Journal of Mathematical Sciences

, Volume 156, Issue 5, pp 813–818 | Cite as

Radial limits of positive solutions to the Darboux equation

  • E. S. Dubtsov
Assume that a positive function u satisfies the Darboux equation
$$\Delta u = \frac{{\left( {\alpha - 1} \right)}} {y}\frac{{\partial u}} {{\partial y}},\quad \alpha > 0,$$
in the upper half-space ℝ + d+1. We study Bloch type conditions which guarantee the following property: for any a ∈ (0, + ∞), the set on which the radial limit of u is equal to a is large in the sense of the Hausdorff dimension. Bibliography: 6 titles.


Russia Positive Function Type Condition Hausdorff Dimension Mathematical Institute 
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  1. 1.
    J. J. Donaire, Conjuntos excepcionales para las classes de Zygmund, Thesis, Barcelona (1995).Google Scholar
  2. 2.
    E. Doubtsov and A. Nicolau, “Symmetric and Zygmund measures in several variables,” Ann. Inst. Fourier (Grenoble), 52, 153–177 (2002).MathSciNetGoogle Scholar
  3. 3.
    E. Doubtsov, “Differentialtion properties of symmetric measures,” Potential Anal., 27, 389–401 (2007).CrossRefMathSciNetGoogle Scholar
  4. 4.
    E. S. Dubtsov, “Derivatives of regular measures,” Algebra Analiz, 19, No. 2, 86–104 (2007).MathSciNetGoogle Scholar
  5. 5.
    P. P. Kargaev, “On positive solutions of the Darboux equation {ie818-01}, in the half-space y > 0,” Algebra Analiz, 8, No. 3, 125–150 (1996).MathSciNetGoogle Scholar
  6. 6.
    J. G. Llorente, “Boundary values of harmonic Bloch functions in Lipschitz domains: a martingale approach,” Potential Anal., 9, 229–260 (1998).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.St.Peterburg Department of the Steklov Mathematical InstituteSt.PetersburgRussia

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