Radial behaviour of Bloch Harmonic functions and their area function
Artur Nicolau · Indiana University Mathematics Journal · 1999
Let u be a harmonic function in the upper half space R n+1 + and A(u) its (truncated) area function.Classical results of Calderón, Stein and Zygmund assert that the following two sets {x ∈ R n : u has non-tangential limit at x}, {x ∈ R n : 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 ∈ R n + .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 ∈ R n : lim y→0 sup |u(x, y)γ(A 2 (u)(x, y))| < ∞} has Hausdorff dimension n.A multiplicative version of this result is also proved.