Steven G. Krantz
We treat the classical concept of domain of holomorphy in $\CC^n$ when the holomorphic functions considered are restricted to lie in some Banach space. Positive and negative results are presented. A new view of the case $n = 1$ is considered.
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We present the envelope of holomorphy of a classical truncated tube domain.
A characterization of $L_h^2$-domains of holomorphy in the class of Hartogs domains in $\Bbb C^2$ is given.
We present various characterizations of $n$-circled domains of holomorphy $G\subset\CC^n$ with respect to some subspaces of $\Cal H^\infty(G)$.
Necessary and sufficient geometric conditions are given for domains with regular boundary points and edges to be domains of holomorphy provided the remainder boundary subset is of zero Hausdorff 1-codimensional measure.
We give a characterization of $L_h^2$-domains of holomorphy with the help of the boundary behavior of the Bergman kernel and geometric properties of the boundary, respectively.
Every connected bounded domain of holomorphy has connected boundary
研究证明多维复空间中连通有界全纯域的边界必连通，给出了全纯域拓扑性质的结论。
Let $X$ be a Riemann domain over $\mathbb C^k\times\mathbb C^\ell$. If $X$ is domain of holomorphy with respect to a family $\mathcal F\subset\mathcal O(X)$, then there exists a pluripolar set $P\subset\mathcal C^k$ such that every slice $X_a$ of $X$ with $a\notin P$ is a domain of holomorphy with respect to the family $\{f|_{X_a}: f\in\mathcal F\}$.
In this paper we investigate invariant domains in $\, Ξ^+$, a distinguished $\,G$-invariant, Stein domain in the complexification of an irreducible Hermitian symmetric space $\,G/K$. The domain $\,Ξ^+$, recently introduced by Krötz and Opdam, contains the crown domain $\,Ξ\,$ and it is maximal with respect to properness of the $\,G$-action. In the tube case, it also contains $\,S^+$, an invariant Stein domain arising from the compactly causal structure of a symmetric orbit in the boundary of $\,Ξ$. We prove that the envelope of holomorphy of an invariant domain in $\,Ξ^+$, which is contained n…
Let $p:X\longrightarrow M$ be a Riemann domain over a connected $n$-dimensional complex submanifold $M$ of $\mathbb C^N$ and let $\mathcal F\subset\mathcal O(X)$ be such that $p\in\mathcal F^N$. Our aim is to discuss relations between the $\mathcal F$-envelope of holomorphy of $(X,p)$ in the sense of Riemann domains over $M$ and the $\mathcal F$-envelope of holomorphy of $X$ in the sense of complex manifolds.
Let Omega be a domain in C^2. We prove the following theorem. If the envelope of holomorphy of Omega is schlicht over Omega, then the envelope is in fact schlicht. We provide examples showing that the theorem is not true in C^n, n>2. Additionally, we show that the theorem cannot be generalized to provide information about domains in C^2 whose envelopes are multiply sheeted.
A new characterization of $L^2$-domains of holomorphy with null thin complements via $L^2$-optimal conditions
本文研究$L^2$最优条件的性质，证明其可推出$L^2$可积全纯函数的$L^2$可除性，给出补集为零薄集的有界$L^2$全纯域的新刻画，解决相关Stein域判定问题并构造特殊的$L^2$最优域。
For a domain $D\subset {\Bbb{C}}^n$ we construct a continuous foliation of $D$ into one real dimensional curves such that any function $f\in {C^1(D)}$ which can be extended holomorphically into some neighborhood of each curve in the foliation will be holomorphic on $D$.
A domain $G\subset \bar\mathbb C$ is the domain of holomorphy of the generating function of a Pólya frequency sequence of order $r$ if and only if it satisfies the following conditions: (A) $G$ contains the point $z=0$, (B) $G$ is symmetric with respect to the real axis, (C) $T=dist(0,\partial G)\in \partial G$.
