TY - JOUR
T1 - Radial behaviour of harmonic Bloch functions and their area function
AU - Nicolau, Artur
PY - 1999/12/1
Y1 - 1999/12/1
N2 - 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.
AB - 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.
KW - Area function
KW - Bloch
KW - Harmonic
KW - Hausdorff dimension
U2 - 10.1512/iumj.1999.48.1662
DO - 10.1512/iumj.1999.48.1662
M3 - Article
SN - 0022-2518
VL - 48
SP - 1213
EP - 1236
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
IS - 4
ER -