Aydin Aytuna | Searcharxiv

arXiv2012

Parabolic Stein Manifolds

An open Riemann surface is called parabolic in case every bounded subharmonic function on it reduces to a constant. Several authors introduced seemingly different analogs of this notion for Stein manifolds of arbitrary dimension. In the first part of this note we compile these notions of parabolicity and give some immediate relations among them. In section 3 we relate some of these notions to the linear topological type of the Fréchet space of analytic functions on the given manifold. In sections 4 and 5 we look at some examples and show, for example, that the complement of the zero set of a Weierstrass polynomial possesses a continuous plurisubharmonic exhaustion function that is maximal off a compact subset.

math.CV↗

arXiv2012

Bohr property of bases in the space of entire functions and its generalizations

We prove that if $(φ_n)_{n=0}^\infty, \; φ_0 \equiv 1, $ is a basis in the space of entire functions of $d$ complex variables, $d\geq 1,$ then for every compact $K\subset \mathbb{C}^d$ there is a compact $K_1 \supset K$ such that for every entire function $f= \sum_{n=0}^\infty f_n φ_n$ we have $\sum_{n=0}^\infty |f_n|\, \sup_{K}|φ_n| \leq \sup_{K_1} |f|.$ A similar assertion holds for bases in the space of global analytic functions on a Stein manifold with the Liouville Property.

math.CV↗