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Separately analytic functions. (English) Zbl 1247.32002

EMS Tracts in Mathematics 16. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-098-2/hbk). ix, 297 p. (2011).
A function defined on a domain \(D\) in \(\mathbb C^n\) is called separately holomorphic if its restrictions to lines parallel to the coordinate axes are holomorphic. In striking contrast to real-variable theory, where separate differentiability does not imply total differentiability, a landmark result by F. Hartogs [“Zur Theorie der analytischen Funktion mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten”, Math. Ann. 62, 1–88 (1906; JFM 37.0444.01)] tells that separately holomorphic functions are holomorphic. The present book gives a comprehensive treatment of separate holomorphy, guiding the reader from Hartogs’ classical result to far reaching, technically demanding generalizations, some of them being very recent. Although the book does not put much emphasis on historical background, it gives a splendid opportunity to review a wealth of ideas and methods developed by complex analysts in the course of one century.
The kind of generalization the book is mostly concerned with is holomorphic extension from crosses. In the easiest relevant case, one looks at domains \(D\subset\mathbb C^p\) and \(G\subset\mathbb C^q\) and subsets \(A\subset D\), \(B\subset G\). We define the cross \(\mathbb X=(A\times G)\cup(D\times B)\) and say that a function is separately holomorphic on \(\mathbb X\) (in short \(f\in{\mathcal O}_s(\mathbb X)\)) if we have \(f(a,\cdot)\in{\mathcal O}(G)\) for every \(a\in A\), and \(f(\cdot,b)\in{\mathcal O}(D)\) for every \(b\in B\). The question is whether such functions admit a holomorphic extension to some uniform domain containing \(\mathbb X\).
A thorough study of this problem is the main content of the first part of the book. Before attacking the question in full generality, the authors start in Chapter 1 with the fundamental theorems of Osgood and Hartogs. For the theorem of Hartogs, two less known proofs, due to Leja and Koseki, are presented, where explicit use of potential theory is avoided. This very classical material is complemented by a discussion of refined problems considered by Hukuhara, Shimoda and Tuichiev. The next two chapters collect necessary tools. Chapter 2 treats pseudoconvexity of Riemann domains and envelopes of holomorphy. Here a certain number of major results are stated without proof, but in a way that an interested reader can easily refer to the authors’ previous book [“Extension of holomorphic functions”. Berlin: de Gruyter (2000; Zbl 0976.32007)]. Chapter 3 reviews topics from pluripotential theory with a focus on relative extremal functions. Chapter 4 presents classical applications like Tuichiev’s and Terada’s theorems and indicates some interesting related problems, like, for instance, problems about separate analyticity along foliations by holomorphic discs which are only smooth in the normal direction.
Chapter 5 treats the classical cross theorem.
Recall that \(A\subset D\) is pluriregular at \(p\in\overline{A}\) if for every open neighborhood \(U\subset D\) of \(p\), the relative extremal function \(h^*_{A\cap U,U}\) of \(A\cap U\) in \(U\) vanishes at \(p\). To exclude obvious obstructions, one assumes that \(A\) and \(B\) are pluriregular at all of their points. Then the main cross theorem tells that any \(f\in{\mathcal O}_s(\mathbb X)\) has a unique holomorphic extension \(\hat{f}\) to the domain
\[ \widehat{\mathbb X}=\big\{(z,w)\in D\times G: h^*_{A,D}(z)+h^*_{B,G}(w)<1\big\} \]
such that we have \(\hat{f}(\widehat{\mathbb X})=f(\mathbb X)\) and the estimate
\[ \big|\hat{f}(z,w)\big|\leq \|f\|_{A\times B}^{1-h^*_{A,D}(z)-h^*_{B,G}(w)} \|f\|_{{\mathbb X}}^{h^*_{A,D}(z)+h^*_{B,G}(w)} \]
for \((z,w)\in\widehat{\mathbb X}\), with sup-norms on the right side. Chapter 6 is concerned with an extension to arbitrary manifolds, which was established by V.-A. Ahn using Poletsky discs. Chapters 7 and 8 establish cross theorems of generalized and boundary crosses.
The second part of the book studies cross extension theorems in the presence of singularities. In preparation, Chapter 9 presents some fundamental results on holomorphic extension with singularities like theorems of Oka-Nishino, Chirka-Sadullaev, Grauert-Remmert and Dloussky. In Chapter 10, much of this is then applied to prove a very general cross theorem with singularities, due to recent work of the authors. The whole monograph is rounded off in Chapter 11 by the Rothstein theorem on separate meromorphy and a cross theorem for meromorphic extension.
As the authors admit themselves, the book is not easy reading. It is however attractive for researchers and serious students since the writing is very clear and complemented by open problems. Needless to say, many results appear for the first time in a monograph.

MSC:

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32A10 Holomorphic functions of several complex variables
32A17 Special families of functions of several complex variables
32D05 Domains of holomorphy
32D10 Envelopes of holomorphy
32D26 Riemann domains
32U15 General pluripotential theory
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