Given a domain Y in a complex manifold X, it is a difficult problem with no general solution to determine whether Y has a schlicht envelope of holomorphy in X, and if it does, to describe the envelope. The purpose of this paper is to tackle the problem with the help of a smooth 1-dimensional foliation F of X with no compact leaves. We call a domain Y in X an interval domain with respect to F if Y intersects every leaf of F in a nonempty connected set. We show that if X is Stein and if F satisfies a new property called quasiholomorphicity, then every interval domain in X has a schlicht envelope…
We give a simple and more elementary proof that the notions of Domain of Holomorphy and Weak Domain of Holomorphy are equivalent. This proof is based on a combination of Baire's Category Theorem and Montel's Theorem. We also obtain generalizations by demanding that the non-extentable functions belong to a particular class of holomorphic functions in the domain. We give an example of a domain in the plane which is a domain of holomorphy with respect to the class of all holomorphic functions but not with respect to the class of holomorphic functions continuously extendable on the closure of the …
This paper examines the broad structure on Stein manifolds and how it generalizes the notion of a domain of holomorphy in $\mathbb C^n$. Along with this generalization, we see that Stein manifolds share key properties from domains of holomorphy, and we prove one of these major consequences. In particular, we investigate an equivalence, similar to domains of holomorphy and pseudoconvexity, on the class of manifolds. Then, we examine the canonical symplectic structure of Stein manifolds inherited from this equivalence, and how its symplectic topology develops.
The bounded domains of holomorphy in~$\mathbf{C}^n$ whose Bergman kernel functions are zero-free form a nowhere dense subset (with respect to a variant of the Hausdorff distance) of all bounded domains of holomorphy.
The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein manifold is identified with a connected component of the set of equivalence classes of analytic discs immersed into the Stein manifold with boundary in the domain. This implies, in particular, that for each of its points the envelope of holomorphy contains an embedded (non-singular) Riemann surface (and also an immersed analytic disc) passing through this point with boundary contained in the natural embedding of the original domain into its envelope of holomorphy. Moreover, it says, that analytic continuation to a nei…
An explicit expression of the Lerch zeta function on maximal domains of holomorphy
该研究给出Lerch zeta函数的显式表达式，实现其多变量到最大全纯域的解析延拓并给出特殊值扩展公式，同时证明Lerch与Apostol函数方程本质一致。
We study the extension of holomorphic functions of bounded type defined on an open subset of a Banach space, to larger domains. For this, we first characterize the envelope of holomorphy of a Riemann domain over a Banach space, with respect to the algebra of bounded type holomorphic functions, in terms of the spectrum of the algebra. We then give a simple description of the envelopes of balanced open sets and relate the concepts of domain of holomorphy and polynomial convexity. We show that for bounded balanced sets, extensions to the envelope are always of bounded type, and that this does not…
In section 1, we show that if $X$ is a Stein normal complex space of dimension n and $D\subset \subset X$ an open subset which is the union of an increasing sequence $D_{1}\subset D_{2}\subset ...\subset D_{n}\subset >...$ of domains of holomorphy in $X$. Then $D$ is a domain of holomorphy. In section 2, we prove that a domain of holomorphy $D$ which is relatively compact in a 2-dimensional normal Stein space $X$ itself is Stein. In section 3, we show that if $X$ is a Stein space of dimension n and $D\subset X$ an open subspace which is the union of an increasing sequence $D_{1}\subset D_{2}\s…
In the present paper, we associate the techniques of the Lewy-Pinchuk reflection principle with the Behnke-Sommer continuity principle. Extending a so-called reflection function to a parameterized congruence of Segre varieties, we are led to studying the envelope of holomorphy of a certain domain covered by a smooth Levi-flat ``hat''. In our main theorem, we show that every C^infty-smooth CR diffeomorphism h: M to M' between two globally minimal real analytic hypersurfaces in C^n (n >1) is real analytic at every point of M if M' is holomorphically nondegenerate. More generally, we establish th…
We discuss domains of holomorphy and several notions of pseudoconvexity (drawing parallels with the corresponding notions from geometric convexity), and present a mostly self-contained solution to the Levi problem. We restrict our attention to domains of $\mathbb C^n$.
