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Pluripolar sets and pseudocontinuation. (English) Zbl 1085.32015

Agranovsky, Mark (ed.) et al., Complex analysis and dynamical systems II. Proceedings of the 2nd conference in honor of Professor Lawrence Zalcman’s sixtieth birthday, Nahariya, Israel, June 9–12, 2003. Providence, RI: American Mathematical Society (AMS); Ramat Gan: Bar-Ilan University (ISBN 0-8218-3709-5/pbk). Contemporary Mathematics 382. Israel Mathematical Conference Proceedings, 385-394 (2005).
Summary: In their recent book [‘Generalized analytic continuation’ (2002; Zbl 1009.30002)], W. T. Ross and H. S. Sharpiro, study various types of generalized analytic continuations (GAC) of meromorphic functions of a complex variable in situations where the classical theory says there is a natural boundary. An important representative of GAC is pseudocontinuation.
In this paper, we study a new type of GAC (not discussed in [loc. cit.]), expressed in terms of a pluripolar hull of the graph of a meromorphic function. In particular, we give an example of a function \(f\in A^\infty(\mathbb{D})\) such that \(f\) does not have a pseudo-continuation across any subset \(E\) of \(\partial\mathbb{D}\) of positive measure, while there is a meromorphic function \(F\) in \(\mathbb{D}_e:=\{|z |>1\}\) such that for every function \(U\) plurisubharmonic on \(\mathbb{C}^2\): if \(U(z,f (z))=-\infty\) on \(\mathbb{D}\) then \(U(z,f(z))=-\infty\) on the unit circle \(\partial \mathbb{D}\), and \(U(z,F(z))=-\infty\) for all \(z\in\mathbb{D}_e\) with \(F(z)\neq\infty\).
For the entire collection see [Zbl 1078.37001].

MSC:

32U15 General pluripotential theory
32U05 Plurisubharmonic functions and generalizations

Citations:

Zbl 1009.30002
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