论文标题
hartogs伴侣和全体形态扩展在任意维度
Hartogs companions and holomorphic extensions in arbitrary dimension
论文作者
论文摘要
我们表明，每一个全态映射$ f \ in \ Mathcal {h}（ω\ setMinus k）$（$ k \subsetΩ\ subset \ subset \ mathbb {c}^n $，带有$ k $ compact，$ ch $ compact，$ω$打开，$ n \ ge2 $），$ n \ ge2 $） f \ in \ Mathcal {h}（ω）$匹配$ f $在打开子集$ c_ {k，ω} \subsetΩ\ setMinus k $上。此外，$ \ tilde f $扩展$ f $，\ emph {if and＆hif} $ \ mathbb {c}^n \ setMinus k $是一个连接的集合；这种等价证明了hartogs kugelsatz的相反含义。 The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for Gâteaux holomorphic maps $f\in\mathcal{H}_\mathrm{G}(Ω\setminus K,Y)$ on finitely open sets in arbitrary complex vector spaces.等效性与$ k \subsetΩ\ subset \ mathbb {c}^n $非常相似，并导致相应的hartogs kugelsatz在任意维度上，并扩展了五种类型的Holomorphy的定理（Gâteaux，Mackey/Mackey/silva，silva，hypoanalytic，frolechet，locy in locally in locy byny douncy dourcyed byny dourcyemphy。我们还表明，矢量值Hartogs Companion的范围$ \ tilde f（ω）$不能留下包含$ f（ω\ setminus k）$的Holomorphy域。我们在\ Mathcal {h} _ \ Mathrm {g}（ω，y）\ cap \ mathcal {c}（\barΩ，y）$上建立了映射$ f \ in \ mathcal {h} _ \ mathrm {g} _ \ mathrm {h} _ \ mathrm {\barΩ，y）$的边界原理。对于$ y = \ mathbb {c} $，该原则指出$ f \ big（\barΩ\ big）= f（\partialΩ）$（因此$ \ sup_ {x \inΩ} | f（x）| = \ sup_ | = \ sup_ {x \ in \ in \ in \partialΩ}}} | f（x）| $）。几个结果需要一个新的身份定理，该定理产生了最大的规范原理和“最大值”半米原理。
We show that every holomorphic map $f\in\mathcal{H}(Ω\setminus K)$ ($K\subsetΩ\subset\mathbb{C}^n$, with $K$ compact, $Ω$ open, and $n\ge2$), has a unique "\emph{Hartogs companion}" $\tilde f\in\mathcal{H}(Ω)$ matching $f$ on an open subset $C_{K,Ω}\subsetΩ\setminus K$. Furthermore, $\tilde f$ extends $f$, \emph{if and only if} $\mathbb{C}^n\setminus K$ is a connected set; this equivalence proves the converse implication from the Hartogs Kugelsatz. The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for Gâteaux holomorphic maps $f\in\mathcal{H}_\mathrm{G}(Ω\setminus K,Y)$ on finitely open sets in arbitrary complex vector spaces. The equivalence is very similar to that for $K\subsetΩ\subset\mathbb{C}^n$ and leads to a corresponding Hartogs Kugelsatz in arbitrary dimension and to extension theorems for five types of holomorphy (Gâteaux, Mackey/Silva, hypoanalytic, Fréchet, locally bounded). We also show that the range $\tilde f(Ω)$ of a vector-valued Hartogs companion cannot leave a domain of holomorphy containing $f(Ω\setminus K)$. We establish a boundary principle for maps $f\in\mathcal{H}_\mathrm{G}(Ω,Y)\cap\mathcal{C}(\barΩ,Y)$ on finitely bounded open sets. For $Y=\mathbb{C}$, the principle states that $f\big(\barΩ\big)=f(\partialΩ)$ (hence $\sup_{x\inΩ}|f(x)|=\sup_{x\in\partialΩ}|f(x)|$). Several results require a new identity theorem, which yields a maximum norm principle and a "max-min" seminorm principle.
