Growth properties of analytic and plurisubharmonic functions of finite order. (English) Zbl 0619.32003

Sigurdsson, Ragnar

Math. Scand. 59, 235-304 (1986).

Let \(f\in {\mathcal O}({\mathbb{C}}^ n)\) be an entire function of finite order \(\rho >0\) and of finite type. In the first part of the paper, the author studies problems related to characterization of the indicator \(i_ f\) and of the limit set of \(\log | f|\) (i.e. the set of all \(p\in PSH({\mathbb{C}}^ n)\) such that there exists a sequence \(t_ j\to +\infty\) with \(t_ j^{-\rho}\log | f(t_ j.)| \to p\) in \({\mathcal D}')\). In particular, he presents the following improvement of the indicator theorem. Let M be a compact neighbourhood of \(O\in {\mathbb{C}}^ n\) and let \(p\in PSH({\mathbb{C}}^ n)\) be positively homogeneous of order \(\rho\). Then there exists \(f\in {\mathcal O}({\mathbb{C}}^ n)\) such that \(i_ f=p\) and \(\int | f|^ 2(1+| z|^ 2)^{-\rho -3n}\exp (-2p_ M)d\lambda <+\infty\), where \(p_ M(z):=\sup \{p(z+w): w\in M\}.\) In the second section, the author investigates the indicator function \(i_{\hat u}\) of the Fourier-Laplace transform of a hyperfunction u with compact support. In particular, he characterizes the set of all \(\zeta \in {\mathbb{C}}^ n\) such that \(i_{\hat u}(\zeta)=H(im\zeta)\), where H is the supporting function of the convex hull of supp(u). The last section is devoted to problems concerning indicator functions \(i_{\hat u}\) and limit sets of \(\log| \hat u|\) for Fourier-Laplace transforms \(\hat u\) of distributions u with compact support.

Reviewer: M.Jarnicki

MSC:

32A15	Entire functions of several complex variables
32U05	Plurisubharmonic functions and generalizations
32A45	Hyperfunctions

Keywords:

asymptotic approximation; plurisubharmonic function; entire function; finite order; finite type; indicator; limit set; Fourier-Laplace transforms; distributions