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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.

We define an extended Cesaro operator Tg with holomorphic symbol g in the unit ball B of C n as T g (f)(z) = ∫ 1 0 f(tz)R g (tz)dt/t, f ∈ H(B), z ∈ B, where R g (z) = Σ n j=1 z j ∂ f /∂ zj is the radial derivative of g. In this paper we characterize those g for which T g is bounded (or compact) on the mixed norm space H p,q (ω).

In the setting of the half-space of the euclidean n-space, we show that every harmonic Bergman function is the radial derivative of a Bergman function with an appropriate norm bound.