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