Radial limits of inner functions and Bloch spaces

Abstract

Let f f be an inner function in the unit ball B n ⊂ C n B_n \subset \mathbb {C}^n , n ≥ 1 n\ge 1 . Assume that \[ sup z ∈ B n | R f ( z ) | ( 1 − | z | 2 ) 1 + β ( 1 − | f ( z ) | 2 ) 2 > ∞ , \sup _{z\in B_n} \frac {|\mathcal {R} f(z)|(1-|z|^2)^{1+\beta }}{\left (1-|f(z)|^2 \right )^2} > \infty , \] where β ∈ ( 0 , 1 ) \beta \in (0,1) and R f \mathcal {R} f is the radial derivative. Then, for all α ∈ ∂ B 1 \alpha \in \partial B_1 , the set { ζ ∈ ∂ B n : f ∗ ( ζ ) = α } \{\zeta \in \partial B_n:\, f^*(\zeta ) =\alpha \} has a non-zero real Hausdorff t 2 n − 1 − β t^{2n-1-\beta } -content, and it has a non-zero complex Hausdorff t n − β t^{n-\beta } -content.