Radial behaviour of harmonic Bloch functions and their area function

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Abstract

Let u be a harmonic function in the upper half space ℝn+1+ and A(u) its (truncated) area function. Classical results of Calderón, Stein and Zygmund assert that the following two sets {x ∈ ℝn : u has non-tangential limit at x}, {x ∈ Rn : A(u)(x) < ∞} can only differ in a set of zero Lebesgue measure. When these sets have zero Lebesgue measure, the Law of the Iterated Logarithm proved by Bañuelos, Klemeš and Moore, describes the maximal non-tangential growth of u(x,y) in terms of its (doubly) truncated area function A(u)(x,y), at almost evey point x ∈ ℝn+. In this paper we show that if u is in the Bloch space and its area function diverges at almost every point, one can prescribe any "reasonable" radial behaviour of u in a set of rays of maximal Hausdorff dimension. More concretely, if γ : [0,∞) → ℝ satisfies certain regularity conditions, the set {x ∈ ℝn : limy→0sup|u(x,y)-γ(A2(u)(x,y))| < ∞} has Hausdorff dimension n. A multiplicative version of this result is also proved.
Original languageEnglish
Pages (from-to)1213-1236
JournalIndiana University Mathematics Journal
Volume48
Issue number4
Publication statusPublished - 1 Dec 1999

Keywords

  • Area function
  • Bloch
  • Harmonic
  • Hausdorff dimension

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