.: On the sharpness of a limiting case of the Sobolev imbedding theorem. Bull. Australian Math. Soc. 3, 369–373 (1970)MATHMathSciNetCrossRefGoogle Scholar
  25. 25.
    Holmstedt, T.: Interpolation of quasi-normed spaces. Math. Scand. 26, 177–199 (1970)MATHMathSciNetGoogle Scholar
  26. 26.
    Hunt, R.: On L(p, q) spaces. Enseign. Math. 12, 249–276 (1966)MATHGoogle Scholar
  27. 27.
    Kerman, R., Milman, M., Sinnamon, G.: On the Brudnyi–Krugljak duality theory of spaces formed by the K-method of interpolation. Rev. Mat. Compl. 20, 367–389 (2007)MATHMathSciNetGoogle Scholar
  28. 28.
    Kerman, R., Pick, L.: Optimal Sobolev imbeddings. Forum Math 18, no. 4, 535–570 (2006)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Kerman, R., Pick, L.: Optimal Sobolev imbedding spaces. Preprint No. MATH-KMA-2005/161, Department of Mathematical Analysis, Fac ulty of Mathematics and Physics, Charles University, Prague, 2005, 1–19, http://www.karlin.mff.cuni.cz/’rokyta/preprint/index.php. [Submitted for pub lication]
  30. 30.
    Kerman, R., Pick, L.: Compactness of Sobolev imbeddings involving rearrangement-invariant norms. Stud. Math. [To appear]Google Scholar
  31. 31.
    Kerman, R., Pick, L.: Explicit formulas for optimal rearrangement-invariant norms in Sobolev imbedding inequalities. (2008) [In preparation]Google Scholar
  32. 32.
    Kondrachov, V.I.: Certain properties of functions in the space L p (Russian, French). Dokl. Akad. Nauk SSSR 48, 535–538 (1945)MathSciNetGoogle Scholar
  33. 33.
    Malý, J., Pick, L.: An elementary proof of sharp Sobolev embeddings. Proc. Am. Math. Soc. 130, 555–563 (2002)MATHCrossRefGoogle Scholar
  34. 34.
    Maz'ya, V.G.: Sobolev Spaces. Springer-Verlag, Berlin-Tokyo (1985)Google Scholar
  35. 35.
    O'Neil, R.: Convolution operators and L(p,q)spaces. Duke Math. J. 30, 129–142 (1963)MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Nelson, E.: The free Markoff field. J. Funct. Anal. 12, 221–227 (1973)Google Scholar
  37. 37.
    Peetre, J.: Espaces d'interpolation et théorème de Soboleff. Ann. Inst. Fourier 16, 279–317 (1966)MATHMathSciNetGoogle Scholar
  38. 38.
    Pick, L.: Supremum operators and optimal Sobolev inequalities. In: Function Spaces, Differential Operators and Nonlinear Analysis, Jpp. 207–219. Math. Inst. AS CR, Prague (2000)Google Scholar
  39. 39.
    Pick, L.: Optimal Sobolev Embeddings. Rudolph-Lipschitz-Vorlesungsreihe no. 43, Rheinische Friedrich-Wilhelms-Universitat Bonn (2002)Google Scholar
  40. 40.
    Pokhozhaev, S.I.: Eigenfunctions of the equation Δu+ λf(u)= 0 (Russian). Dokl. Akad. Nauk SSSR 165, 36–39 (1965); English transl.: Sov. Math., Dokl. 6, 1408–1411 (1965)MathSciNetGoogle Scholar
  41. 41.
    Rellich, F.: Ein Satz über mittlere Konvergenz. Gött. Nachr. 30–35 (1930)Google Scholar
  42. 42.
    Sawyer, E.: Boundedness of classical operators on classical Lorentz spaces. Stud. Math. 96, 145–158 (1990)MATHMathSciNetGoogle Scholar
  43. 43.
    Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MATHMathSciNetGoogle Scholar
  45. 45.
    Yudovich, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations (Russian). Dokl. Akad. Nauk SSSR 138, 805–808 (1961); English transl.: Sov. Math., Dokl. 2, 746–749 (1961)MathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Luboš Pick
    • 1
  1. 1.Charles UniversityPraha 8

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