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Radial behaviour of harmonic Bloch functions and their area function
Nicolau, Artur (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Data: 1999
Resum: 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, Klemeˇs 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→0 sup |u(x, y) − γ(A2(u)(x, y))| < ∞} has Hausdorff dimension n. A multiplicative version of this result is also proved.
Drets: Tots els drets reservats.
Llengua: Anglès
Document: article ; publishedVersion
Matèria: Harmonic ; Bloch ; Area function ; Hausdorff dimension
Publicat a: Indiana University mathematics journal, Vol. 48, No. 4 (1999) , p. 1213-1236, ISSN 0022-2518

DOI: 10.1512/iumj.1999.48.1662

24 p, 199.8 KB

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