A REMARK ON THE DIMENSION OF THE BERGMAN SPACE OF SOME HARTOGS DOMAINS

Abstract. Let D be a Hartogs domain of the form D = {(z, w) ∈ C × CN: ‖w ‖ < e−u(z) } where u is a subharmonic function on C. We prove that the Bergman space L2 h (D) of holomorphic and square integrable functions on D is either trivial or infinite dimensional. 1. introduction Let L 2 h (Ω) denote the Bergman space of a domain Ω ⊂ CN, i.e. the space of square integrable and holomorphic functions on Ω. We are interested in the following open question (see e.g. [Jar–Pfl], [Pfl-Zwo]): is there a pseudoconvex domain with finite dimensional and nontrivial Bergman space? J. Wiegerinck (cf. [Wie]) gave examples of Reinhardt domains in C 2 such that their Bergman spaces were finite dimensional but nontrivial. Those domains, however, are not pseudoconvex. What is more, there exists a simple geometric characterization of pseudoconvex Reinhardt domains: if a logarithmic image of such a domain contains a real affine line then its Bergman space is {0}, otherwise it is infinite dimensional (cf. [Zwo 1, Zwo 2]). It is known that a Bergman space for any subdomain of C is also either infinite dimensional or trivial (cf. [Skw], [Wie]). We consider Hartogs domains Dϕ of the form Dϕ = Dϕ(G) = {(z, w) ∈ G × C N: ‖w ‖ < e −ϕ(z) } ⊂ C M × C N, where G is a domain in C M, ϕ ∈ PSH(G) and ‖ · ‖ denotes the maximum norm. We use the maximum norm for convenience but the same results hold for any C– norm. (If ˜ Dϕ is such a domain defined for other C–norm then one just needs to take Dϕ+c1 ⊂ ˜ Dϕ ⊂ Dϕ+c2 for suitable constants c1, c2.) We believe that the answer for this question is negative, at least for Hartogs domains. Even though in this paper we are dealing with domains with one dimensional basis, we think that the main idea (cf. Proposition 3.3) and some techniques of the proofs could also be used at least in some multi-dimensional cases with the help of advanced pluripotential theory.