Stein manifolds M for which O(M) has the property Ω-tilda (Dedicated to Mikhail Mikhaylovich Dragilev on the occation of his 90th birthday)

In this note, we consider the linear topological invariant Ω-tilda for Fréchet spaces of global analytic functions on Stein manifolds. We show that O(M), for a Stein manifold M, enjoys the property Ω-tilda if and only if every compact subset of M lies in a relatively compact sublevel set of a bounded plurisubharmonic function defined on M. We also look at some immediate implications of this characterization

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 these different definitions. In section 3 we relate some of these notions to the linear topological type of the Fr´echet 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

S-parabolic manifolds

A Stein manifold is called S-Parabolic in case there exits a special plurisubharmonic exhaustion function that is maximal outside a compact set. If a continuous special plurisubharmonic exits then we will call the manifold S*-Parabolic: In one dimensional case these notions are equivalent. However in several variables the question as to weather these notions coincide seems open. In this note we establish an interrelation between these two notions

On lelong-bremermann lemma

The main theorem of this note is the following refinement of the well-known Lelong-Bremermann Lemma: Let u be a continuous plurisubharmonic function on a Stein manifold. of dimension n. Then there exists an integer m C-m, s = 1, 2,..., such that the sequence of functions u(s) (z) = 1/p(s) max (ln vertical bar g(j)((s)) (z)vertical bar : j = 1,..., m converges to u uniformly on each compact subset of Omega. In the case when Omega is a domain in the complex plane, it is shown that one can take m = 2 in the theorem above (Section 3); on the other hand, for n-circular plurisubharmonic functions in C-n the statement of this theorem is true with m = n + 1 (Section 4). The last section contains some remarks and open questions

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 Fr\'echet 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.Comment: Corrected typos. Added reference

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

We prove that if $(\varphi_n)_{n=0}^\infty, \; \varphi_0 \equiv 1,$ 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 $f= \sum_{n=0}^\infty f_n \varphi_n$ we have $\sum_{n=0}^\infty |f_n|\, \sup_{K}|\varphi_n| \leq \sup_{K_1} |f|.$ A similar assertion holds for bases in the space of global analytic functions on a Stein manifold with the Liouville Property.Comment: This version is accepted for publication in the Bulletin of the London Mathematical Societ

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 these different definitions. 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 section 4 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