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 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.

2013

In this note, we consider the linear topological invariant e for Fréchet spaces of global analytic functions on Stein manifolds. We show that O (M) ; for a Stein manifold M; enjoys the property e if and only if every compact subset of M lies in a relatively compact sublevel set of a bounded plurisubharmonic function de ned on M: We also look at some immediate implications of this characterization. 1. Introduction Spaces of analytic functions, regarded as an important class of nuclear Fréchet spaces contributed amply to the development of the structure theory of Fréchet spaces. A profound example is the pioneering result of Dragilev [6] on the absoluteness of bases in the space of analytic functions on the unit disc with the usual topology. This paved the way to the far-reaching theorem of Dynin-Mitiagin [7] on the absoluteness of bases in every nuclear Fréchet space. Many more examples could readily be provided. Of course this in‡uence has not been one-sided. Techniques and concepts...

In this note, we consider the linear topological invariant {\Omega}-tilda for Fr\'echet spaces of global analytic functions on Stein manifolds. We show that O(M), for a Stein manifold M, enjoys the property {\Omega}-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.

Manuscripta Mathematica, 1988

We give a characterization of Stein manifolds M for which the space of analytic functions, 0(M), is isomorphic as Fr~chet spaces to the space of analytic functions on a polydisc interms of the existence of a plurisubharmonic function on M with certain properties. We discuss some corollaries of this result and give some examples.

PLURICOMPLEX GREEN FUNCTONS ON STEIN MANIFOLDS AND CERTAIN LINEAR TOPOLOGICAL INVARIANTS, 2023

In this paper we explore the existence of pluricomplex Green functions for Stein manifolds from a functional analysis point of view. For a Stein manifold M , we will denote by O (M) the Fréchet space of analytic functions on M equipped with the topology of uniform convergence on compact subsets. In the first section, we examine the relationship between existence of pluricomplex Green functions and the diametral dimension of O (M). This led us to consider negative plurisubharmonic functions on M with a nontrivial relatively compact sublevel set (semi-proper). In section 2, we characterize Stein manifolds possessing a semi-proper negative plurisubharmonic function through a local, controlled approximation type condition, which can be considered as a local version of the linear topological invariant Ω of D. Vogt. In Section 3 we look into pluri-Greenian and locally uniformly pluri-Greenian complex manifolds introduced by E. Poletsky. We show that a complex manifold is locally uniformly pluri-Greenian if and only if it is pluri-Greenian and give a characterization of locally uniformly pluri-Greenian Stein manifolds in terms of the notions introduced in Section 2.

2021

In this paper, we explore the existence of pluricomplex Green functions for Stein manifolds from a functional analysis point of view. For a Stein manifold M, we will denote by O(M) the Fr\\'echet space of analytic functions on M equipped with the topology of uniform convergence on compact subsets. In the first section, we examine the relationship between the existence of pluricomplex Green functions and the diametral dimension of O(M). This led us to consider negative plurisubharmonic functions on M with a nontrivial relatively compact sublevel set (semi-proper). In section 2, we characterize Stein manifolds possessing a semi-proper negative plurisubharmonic function through a local version of the linear topological invariant Omega-Tilda, of D.Vogt. In section 3 we look into pluri-Greenian complex manifolds introduced by E.Poletsky. We show that a complex manifold is locally uniformly pluri-Greenian if and only if it is pluri-Greenian and give a characterization of locally unifor...

The Michigan Mathematical Journal, 2008

We prove that a relatively compact pseudoconvex domain with smooth boundary in an almost complex manifold admits a bounded strictly plurisubharmonic exhaustion function. We use this result in order to study convexity and hyperbolicity properties of these domains and the contact geometry of their boundaries.

Pacific Journal of Mathematics, 1987

It is known that for 0 < p < oo the Hardy space H p contains a residual set of functions, each of which has range equal to the whole plane at every boundary point of the unit disk. With quite new general techniques, we are able to show that this result holds for numerous other spaces. The space BMOA of analytic functions of bounded mean oscillation, the Bloch spaces, the Nevanlinna space and the Dirichlet spaces D a f or 0 < a < 1/2 are examples. Our methods involve hyperbolic geometry, cluster set analysis and the "depth" function which we have used previously for determining geometric properties of the image surfaces of functions. Denote by D(a, r) the open disc in C centered at a and of radius r. Denote by D the unit disc D(0,l) and let Δ(a,r) = D Π D{a,r) for a e 3D. Brown and Hansen [4] proved that each Hardy space H p 9

Revista de la Unión Matemática Argentina

In this paper, we establish some variants of Stein's theorem, which states that a non-negative function belongs to the Hardy space H 1 (T) if and only if it belongs to L log L(T). We consider Bergman spaces of holomorphic functions in the upper half plane and develop avatars of Stein's theorem and relative results in this context. We are led to consider weighted Bergman spaces and Bergman spaces of Musielak-Orlicz type. Eventually, we characterize bounded Hankel operators on A 1 (C +).

In this manuscript, we study some hypersurfaces of Stein manifolds. A Stein manifold (M, J, g) is a complex manifold M with a complex structure J, a Kähler metric g and a fundamental form µ = i∂∂ ρ, where ρ : M → R is a smooth strictly plurisubharmonic exhaustion. We study the biconservativity condition on real hypersurfaces and Lagrangian submanifolds of Stein manifolds.