Skip to main content
SpringerLink
Log in

Morrey spaces, their duals and preduals

  • Published:
Revista Matemática Complutense Aims and scope Submit manuscript

Abstract

This survey deals with Morrey spaces, their duals and preduals in the framework of tempered distributions on Euclidean spaces. We concentrate on basic assertions (including density and non-separability), duality, embeddings (including relations to distinguished Besov spaces) and applications to Calderón–Zygmund operators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, D.R.: A note on Riesz potentials. Duke Math. J. 42, 765–778 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  2. Adams, D.R.: A note on Choquet integrals with respect to Hausdorff capacity. In: Function Spaces and Applications. Proc., Lund, 1986, Lecture Notes in Mathematics, vol. 1302, pp. 115–124. Springer, Berlin (1988)

  3. Adams, D.R., Xiao, J.: Nonlinear potential analysis on Morrey spaces and their capacities. Indiana Univ. Math. J. 53, 1629–1663 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  4. Adams, D.R., Xiao, J.: Morrey spaces in harmonic analysis. Ark. Mat. 50, 201–230 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  5. Alvarez, J.: Continuity of Calderón-Zygmund type operators on the predual of a Morrey space. In: Clifford Algebras in Analysis and Related Topics, pp. 309–319. CRC Press, Boca Raton (1996)

  6. Banach, S.: Sur les fontionnelles linéaires. Studia Math. 1(211–216), 223–239 (1929)

    MATH  Google Scholar 

  7. Bennett, C., Sharpley, R.: Interpolation of Operators. Academic Press, Boston (1988)

    MATH  Google Scholar 

  8. Besov, O.V., Il’in, V.P., Nikol’skij, S.M.: Integral representations of functions and embedding theorems (Russian). Nauka, Moskva, 1975, 2nd ed. 1996; Engl. transl: Wiley, New York, 1978/79

  9. Brudnyi, Y.: Spaces defined by local approximations. Tr. Mosk. Mat. Obshch. 24, 69–132 (1971) (Russian); Engl. transl.: Trans. Mosc. Math. Soc. 24, 73–139 (1971)

    Google Scholar 

  10. Brudnyi, Y.: Sobolev spaces and their relatives: local polynomial approximation approach. In: Sobolev Spaces in Mathematics, II, pp. 31–68. Springer, New York (2009)

  11. Calderón, A.P., Zygmund, A.: On the existence of certain singular integrals. Acta Math. 88, 85–139 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  12. Campanato, S.: Proprietà di hölderianità di alcune classi die funzioni. Ann. Scuola Norm. Sup. Pisa 17, 175–188 (1963)

    MATH  MathSciNet  Google Scholar 

  13. Campanato, S.: Proprietà di una famiglia di spazi funzionali. Ann. Scuola Norm. Sup. Pisa 18, 137–160 (1964)

    MATH  MathSciNet  Google Scholar 

  14. Cobos, F., Fernández-Cabrera, L.M., Manzano, A., Martínez, A.: Logarithmic interpolation spaces between quasi-Banach spaces. Z. Anal. Anwend. 26, 65–86 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Duoandikoetxea, J.: Fourier Analysis. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  16. Duoandikoetxea, J., Rubio de Francia, J.L.: Maximal and singular integral operators via Fourier transform estimates. Invent. Math. 84, 541–561 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  17. Edmunds, D.E., Evans, W.D.: Hardy Operators, Function Spaces and Embeddings. Springer, Berlin (2004)

    Book  MATH  Google Scholar 

  18. Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  19. Gogatishvili, A., Mustafayev, R.Ch.: New pre-dual space of Morrey space. J. Math. Anal. Appl. 397(2), 678–692 (2013)

    Google Scholar 

  20. Kalita, E.A.: Dual Morrey spaces. Dokl. Akad. Nauk 361, 447–449 (1998) (Russian); Engl. transl. Dokl. Math. 58, 85–87 (1998)

  21. Karadzhov, G.E., Milman, M.: Extrapolation theory: new results and applications. J. Approx. Theory 133, 38–99 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kufner, A., John, O., Fučik, S.: Function Spaces. Academia, Prague (1977)

    MATH  Google Scholar 

  23. Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126–166 (1938)

