Skip to main content
SpringerLink
Log in
Book cover

Spectral Theory, Function Spaces and Inequalities pp 69–89Cite as

Dichotomy in Muckenhoupt Weighted Function Space: A Fractal Example

  • Conference paper
  • First Online:

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 219))

Abstract

We study dichotomy questions for weighted function spaces of Besov and Triebel–Lizorkin type where the weight is related to the distance of some point to a d-set Г, 0 < d < n, and the trace is taken on G. We can prove that – depending on the function space, the weight and the set Г – there occurs an alternative: either the trace on Г exists, or smooth functions compactly supported outside Г are dense in the space. Such a phenomenon is called dichotomy. The paper is based on trace results in [33] and the atomic decomposition in [18] and perfectly fits together with corresponding unweighted results in [51].

Mathematics Subject Classification (2010). 46E35.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Abels, M. Krbec, and K. Schumacher. On the trace space of a Sobolev space with a radial weight. J. Funct. Spaces Appl., 6(3):259–276, 2008.

    MathSciNet  MATH  Google Scholar 

  2. M. Bownik. Atomic and molecular decompositions of anisotropic Besov spaces. Math. Z., 250(3):539–571, 2005.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  8. V.I. Burenkov and M.L. Gol’dman. Extension of functions from Lp. Trudy Mat. Inst. Steklov., 150:31–51, 1979. Studies in the theory of differentiable functions of several variables and its applications, VII.

    Google Scholar 

  9. A.M. Caetano and D.D. Haroske. Traces for Besov spaces on fractal h-sets and dichotomy results. Preprint.

    Google Scholar 

  10. D.E. Edmunds and H. Triebel. Spectral theory for isotropic fractal drums. C. R. Acad. Sci. Paris, 326(11):1269–1274, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  11. K.J. Falconer. The geometry of fractal sets. Cambridge Univ. Press, Cambridge, 1985.

    Google Scholar 

  12. R. Farwig and H. Sohr. Weighted Lq-theory for the Stokes resolvent in exterior domains. J. Math. Soc. Japan, 49(2):251–288, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  13. M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34(4):777–799, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Frazier and B. Jawerth. A discrete transform and decomposition of distribution spaces. J. Funct. Anal., 93(1):34–170, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Frazier and S. Roudenko. Matrix-weighted Besov spaces and conditions of Ap type for 0 < p ≤ 1. Indiana Univ. Math. J., 53(5):1225–1254, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  16. J. García-Cuerva and J. L. Rubio de Francia. Weighted norm inequalities and related topics, volume 116 of North-Holland Mathematics Studies. North-Holland, Amsterdam, 1985.

    Google Scholar 

  17. M.L. Gol’dman. Extension of functions in Lp(R m) to a space of higher dimension. Mat. Zametki, 25(4):513–520, 1979.

    Google Scholar 

  18. D.D. Haroske and I. Piotrowska. Atomic decompositions of function spaces with Muckenhoupt weights, and some relation to fractal analysis. Math. Nachr., 281(10):1476–1494, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  19. D.D. Haroske and H.-J. Schmeiser. On trace spaces of function spaces with a radial weight: the atomic approach. Complex Var. Elliptic Equ., 55(8–10):875–896, 2010.

    Article  MathSciNet  MATH  Google Scholar 

  20. D.D. Haroske and L. Skrzypczak. Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, I. Rev. Mat. Complut., 21(1):135–177, 2008.

    MathSciNet  MATH  Google Scholar 

  21. D.D. Haroske and L. Skrzypczak. Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights. Annales Academia Scientiarum Fennica, 36(1):111–138, 2011.

    Article  MathSciNet  MATH  Google Scholar 

  22. D.D. Haroske and L. Skrzypczak. Entropy numbers of embeddings of function spaces with Muckenhoupt weights, III. Some limiting cases. J. Funct. Spaces Appl., 9(2):129–178, 2011.

    MathSciNet  Google Scholar 

  23. A. Jonsson and H. Wallin. Function spaces on subsets of Rn. Math. Rep. Ser. 2, No.1, xiv + 221 p., 1984.

    Google Scholar 

  24. P. Mattila. Geometry of sets and measures in euclidean spaces. Cambridge Univ. Press, Cambridge, 1995.

    Google Scholar 

  25. S.D. Moura. Function spaces of generalised smoothness. Dissertationes Math., 398:88 pp., 2001.

    Google Scholar 

  26. S.D. Moura. Function spaces of generalised smoothness, entropy numbers, applications. PhD thesis, Universidade de Coimbra, Portugal, 2002.

    Google Scholar 

  27. B. Muckenhoupt. Hardy’s inequality with weights. Studia Math., 44:31–38, 1972.

    MathSciNet  MATH  Google Scholar 

  28. B. Muckenhoupt. Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc., 165:207–226, 1972.

    Article  MathSciNet  MATH  Google Scholar 

  29. B. Muckenhoupt. The equivalence of two conditions for weight functions. Studia Math., 49:101–106, 1973/74.

    Google Scholar 

  30. S.M. Nikol’skiî. Approximation of functions of several variables and embedding theorems. Nauka, Moscow, second, revised and supplemented edition, 1977. Russian; First edition 1969. English Transl.: Grundlehren der Mathematischen Wissenschaften, vol. 205, Springer, New York-Heidelberg, 1975.

