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We prove that the Berezin transform is L p regular on a large class of domains in C n and not L 2 regular on the Hartogs triangle. [ABSTRACT FROM AUTHOR]

Let Ω be a smooth bounded pseudoconvex domain in ℂ n . We give several characterizations for the closure of Ω to have a Stein neighbourhood basis in the sense that Ω has a defining function ρ such that {z ∈ ℂ n  : ρ(z) < ϵ} is pseudoconvex for sufficiently small ϵ > 0. We also show that this condition is invariant under proper holomorphic maps that extend smoothly up to the boundary. [ABSTRACT FROM AUTHOR]