    Article  MathSciNet  Google Scholar 

  24. Nikol’skij, S.M.: On a property of \(H^{(r)}_p\) classes (Russian). Ann. Univ. Sci. Budapest, Sect. Math. 3–4, 205–216 (1960/61). In: Collected Papers, vol. 2, pp. 228–239. ’Function Spaces’, Moscow, Nauka (2007)

  25. Peetre, J.: On convolution operators leaving \(L^{p, \lambda }\) spaces invariant. Ann. Mat. Pura Appl. 72, 295–304 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  26. Peetre, J.: On the theory of \(\cal L_{p, \lambda }\) spaces. J. Funct. Anal. 4, 71–87 (1969)

  27. Piccinini, L.C.: Proprietà di inclusione e interpolazione tra spazi di Morrey e loro generalizzazioni, pp. 1–153. Tesi di perfezionamento. Publicazione Scuola Normale Superiore, Classe di Scienze, Pisa (1969)

  28. Pick, L., Kufner, A., John, O., Fučik, S.: Function spaces, vol. 1, 2nd edn. De Gruyter, Berlin (2013)

  29. Pietsch, A.: History of Banach Spaces and Linear Operators. Birkhäuser, Boston (2007)

    MATH  Google Scholar 

  30. Rafeiro, H., Samko, N., Samko, S.: Morrey-Campanato spaces: an overview. In: Karlovich, Yi., Rodino, L., Silbermann, B., Spitkovsky, IM. (eds.) Operator Theory, Pseudo-Differential Equations, and Mathematical Physics. Advances and Applications, vol. 228, pp. 293–323. Springer, Basel (2013)

  31. Rosenthal, M.: Mapping properties of operators in Morrey spaces and wavelet isomorphisms in related Morrey smoothness spaces. PhD-Thesis, Jena (2013)

  32. Rosenthal, M., Triebel, H.: Calderón-Zygmund operators in Morrey spaces. Rev. Mat. Complut. 27, 1–11 (2014)

    Google Scholar 

  33. Rudin, W.: Functional Analysis, 2nd edn. McGraw-Hill, New York (1991)

    MATH  Google Scholar 

  34. Sickel, W., Triebel, H.: Hölder inequalities and sharp embeddings in function spaces of \(B^s_{pq}\) and \(F^s_{pq}\) type. Z. Anal. Anwend. 14, 105–140 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  35. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  36. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  37. Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Academic Press, San Diego (1986)

    MATH  Google Scholar 

  38. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978) (2nd ed., Barth, Heidelberg, 1995)

  39. Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  40. Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)

    Book  MATH  Google Scholar 

  41. Triebel, H.: Higher Analysis. J.A. Barth, Leipzig (1992)

    MATH  Google Scholar 

  42. Triebel, H.: Theory of Function Spaces III. Birkhäuser, Basel (2006)

    MATH  Google Scholar 

  43. Triebel, H.: Local Function Spaces, Heat and Navier–Stokes Equations. European Mathematical Society Publishing House, Zürich (2013)

    Book  MATH  Google Scholar 

  44. Triebel, H.: Characterizations of some function spaces in terms of Haar wavelets. Commentationes Math. 53, 35–53 (2013)

    Google Scholar 

  45. Yang, D., Yuan, W.: A new class of function spaces connecting Triebel–Lizorkin spaces and \(Q\) spaces. J. Funct. Anal. 255, 2760–2809 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  46. Yang, D., Yuan, W.: Dual properties of Triebel–Lizorkin-type spaces and their applications. Z. Anal. Anwend. 30, 29–58 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  47. Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

  48. Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics 2005. Springer, Heidelberg (2010)

  49. Zorko, C.T.: Morrey spaces. Proc. Am. Math. Soc. 98, 586–592 (1986)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We wish to thank the referees for careful reading and suggestions.

Author information

Authors and Affiliations

  1. Mathematisches Institut, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität Jena, 07737 , Jena, Germany

    Marcel Rosenthal & Hans Triebel

Authors
  1. Marcel Rosenthal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hans Triebel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hans Triebel.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rosenthal, M., Triebel, H. Morrey spaces, their duals and preduals. Rev Mat Complut 28, 1–30 (2015). https://doi.org/10.1007/s13163-013-0145-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13163-013-0145-z

Keywords

Mathematics Subject Classification (2010)

Search

Navigation