A Herglotz-type representation for vector-valued holomorphic mappings on the unit ball of

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• 作者：Jerry R. Muir Jr. ^{jerry.muir@scranton.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Herglotz representation ; Carathé ; odory class ; Cauchy kernel ; Probability measures ; Holomorphic mappings
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 August 2016
• 年：2016
• 卷：440
• 期：1
• 页码：127-144
• 全文大小：415 K

文摘

The generalization of the Carathéodory class, those analytic functions on the open unit disk having positive real part and taking the value 1 at the origin, to the open unit ball B$\mathbb{B}$ of Cⁿ${\mathbb{C}}^n$ is the family M$\mathcal{M}$ of all holomorphic mappings f:B→Cⁿ$f:\mathbb{B}\to {\mathbb{C}}^n$ such that f(0)=0$f(0)=0$, Df(0)=I$Df(0)=I$, and Re〈f(z),z〉>0$\mathrm{Re}〈f(z),z〉>0$ for all z∈B∖{0}$z\in \mathbb{B}\setminus \{0\}$, where Df is the Fréchet derivative of f, I   is the identity operator on Cⁿ${\mathbb{C}}^n$, and 〈⋅,⋅〉$〈\cdot ,\cdot 〉$ is the Hermitian inner product in Cⁿ${\mathbb{C}}^n$. We present an integral representation for functions in the class M$\mathcal{M}$ in terms of probability measures on the unit sphere S=∂B$\mathbb{S}=\partial \mathbb{B}$ similar to the well-known Herglotz representation of the Carathéodory class. This representation follows, in part, from a new integral formula of Cauchy type that reproduces a continuous $f:\bar{\mathbb{B}}\to {\mathbb{C}}^n$ whose restriction to B$\mathbb{B}$ is holomorphic by using a fixed vector-valued kernel and the scalar values 〈f(u),u〉$〈f(u),u〉$, u∈S$u\in \mathbb{S}$. Not every probability measure on S$\mathbb{S}$ corresponds to a mapping in M$\mathcal{M}$, and we conclude by examining several additional properties the representing measures must satisfy.