Hyperconvexity and Bergman completeness

The following text is a modified and updated version of the problem collection , which was written in 1993 but became publicly available only in 1995. It was a survey of various open problems; a general survey of the field was provided in [41, 42] in 1998, written in 1995 and 1996, respectively. Since then, a number of new developments have taken place, which in their turn have led to new questions. We feel it is time to update the problem collection.

Ann. Acad. Sci. Fenn. Math, 2008

We obtain several new characterizations for the standard weighted Bergman spaces A p α on the unit ball of C n in terms of the radial derivative, the holomorphic gradient, and the invariant gradient.

Tohoku Mathematical Journal, 1996

We show that the Bergman kernel function, associated to pseudoconvex domains of finite type with the property that the Levi form of the boundary has at most one degenerate eigenvalue, is a standard kernel of Calderón-Zygmund type with respect to the Lebesgue measure. As an application, we show that the Bergman projection on these domains preserves some of the Lebesgue classes.

Annales Polonici Mathematici, 2003

Sharp geometrical lower and upper estimates are obtained for the Bergman kernel on the diagonal of a convex domain D ⊂ C n which does not contain complex lines. It is also proved that the ratio of the Bergman and Carathéodory metrics of D does not exceed a constant depending only on n.

Bulletin of the American Mathematical Society, 1981

If D is a bounded open subset of C", the set H = {ƒ: D-> C| ƒ is holomorphic and S D \f\ 2 < +°°} is a separable infinite-dimensional Hubert space relative to the inner product <ƒ, g) = f D fg. The completeness of H can be seen from Cauchy integral estimates. Similar estimates show that for any p E D the functional ƒ H* ƒ(/?),ƒ£ H, is continuous. Thus there is a unique element K D (z, p) E f/ (as a function of z) such that, f(p) = f f(z)K D (z, p)dV(z) for all ƒ G H. The function K D is called the Bergman kernel function. If {y i }™ = , l is an orthonormal basis for f/ then K D (z, p) = ^.^.(z)^/?). The convergence of the series is absolute, uniformly on compact subsets ofD x D. For any z ED, K D (z, z) > 0 and log K D (z, z) is a real analytic function on D. The Hermitian form 3 2 ^âTÂF log K D(Z > z^dz i 0 dz~j i,j oz i oz j

The Geometry of Complex Domains, 2011

Journal of Operator Theory, 2002

In this paper we develop an overconvergence result in the context of polynomial approximation in the mean, primarily for certain weighted area measures. We then explore applications of this result pertaining to the existence and variety of cyclic vectors for the shift on Bergman spaces.

Proceedings of the American Mathematical Society

The boundary behavior of the Bergman metric near a convex boundary point z 0 of a pseudoconvex domain D ⊂ C n is studied; it turns out that the Bergman metric at points z ∈ D in direction of a fixed vector X 0 ∈ C n tends to infinite, when z is approaching z 0 , if and only if the boundary of D does not contain any analytic disc through z 0 in direction of X 0 .

Banach Journal of Mathematical Analysis, 2014

Let D be the open unit disk with its boundary ∂D in the complex plane C and dA(z) = 1 π dxdy, the normalized area measure on D. Let L 2 a (D, dA) be the Bergman space consisting of analytic functions on D that are also in L 2 (D, dA). In this paper we obtain certain distance estimates for bounded linear operators defined on the Bergman space. 1 (1−zw) 2. The function K(z, w) is called the Bergman kernel of D or the

Indiana University Mathematics Journal, 2008

arXiv: Complex Variables, 2014

The Kohn-Nireberg domains are unbounded domains in the complex Euclidean space of dimension 2 upon which many outstanding questions are yet to be explored. The primary aim of this article is to demonstrate that the Bergman and Caratheodory metrics of any Kohn-Nirenberg domains are positive and complete.

Glasgow Mathematical Journal, 2009

It was shown in [2] that a holomorphic function f in the unit ball B n of C n belongs to the weighted Bergman space A p α , p > n + 1 + α, if and only if the function |f

Rivista di Matematica della Università di Parma, 2001

In this paper we study the set of self-Bergmann metrics on the Riemann sphere endowed with the Fubini-Study metric and we extend a theorem of Tian to the case of the punctured plane endowed with a natural flat metric.

Nagoya mathematical journal

Some results for the Bergman functions in unbounded domains are shown. In particular, a class of unbounded Hartogs domains, which are Bergman complete and Bergman exhaustive, is given.

The purpose of this paper is to prove $L^p$-Sobolev and H\&quot;older estimates for the Bergman projection on both pseudoconvex domains of finite type and a large class of pseudoconvex domains of infinite type.