Capacity distortion by inner functions in the unit ball of \(\mathbb{C}^n\)
The main result of this paper is: Theorem 1. Let \(f\) be an inner function in the unit ball of \(\mathbb{C}^n\), \(n\geq 1\), \(f(0)=0\); let \(E\) be a Borel subset of the boundary of the unit disk. Then: (i) if \(0<\alpha <2\) (if \(n=1\) then \(\alpha=0)\) then \[ \text{cap}_{2n-2+ \alpha} \bigl(f^{-1} (E)\bigr)\geq C(n,\alpha) \text{cap}_\alpha (E); \] (ii) if \(\alpha=0\), \(n>1\) then \[ {1\over\text{cap}_{2n-2} \bigl(f^{-1} (E)\bigr)}\leq C(n) \left(1+\log {1\over \text{cap}_0 (E)} \right), \] where \(\text{cap}_\alpha\) denotes \(\alpha\)-dimensional Riesz capacity \((\text{cap}_0\) denotes logarithmic capacity) with respect to the distance in the unit sphere given by \(d(a,b)= |1-\sum^n_{j=1} a_j\overline {b_j} |^{1/2}\).
scientific article; zbMATH DE number 1046971