Little Bloch functions, symmetric pluriharmonic measures, and Zygmund's dichotomy (Q1968732)

Let \(B=B_n\) denote the unit ball of \(\mathbb{C}^n\). The little Bloch space \(\mathcal{B}_0(B)\) consists of functions \(f\) holomorphic in \(B\) and such that \[ \lim\limits_{|z|\to 1^-}(1-|z|^2)|\mathcal{R}f(z)|=0 , \] where \(\mathcal{R}=\sum_{j=1}^n z_j \partial/\partial z_j\). The author investigates elements of the algebra \(H^\infty(B_n)\cap\mathcal{B}_0(B_n)\), \(n\geq 2\) (as usual, \(H^\infty\) stands for the bounded holomorphic functions). Special attention is given to the inner functions in \(\mathcal{B}_0\). The main objects of the present paper are pluriharmonic symmetric measures. Let \(S=\partial B\) be the unit sphere, and let \(\sigma\) be the normalized Lebesgue measure on \(S\). A measure \(\mu\in M(S)\) is said to be pluriharmonic if the Poisson integral \(P[\mu]\) is a pluriharmonic function in the ball. It is proved that every non-negative upper semi-continuous function \(\psi\) can be corrected by a positive singular measure \(\nu\) in such a way that \(\psi\sigma+\nu\) is a pluriharmonic measure and the slice-measures \(\mu_\zeta\), \(\zeta\in S\), are uniformly symmetric. In particular, there exist non-constant little Bloch inner functions in the complex ball. In the paper, a special Riesz pair \((R,a)=(\{R(j,L)\}_{j, L},\{a_k\}_{k=1}^\infty)\) is constructed such that the analogue of the Zygmund dichotomy holds for the corresponding generalized pluriharmonic Riesz products.