$$L_h^2$$ L h 2 -Functions in Unbounded Balanced Domains

Abstract

We investigate problems related with the existence of square integrable holomorphic functions on (unbounded) balanced domains. In particular, we solve the problem of Wiegerinck for balanced domains in dimension two. We also give a description of L 2 h -domains of holomorphy in the class of balanced domains and present a purely algebraic criterion for homogeneous polynomials to be square integrable in a pseudoconvex balanced domain in C 2 . This allows easily to decide which pseudoconvex balanced domain in C 2 has a positive Bergman kernel and which admits the Bergman metric.

Generalizations of the Higher Dimensional Suita Conjecture and Its Relation with a Problem of Wiegerinck

“…Note that the non-triviality of the space L 2 h (I D (w)) in the case n = 2 is precisely described in [16].…”

“…, k − 1. We use a theorem of Donelly-Feffermann (see [10] or Theorem 2.2 in [4]) with the following data (16) ϕ…”

“…). Making use of the result from [16] we get the following partial solution of the problem of Wiegerinck (see [18]). Corollary 6.…”

On a theorem of Wiegerinck

“…Despite of a lot of work and partial results (cf. [BZ20,GHH17,J12,PW07,PW17], as far as we can tell, the conjecture is still open.…”

On the dimension of Bergman spaces on $\mathbb{P}^1$

“…Through work of Carleson [4] and Wiegerinck [18] the dimension of the Bergman space of an open set in the complex plane is completely characterized by the polarity of the complement, see the equivalences (a)-(c) in Theorem 1.1. Wiegerinck [18] also constructs domains in C 2 whose Bergman spaces are nontrivial but finite dimensional; see [12,6,10,7,13,3] for further partial results on the dimension of Bergman spaces of open sets in higher dimensions.…”