六块域（hexablock）的函数理论及其在四块域（tetrablock）与欧几里得双球上的应用
Function theory of the hexablock and applications to the tetrablock and Euclidean biball
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中文总结 AI 辅助
本文研究六块域$\u210d$的函数论核心问题，确定其Schur-Agler类与实现公式，证明相关插值、延拓及Toeplitz日冕定理，并将结果应用于欧几里得双球，同时推导出四块域的已有结论。
中文摘要 AI 辅助
实现问题、插值问题、延拓问题与Toeplitz日冕问题是$\u2102^d$中区域函数论的核心主题。本文针对六块域$\u210d$——一个源于$H^{\u221e}$控制理论中$μ$-综合特殊情形的$\u2102^4$中的区域——研究这四类问题。我们确定了$\u210d$的Schur-Agler类$SA(\u210d)$，并得到了$\u210d$的实现公式。借助该实现公式，我们陈述并证明了六块域的插值定理、延拓定理与Toeplitz日冕定理。作为应用，我们得到了$\u2102^2$中欧几里得单位球的类似定理。此外，我们还将四块域$\u214e$——另一个与$μ$-综合相关的区域——已有的同类结果作为六块域理论的推论，通过Schur-Agler类$SA(\u210d)$与$\u210d$上的容许核重新推导得到。
英文摘要
The realization, interpolation, extension and Toeplitz corona problems are amongst the central themes in function theory of a domain in $\mathbb{C}^d$. The present work addresses these four problems for the hexablock $\mathbb{H}$, a domain in $\mathbb{C}^4$ arising in connection with a special case of $μ$-synthesis in $H^{\infty}$ control theory. We determine Schur-Agler class $SA(\mathbb{H})$ for $\mathbb{H}$ and find a realization formula for $\mathbb H$. With the help of this realization formula we state and prove interpolation, extension and Toeplitz corona theorems for the hexablock. As an application, we obtain analogous theorems for the Euclidean unit ball in $\mathbb{C}^2$. Moreover, we recover the existing same results for the tetrablock $\mathbb E$, another domain associated with the $μ$-synthesis, as consequences of the hexablock theory and in terms of the Schur-Agler class $SA(\mathbb{H})$ and admissible kernels on $\mathbb{H}$.
