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Complex interpolation of weighted Besov and Lizorkin-Triebel spaces

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Abstract

We study complex interpolation of weighted Besov and Lizorkin-Triebel spaces. The used weights w 0, w 1 are local Muckenhoupt weights in the sense of Rychkov. As a first step we calculate the Calderón products of associated sequence spaces. Finally, as a corollary of these investigations, we obtain results on complex interpolation of radial subspaces of Besov and Lizorkin-Triebel spaces on ℝd.

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References

  1. Andersen, K. F., John, R. T.: Weighted inequalities for vector-valued maximal functions and singular integrals. Studia Math., 69, 19–31 (1980/81)

    MATH  MathSciNet  Google Scholar 

  2. Bennett, C., Sharpley, R.: Interpolation of Operators, Academic Press, Boston, 1988

    MATH  Google Scholar 

  3. Bergh, J., Löfström, J.: Interpolation Spaces. An Introduction, Springer-Verlag, New York, 1976

    Book  MATH  Google Scholar 

  4. Bownik, M.: Duality and interpolation of anisotropic Triebel-Lizorkin spaces. Math. Z., 259(1), 131–169 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Bownik, M., Ho, K.-P.: Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces. Trans. Amer. Math. Soc., 358(4), 1469–1510 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brudnyi, Yu. A., Kruglyak, N. Ya.: Interpolation Functors and Interpolation Spaces, North Holland, Amsterdam, 1991

    MATH  Google Scholar 

  7. Bui, H.-Q.: Weighted Besov and Triebel spaces: Interpolation by the real method. Hiroshima Math. J., 12(3), 581–605 (1982)

    MATH  MathSciNet  Google Scholar 

  8. Bui, H.-Q.: Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures. J. Funct. Anal., 55(1), 39–62 (1984)

    Article  MathSciNet  Google Scholar 

  9. Bui, H.-Q., Paluszyński, M., Taibleson, M. H.: A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math., 119(3), 219–246 (1996)

    MATH  MathSciNet  Google Scholar 

  10. Bui, H.-Q., Paluszyński, M., Taibleson, M. H.: Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case q < 1. J. Fourier Anal. Appl., 3, 837–846 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  11. Calderón, A. P.: Intermediate spaces and interpolation, the complex method. Studia Math., 24, 113–190 (1964)

    MATH  MathSciNet  Google Scholar 

  12. Cho, Y., Ozawa, T.: Sobolev inequalities with symmetry. Commun. Contemp. Math., 11, 355–365 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Duoandikoetxea, J.: Fourier Analysis, Graduate Studies in Math. 29, AMS, Providence, Rhode Island, 2001

    MATH  Google Scholar 

  14. Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces. J. Funct. Anal., 93, 34–170 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  15. García-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics, Graduate Studies in Math. 29, North-Holland, Amsterdam, 1985

    MATH  Google Scholar 

  16. Haroske, D. D., Piotrowska, I.: Atomic decompositions of function spaces with Muckenhoupt weights; an example from fractal geometry. Math. Nachr., 281, 1476–1494 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. Haroske, D. D., Skrzypczak, L.: Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, I. Rev. Mat. Complut., 21, 135–177 (2008)

    MATH  MathSciNet  Google Scholar 

  18. Izuki, M., Sawano, Y.: Wavelet bases in the weighted Besov and Triebel-Lizorkin spaces with A p/loc-weights. J. Approx. Theory, 161, 656–673 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kahane, J.-P., Lemarie-Rieuseut, P.-G.: Fourier Series and Wavelets, Gordon and Breach Publ., Amsterdam, 1995

    Google Scholar 

  20. Kalton, N., Mayboroda, S., Mitrea, M.: Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin spaces and applications to problems in partial differential equations, Interpolation theory and applications. Contemporary Math., 445, 121–177 (2007)

    Article  MathSciNet  Google Scholar 

  21. Kalton, N., Mitrea, M.: Stability results on interpolation scales of quasi-Banach spaces and applications. Trans. Amer. Math. Soc., 350, 3903–3922 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kokilashvili, V. N.: Maximal inequalities and multipliers in weighted Lizorkin-Triebel spaces. Dokl. Akad. Nauk. SSSR, 239(1), 42–45 (1978)

    MathSciNet  Google Scholar 

  23. Krein, S. G., Petunin, Yu. I., Semenov, E. M.: Interpolation of Linear Operators, Nauka, Moscow, 1978, Engl. translation AMS, Providence, R.I., 1982

    Google Scholar 

  24. Lions, P. L.: Symmetrie et compacité dans les espaces de Sobolev. J. Funct. Anal., 49, 315–334 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  25. Maligranda, L.: Orlicz Spaces and Interpolation, Seminars in Math., Dept. of Math., Univ. Estadual de Campinas, Campinas, 1989

