论文标题
内部功能的临界值
Critical values of inner functions
论文作者
论文摘要
令$ \ mathscr j $为有限熵的内部功能的空间，并具有稳定收敛的拓扑结构。我们证明，\ mathscr j $中的内部函数$ f \在单位磁盘中具有径向限制（实际上是最小的罚款限制），$σ（f'）$ a.e.点在单位圆上。我们用它来表明奇异值测量$ν（f）= \ sum_ {c \ in \ text {crit} f}（1- | c |）\cdotΔ__{f（c） + f（c） + f _*（σ（f'）$在$ f $中连续差异。我们的分析涉及贝林 - 卡利森集合与角衍生物之间的惊人联系。
Let $\mathscr J$ be the space of inner functions of finite entropy endowed with the topology of stable convergence. We prove that an inner function $F \in \mathscr J$ possesses a radial limit (and in fact, a minimal fine limit) in the unit disk at $σ(F')$ a.e. point on the unit circle. We use this to show that the singular value measure $ν(F) = \sum_{c \in \text{crit } F} (1-|c|) \cdot δ_{F(c)} + F_*(σ(F'))$ varies continuously in $F$. Our analysis involves a surprising connection between Beurling-Carleson sets and angular derivatives.
