Abstract. A sufficient condition for the infinite dimensionality of the Bergman space of a pseudoconvex domain is given. This condition holds on any pseudoconvex domain that has at least one smooth boundary point of finite type in the sense of D'Angelo.

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Let Ω ⊂ C be an open set. We show that ∂ has closed range in L 2 (Ω) if and only if the Poincaré-Dirichlet inequality holds. Moreover, we give necessary and sufficient potential-theoretic conditions for the ∂-operator to have closed range in L 2 (Ω). We also give a new necessary and sufficient potential-theoretic condition for the Bergman space of Ω to be infinite dimensional.for all u ∈ L 2 (Ω) with ∂u ∈ L 2 (Ω) and u orthogonal to the kernel of ∂, see [6, Theorem 1.1.1]. The kernel of ∂ is the closed subspace of L 2 (Ω) consisting of Date: October 30, 2019.

We investigate the question of existence of plurisubharmonic defining functions for smoothly bounded, pseudoconvex domains in C 2 . In particular, we construct a family of simple counterexamples to the existence of plurisubharmonic smooth local defining functions. Moreover, we give general criteria equivalent to the existence of plurisubharmonic smooth defining functions on or near the boundary of the domain. These equivalent characterizations are then explored for some classes of domains.

Abstract.Let Ω be an unbounded domain in R × R d . A positive harmonic function u on Ω that vanishes on the boundary of Ω is called a Martin function. In this note, we show that, when Ω is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition Ω has certain symmetry with respect to the t-axis, and ∂Ω is sufficiently flat, then the maximum of any Martin function along a slice Ω ∩ ({t} × R d ) is attained at (t, 0).