Skip to main content
SpringerLink
Log in

Atomic and molecular decompositions of anisotropic Besov spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

An Erratum to this article was published on 02 February 2013

Abstract.

In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic φ-transforms of Frazier and Jawerth [19, 21] to non-isotropic settings.

Several results of isotropic theory of Besov spaces are recovered for weighted anisotropic Besov spaces. We show that these spaces are characterized by the magnitude of the φ-transforms in appropriate sequence spaces. We also prove boundedness of an anisotropic analogue of the class of almost diagonal operators and we obtain atomic and molecular decompositions of weighted anisotropic Besov spaces, thus extending isotropic results of Frazier and Jawerth [21].

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral representations of functions and imbedding theorems. Vol. I and II, V. H. Winston & Sons, Washington, D.C., 1979

  2. Bownik, M.: A characterization of affine dual frames in L 2(ℝn). Appl. Comput. Harmon. Anal. 8, 203–221 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bownik, M.: Anisotropic Hardy spaces and wavelets. Mem. Am. Math. Soc. 164(781), 122pp (2003)

    MathSciNet  Google Scholar 

  4. Bownik, M., Ho, K.-P.: Atomic and Molecular Decompositions of Anisotropic Triebel-Lizorkin Spaces. Trans. Am. Math. Soc. (to appear)

  5. Buckley, S.M., MacManus, P.: Singular measures and the key of G. Publ. Mat. 44, 483–489 (2000)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Bui, H.-Q.: Weighted Young’s inequality and convolution theorems on weighted Besov spaces. Math. Nachr. 170, 25–37 (1994)

    Article  MathSciNet  MATH  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, 219–246 (1996)

    MathSciNet  MATH  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)

  11. Calderón, A.P., Torchinsky, A.: Parabolic maximal function associated with a distribution. Adv. Math. 16, 1–64 (1975)

    Article  MATH  Google Scholar 

  12. Calderón, A.P., Torchinsky, A.: Parabolic maximal function associated with a distribution II. Adv. Math. 24, 101–171 (1977)

    Article  MATH  Google Scholar 

  13. Coifman, R.R.: A real variable characterization of H p. Studia Math. 51, 269–274 (1974)

    MathSciNet  MATH  Google Scholar 

  14. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Dintelmann, P.: Classes of Fourier multipliers and Besov-Nikolskij spaces. Math. Nachr. 173, 115–130 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dintelmann, P.: On Fourier multipliers between Besov spaces with 0<p 0≤ min {1,p 1}. Anal. Math. 22, 113–123 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  17. Farkas, W.: Atomic and subatomic decompositions in anisotropic function spaces. Math. Nachr. 209, 83–113 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups. Princeton University Press, Princeton, N.J., 1982

  19. Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana U. Math. J. 34, 777–799 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Frazier, M., Jawerth, B.: The φ-transform and applications to distribution spaces. Lecture Notes in Math., 1302, Springer, Berlin Heidelberg, 1988, pp. 223–246

  21. Frazier, M., Jawerth, B.: A Discrete Transform and Decomposition of Distribution Spaces. J. Funct. Anal. 93, 34–170 (1989)

    Article  MathSciNet  Google Scholar 

  22. Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Ser., 79, American Math. Society, 1991

  23. García-Cuerva, J., Rubio de Francia, J.L.: Weighted Norm Inequalities and Related Topics. North-Holland, 1985

  24. Lemarié-Rieusset, P.-G.: Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions. Rev. Mat. Iberoamericana 10, 283–347 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lemarié-Rieusset, P.-G.: Recent developments in the Navier-Stokes problem. Chapman & Hall/CRC, 2002

  26. Nazarov, F., Treil, S.: The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis. Algebra i Analiz 8, 32–162 (1996)

    MATH  Google Scholar 

  27. Peetre, J.: New thoughts on Besov spaces. Duke University Mathematics Series, No. 1, Mathematics Department, Duke University, Durham, N.C., 1976

  28. Plancherel, M., Pólya, G.: Fonctions entières et intégrales de Fourier multiples. Comment. Math. Helv. 9, 224–248 (1937)

    Article  MATH  Google Scholar 

  29. Roudenko, S.: Matrix-weighted Besov spaces. Trans. Am. Math. Soc. 355, 273–314 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  30. Rychkov, V.S.: On a theorem of Bui, Paluszyński, and Taibleson. Proc. Steklov Inst. Math. 227, 280–292 (1999)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Schmeisser, H.-J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. John Wiley & Sons, 1987

  33. Volberg, A.: Matrix A p weights via S-functions. J. Am. Math. Soc. 10, 445–466 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Triebel, H.: Theory of Function Spaces. Monographs in Math., 78, Birkhäuser, Basel, 1983

  35. Triebel, H.: Theory of function spaces II. Monographs in Math., 84, Birkhäuser, Basel, 1992

  36. Triebel, H.: Wavelet bases in anisotropic function spaces. Preprint, 2004

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Oregon, Eugene, OR, 97403–1222, USA

    Marcin Bownik

Authors
  1. Marcin Bownik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcin Bownik.

Additional information

The author was partially supported by the NSF grant DMS-0441817.

An erratum to this article can be found online at http://dx.doi.org/10.1007/s00209-012-1130-9.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bownik, M. Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z. 250, 539–571 (2005). https://doi.org/10.1007/s00209-005-0765-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-005-0765-1

Mathematics Subject Classification (2000):

Search

Navigation