    Google Scholar 

  26. Mendez, O., Mitrea, M.: The Banach envelopes of Besov and Triebel-Lizorkin spaces and applications to partial differential equations. J. Fourier Anal. Appl., 6, 503–531 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Meyer, Y.: Wavelets and Operators, Cambridge Univ. Press, Cambridge, 1992

    MATH  Google Scholar 

  28. Nilsson, P.: Interpolation of Banach lattices. Studia Math., 82, 135–154 (1985)

    MATH  MathSciNet  Google Scholar 

  29. Peetre, J.: New Thoughts on Besov Spaces, Duke Univ. Press, Durham, 1976

    MATH  Google Scholar 

  30. Rychkov, V. S.: Littlewood-Paley theory and function spaces with A locp weights. Math. Nachr., 224, 145–180 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  31. Shestakov, V. A.: On complex interpolation of Banach spaces of measurable functions. Vestnik Leningrad Univ. Math., 19, 64–68 (1974)

    Google Scholar 

  32. Sickel, W., Skrzypczak, L.: Radial subspaces of Besov and Lizorkin-Triebel spaces: extended Strauss lemma and compactness of embeddings. J. Fourier Anal. Appl., 6, 639–662 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sickel, W., Skrzypczak, L.: On the interplay of regularity and decay in case of radial functions II. Homogeneous spaces. J. Fourier Anal. Appl., 18, 548–582 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sickel, W., Skrzypczak, L., Vybíral, J.: On the interplay of regularity and decay in case of radial functions I. Inhomogeneous spaces. Comm. Contemp. Math., 14(1), (2012)

    Google Scholar 

  35. Sickel, W., Skrzypczak, L., Vybíral, J.: Complex interpolation of weighted Besov- and Lizorkin-Triebel spaces (long version), unpublished, manuscript, http://arxiv.org/abs/1212.1614

  36. Skrzypczak, L.: Rotation invariant subspaces of Besov and Triebel-Lizorkin spaces: compactness of embeddings, smoothness and decay properties. Revista Mat. Iberoamericana, 18, 267–299 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  37. Stein, E. M.: Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, 1993

    MATH  Google Scholar 

  38. Strauss, W. A.: Existence of solitary waves in higher dimensions. Comm. Math. Physics, 55, 149–162 (1977)

    Article  MATH  Google Scholar 

  39. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators, North Holland, Amsterdam, 1978

    Google Scholar 

  40. Triebel, H.: Theory of Function Spaces, Birkhäuser, Basel, 1983

    Book  Google Scholar 

  41. Triebel, H.: Theory of Function Spaces II, Birkhäuser, Basel, 1992

    Book  MATH  Google Scholar 

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

    MATH  Google Scholar 

  43. Wik, I.: On Muckenhoupt’s classes of weight functions. Studia Math., 94, 245–255 (1989)

    MATH  MathSciNet  Google Scholar 

  44. Wojciechowska, A.: Local means and wavelets in function spaces with local Muckenhoupt weights, In: Function Spaces IX, Banach Center Public. 92, Warszawa, 2011, 399–412

    Article  MathSciNet  Google Scholar 

  45. Wojciechowska, A.: Multidimensional wavelet bases in Besov and Lizorkin-Triebel spaces, PhD-thesis, Adam Mickiewicz University Poznań, Poznań, 2012

    Google Scholar 

  46. Wojtaszczyk, P.: A Mathematical Introduction to Wavelets, Cambridge Univ. Press, Cambridge, 1997

    Book  MATH  Google Scholar 

  47. Yang, D., Yuan, W., Zhuo, C.: Complex interpolation on Besov-type and Triebel-Lizorkin-type spaces. Anal. Appl. (Singap.), 11(5), 1350021, 45 pp. (2013)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematisches Institut, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, 07743, Jena, Germany

    Winfried Sickel

  2. Faculty of Mathematics and Computer Science, Adam Mickiewicz University Poznań, Ul. Umultowska 87, 61-614, Poznań, Poland

    Leszek Skrzypczak

  3. Department of Mathematics, Technical University Berlin, Secretary office MA 4-1 Street of 17. June 136, 10623, Berlin, Germany

    Jan Vybíral

Authors
  1. Winfried Sickel
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  2. Leszek Skrzypczak
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  3. Jan Vybíral
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Correspondence to Winfried Sickel.

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Supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin

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Sickel, W., Skrzypczak, L. & Vybíral, J. Complex interpolation of weighted Besov and Lizorkin-Triebel spaces. Acta. Math. Sin.-English Ser. 30, 1297–1323 (2014). https://doi.org/10.1007/s10114-014-2762-y

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  • DOI: https://doi.org/10.1007/s10114-014-2762-y

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