Radial limits of inner functions and Bloch spaces

Doubtsov, Evgueni

doi:10.1090/S0002-9939-08-09215-0

Radial limits of inner functions and Bloch spaces
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Proc. Amer. Math. Soc. 136 (2008), 2177-2182 Request permission

Abstract:

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

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