A two-dimensional counterexample for Kobayashi completeness of balanced domains
针对Jarnicki和Pflug提出的二维平衡域Kobayashi完备性问题，在$\mathbb C^2$中构造了具有连续Minkowski函数但不完备的有界平衡全纯域反例。
We extend the concept of a finite dimensional {\it holomorphic homogeneous regular} (HHR) domain and some of its properties to the infinite dimensional setting. In particular, we show that infinite dimensional HHR domains are domains of holomorphy and determine completely the class of infinite dimensional bounded symmetric domains which are HHR. We compute the greatest lower bound of the squeezing function of all HHR bounded symmetric domains, including the two exceptional domains. We also show that uniformly elliptic domains in Hilbert spaces are HHR.
Grauert showed that the existence of a complete Kähler metric does not characterize domains of holomorphy by constructing such metrics on the complements of complex analytic sets in a domain of holomorphy. In this note, we study the holomorphic sectional curvatures of such metrics in two prototype cases namely, $\mathbb{C}^n \setminus \{0\}, n \ge 2$ and $\mathbb{B}^N \setminus A$, $N \ge 2$ and $A \subset \mathbb{B}^N$ is a hyperplane of codimension at least two. This is done by computing the Gaussian curvature of its restriction to the leaves of a suitable holomorphic foliation of these two …
In this paper we study Carleson and reverse Carleson measures on holomorphic function spaces on a homogeneous Siegel domain of Type II. We prove several necessary conditions and sufficient conditions in order for a measure $μ$ to be Carleson and reverse Carleson on mixed-normed weighted Bergman spaces.
Exploring Holomorphic Retracts
本文系统刻画全纯域的全纯收缩，重点研究有界平衡伪凸域及其乘积，推广Mazet引理，证明非凸域的收缩不存在性，并应用于并集问题与双全纯不等价判定。
Given a domain of holomorphy $D$ in $\mathbb{C}^N$, $N\geq 2$, we show that the set of holomorphic functions in $D$ whose cluster sets along any finite length paths to the boundary of $D$ is maximal, is residual, densely lineable and spaceable in the space $\mathcal{O}(D)$ of holomorphic functions in $D$. Besides, if $D$ is a strictly pseudoconvex domain in $\mathbb{C}^N$, and if a suitable family of smooth curves $γ(x,r)$, $x\in bD$, $r\in [0,1)$, ending at a point of $bD$ is given, then we exhibit a spaceable, densely lineable and residual subset of $\mathcal{O}(D)$, every element $f$ of whi…
We study the holomorphic extendibility of $\text{Op}(p)u$, when $p$ is an analytic symbol, and explicit information is available on the domains of holomorphic extendibility of both $p$ and $u$. By a contour deformation argument, we obtain a precise local estimate of the domain of holomorphy of $\text{Op}(p)u$ in terms of the information on $p$ and $u$.
Let $D\subset \C^n,$ $G\subset \C^m$ be pseudoconvex domains, let $A$ (resp. $B$) be an open subset of the boundary $\partial D$ (resp. $\partial G$) and let $X$ be the 2-fold cross $((D\cup A)\times B)\cup (A\times(B\cup G)).$ Suppose in addition that the domain $D$ (resp. $G$) is {\it locally $\mathcal{C}^2$ smooth on $A$} (resp. $B$). We shall determine the "envelope of holomorphy" $\hat{X}$ of $X$ in the sense that any function continuous on $X$ and separately holomorphic on $(A\times G) \cup (D\times B)$ extends to a function continuous on $\hat{X}$ and holomorphic on the interior of $\ha…
On the Theory of Bergman Spaces on Homogeneous Siegel Domains
针对齐性Siegel域上混合范数Bergman空间现有两种研究思路下结果难对比的问题，本文梳理已有成果并在一种框架下补充了包含关系、对偶性等多方面结论，完善了相关理论。
We show that several spaces of holomorphic functions on a Riemann domain over a Banach space, including the nuclear and Hilbert-Schmidt bounded type, are locally $m$-convex Fréchet algebras. We prove that the spectrum of these algebras has a natural analytic structure, which we use to characterize the envelope of holomorphy. We also show a Cartan-Thullen type theorem.
Records from the S-Matrix Marathon: Tasty Bits of Several Complex Variables
本文以四讲概述多复变函数基础，围绕全纯域刻画，联系边界几何、CR几何与$\bar\partial$问题，并讨论全纯函数的边值与延拓。
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