References for Analytic Continuation in Several Variables

Question

I need analytic continuation for several complex variables for my research. Could someone please recommend me some good resources for the same?

Answer 1

Since this interesting question has not, until now, attracted any answer, I'll try to give a few suggestions, in order to help.

The question itself is ambiguous, since in several complex variable, "analytic continuation" may have two quite different meanings. Precisely, it can mean

1. the extension of the domain of existence of an analytic object (a manifold, a function or more general objects) from a known domain of existence to a larger one or
2. the solution of the Dirichlet or (nowadays more and more frequently) Cauchy problems for a holomorphic function of several complex variable.

These two meaning are not totally unrelated: however, they are not the same thing at all.

Pertaining to the first meaning are the researches starting from Hartogs celebrated theorem and dealing, for example with envelopes of holomorphy and so on: roughly speaking, given an analytic object defined on a given open set, it is asked if this is the larger possible domain of definition of the given object. From my point of view, the most accessible and complete reference on these matters is the monograph [2] (there exists also a second edition, a copy of which unfortunately I do not own). The methods are analytic in character (more precisely potential theoretic) and the authors do their best in order to give a masterly exposition of the subject, by explaining carefully and providing the historical context of each topic.

The second meaning involves a connected set on the boundary of a domain (typically the whole $2n-1$-dimensional "boundary" in $\Bbb R^{2n} \equiv \Bbb C^{n}$ or a strict subset of it is given, but also lower codimension choices are possible): roughly speaking, assigned the "trace" (in some sense) of a (generalized) function on this set, it is asked to find a holomorphic function of several variable which is

• defined on the domain and
• whose "trace" on the boundary agrees with the assigned one.

In my opinion, a masterful reference on these topic is the monograph [1]: it is strongly analytic in character, as it uses concrete integral representations (like the Martinelli-Bochner integral) for holomorphic functions of several variables and goes trough by providing existence theorems with necessary and sufficient conditions for the solutions of the posed problem and so on. Even in this case the authors provide a masterly exposition of the subject.

Beware! Nor reference [1] nor [2] are elementary in character. They are masterly exposition but are nevertheless demanding time and attention to the serious scholar. and Well' my two cents.

References

[1] Alexander M. Kytmanov, Simona G. Myslivets, Multidimensional integral representations. Problems of analytic continuation (English), Cham: Springer, pp. xiii+225 (2015), ISBN 978-3-319-21658-4/hbk; 978-3-319-21659-1/ebook, MR3381727, Zbl 1341.32001.

[2] Marek Jarnicki, Peter Pflug, Extension of holomorphic functions, (English) De Gruyter Expositions in Mathematics, 34, Berlin: De Gruyter, pp. x+487 (2000), ISBN: 3-11-015363-7, MR1797263, Zbl 0976.32007.