Skip to main content
SpringerLink
Log in

Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

Abstract

If the Poisson integral of the unit disc is replaced by its square root, it is known that normalized Poisson integrals of L p and weak L p boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Rönning and the author, respectively. In this paper we characterize the approach regions for boundary functions in two general classes of Orlicz spaces. The first of these classes contains spaces L Φ having the property L ∞ ⊂ L Φ L p, 1 ⩽ p > ∞. The second contains spaces L Φ that resemble L p spaces.

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. H. Aikawa: Harmonic functions having no tangential limits. Proc. Am. Math. Soc. 108 (1990), 457–464.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Bellow, R. L. Jones: A Banach principle for L ∞. Adv. Math. 120 (1996), 155–172.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Brundin: Approach regions for the square root of the Poisson kernel and weak L p boundary functions. Preprint 1999:56. Department of Mathematics, Göteborg University and Chalmers University of Technology, 1999.

  4. M. Brundin: Approach Regions for L p potentials with respect to fractional powers of the Poisson rernel of the halfspace. Department of Mathematics. Göteborg University and Chalmers University of Technology, 2001.

  5. M. Brundin: Approach regions for L p potentials with respect to the square root of the Poisson kernel. Math. Scand. To appear.

  6. F. Di Biase: Fatou Type Theorems: Maximal Functions and Approach Regions. Birkhäuser-Verlag, Boston, 1998.

    MATH  Google Scholar 

  7. P. Fatou: Séries trigonométriques et séries de Taylor. Acta Math. 30 (1906), 335–400.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. E. Littlewood: On a theorem of Fatou. J. London Math. Soc. 2 (1927), 172–176.

    Article  Google Scholar 

  9. A. Nagel, E. M. Stein: On certain maximal functions and approach regions. Adv. Math. 54 (1984), 83–106.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces. Monographs and Textbooks in Pure and Applied Mathematics, Vol. 146. Marcel Dekker, New York, 1991.

    MATH  Google Scholar 

  11. J.-O. Rönning: Convergence results for the square root of the Poisson kernel. Math. Scand. 81 (1997), 219–235.

    MATH  MathSciNet  Google Scholar 

  12. H. A. Schwarz: Zur Integration der partiellen Differentialgleichung \(\frac{{\partial ^2 u}}{{\partial x^2 }} + \frac{{\partial ^2 u}}{{\partial y^2 }} = 0\). J. Reine Angew. Math. 74 (1872), 218–253.

    Google Scholar 

  13. P. Sjögren: Une remarque sur la convergence des fonctions propres du laplacien à valeur propre critique. In: Théorie du potentiel (Orsay, 1983). Springer-Verlag, Berlin, 1984, pp. 544–548.

    Chapter  Google Scholar 

  14. P. Sjögren: Approach regions for the square root of the Poisson kernel and bounded functions. Bull. Austral. Math. Soc. 55 (1997), 521–527.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Karlstad University, 651 88, Karlstad, Sweden

    M. Brundin

Authors
  1. M. Brundin
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brundin, M. Approach regions for the square root of the Poisson kernel and boundary functions in certain Orlicz spaces. Czech Math J 57, 345–365 (2007). https://doi.org/10.1007/s10587-007-0064-6

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-007-0064-6

Keywords

Search

Navigation