    Google Scholar 

  31. J. Peetre. The trace of Besov space – a limiting case. Technical report, University Lund, Sweden, 1975.

    Google Scholar 

  32. J. Peetre. A counterexample connected with Gagliardo’s trace theorem. Comment. Math. Special Issue, 2:277–282, 1979. Special issue dedicated to W_ladys_law Orlicz on the occasion of his seventy-fifth birthday.

    Google Scholar 

  33. I. Piotrowska. Traces on fractals of function spaces with Muckenhoupt weights. Funct. Approx. Comment. Math., 36:95–117, 2006.

    MathSciNet  MATH  Google Scholar 

  34. I. Piotrowska. Weighted function spaces and traces on fractals. PhD thesis, Friedrich-Schiller-Universität Jena, Germany, 2006.

    Google Scholar 

  35. C.A. Rogers. Hausdorff measures. Cambridge Univ. Press, London, 1970.

    Google Scholar 

  36. S. Roudenko. Matrix-weighted Besov spaces. Trans. Amer. Math. Soc., 355:273–314, 2002.

    Article  MathSciNet  Google Scholar 

  37. S. Roudenko. Duality of matrix-weighted Besov spaces. Studia Math., 160(2):129–156, 2004.

    Article  MathSciNet  MATH  Google Scholar 

  38. V.S. Rychkov. Littlewood-Paley theory and function spaces with Aloc p weights. Math. Nachr., 224:145–180, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  39. C. Schneider. Besov spaces with positive smoothness. PhD thesis, Universität Leipzig, 2009.

    Google Scholar 

  40. C. Schneider. Trace operators in Besov and Triebel-Lizorkin spaces. Z. Anal. Anwendungen, 29(3):275–302, 2010.

    Article  MATH  Google Scholar 

  41. W. Sickel and H. Triebel. Hölder inequalities and sharp embeddings in function spaces of Bs p,q and Fs p,q type. Z. Anal. Anwendungen, 14:105–140, 1995.

    MathSciNet  MATH  Google Scholar 

  42. E.M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, 1993.

    Google Scholar 

  43. J.-O. Strömberg and A. Torchinsky. Weighted Hardy spaces, volume 1381 of Lecture Notes in Mathematics. Springer, Berlin, 1989.

    Google Scholar 

  44. A. Torchinsky. Real-variable methods in harmonic analysis, volume 123 of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 1986.

    Google Scholar 

  45. H. Triebel. Interpolation theory, function spaces, differential operators. North-Holland, Amsterdam, 1978.

    Google Scholar 

  46. H. Triebel. Theory of function spaces. Birkhäuser, Basel, 1983.

    Google Scholar 

  47. H. Triebel. Theory of function spaces II. Birkhäuser, Basel, 1992.

    Google Scholar 

  48. H. Triebel. Fractals and spectra. Birkhäuser, Basel, 1997.

    Google Scholar 

  49. H. Triebel. The structure of functions. Birkhäuser, Basel, 2001.

    Google Scholar 

  50. H. Triebel. Theory of function spaces III. Birkhäuser, Basel, 2006.

    Google Scholar 

  51. H. Triebel. The dichotomy between traces on d-sets Γ in Rn and the density of D(Rn \ Γ) in function spaces. Acta Math. Sinica, 24(4):539–554, 2008.

    Article  MathSciNet  MATH  Google Scholar 

  52. H. Triebel. Function Spaces and Wavelets on domains. EMS Tracts in Mathematics (ETM). European Mathematical Society (EMS), Zürich, 2008.

    Google Scholar 

  53. H. Triebel and H. Winkelvos. Intrinsic atomic characterizations of function spaces on domains. Math. Z., 221(4):647–673, 1996.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematical Institute, Friedrich-Schiller-University Jena, D-07737, Jena, Germany

    Dorothee D. Haroske

Authors
  1. Dorothee D. Haroske
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dorothee D. Haroske .

Editor information

Editors and Affiliations

  1. School of Computer Science, Cardiff University, The Parade 5, Cardiff, CF24 3AA, United Kingdom

    B. Malcolm Brown

  2. Dept. Mathematics, Ohio State University, West 18th Ave. 231, Columbus, 43210-1174, Ohio, USA

    Jan Lang

  3. School of Mathematics, Statistics &, Actuarial Science, University of Kent, Canterbury, Kent, CT2 7NF, United Kingdom

    Ian G. Wood

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer Basel AG

About this paper

Cite this paper

Haroske, D.D. (2012). Dichotomy in Muckenhoupt Weighted Function Space: A Fractal Example. In: Brown, B., Lang, J., Wood, I. (eds) Spectral Theory, Function Spaces and Inequalities. Operator Theory: Advances and Applications(), vol 219. Springer, Basel. https://doi.org/10.1007/978-3-0348-0263-5_5

Download citation

Publish with us

Policies and ethics

Search

Navigation