Boundary properties of Blaschke-type products (Q1804155)

The authors consider a Blaschke-type product introduced by M. M. Dzhrbashyan in 1945 (see [Dokl. Akad. Nauk SSSR 3, 3-9 (1945)]). For the lower half plane \(G^{(-)} = \{w : \text{Im} w < 0\}\), if \(- 1 < \alpha < \infty\) and \(\sum_k |v_k |^{1 + \alpha} < \infty\), this product is \(B_\alpha (w, \{w_k\}) = \prod_k b_\alpha (w,w_k)\), where \(b_\alpha (w,w_k) = \exp \{- \int^{2 |v_k |}_0 {\tau^\alpha d \tau \over [\tau + i (w - w_k)]^{1 + \alpha}}\}\), and \(w_k = u_k + iv_k\). Note that, for \(\alpha = 0\), \(B_0 (w, \{w_k\})\) is the usual Blaschke product in the lower half plane with zeros \(\{w_n\}\). Similarly, for the unit disk \(D = \{z : |z |\leq 1\}\), if \(- 1 < \alpha < \infty\) and \(\sum_k (1 - |z_k |)^{1 + \alpha} < \infty\), the product is \[ \pi_\alpha \bigl( z, \{z_k\} \bigr) = \prod_k a_\alpha (z,z_k), \] where \[ a_\alpha (z,z_k) = \exp \left\{ - \int^1_{|z_k |^2} {(1 - \tau)^\alpha \over (1 - z \tau/z_k)^{1 + \alpha}} {d \tau \over \tau} \right \}. \] In the case \(\alpha = 0\), \(\pi_0 (z, \{z_k\})\) is the usual Blaschke product with zeros \(\{z_k\}\). The main results of the paper state that, under appropriate conditions on the sequence \(\{w_k\}\) and \(\{z_k\}\) and the number \(\alpha\), the Blaschke-type product has nonzero finite angular limits at all points of the boundary, except for a ``small'' set, and, in addition, if \(\alpha\geq 0\) and the sequences \(\{w_k\}\) and \(\{z_k\}\) satisfy the usual Blaschke condition on their respective domains, then the ``small'' set is of measure zero. Specifically, for the unit disk, the exact result is as follows. If \(0 < \gamma < 1\) and \(\gamma - 1 < \alpha < \infty\), and if \(\sum_k (1 - |z_k |)^\gamma < \infty\), then \(\pi_\alpha (z, \{z_k\})\) has a nonzero finite angular limit at each point of \(T = \{z : |z |= 1\}\), except possibly on a set with zero \(\gamma\)-capacity. In addition, if \(\alpha \geq 0\) and \(\sum_k (1 - |z_k |) < \infty\), then \(\prod_\alpha (z, \{z_k\})\) has a nonzero finite angular limit at all points of \(T\), except possibly a set of linear measure zero. A similar result holds for the function \(B_\alpha (w, \{w_k\})\) on the region \(G^{(-)}\).