Domains of finite type and Hölder continuity of the Perron-Bremermann function

Abstract

Abstract. Let Ω be a smoothly bounded domain in C n such that 0 ∈ ∂Ω. We give a bound for the type of ∂Ω at 0 in terms of the Hölder exponent of its Perron-Bremermann function with simple boundary data. We then use this to show that a smoothly bounded domain in C 2 is pseudoconvex of finite type if and only if its Perron-Bremermann function corresponding to Hölder continuous boundary data is Hölder continuous.

The complex Monge–Ampère equation on weakly pseudoconvex domains

“…• The L p data: Guedj, Kolodziej and Zeriahi prove in [6] that if ψ ∈ L p (Ω) with p > 1 and ϕ ∈ C 1,1 (bΩ) then u ∈ C γ ( Ω) for any γ < γ p := 2 qn+1 where 1 q + 1 p = 1. When Ω is no longer strongly pseudoconvex but has a certain "finite type", there are some known results for this problem due to Blocki [3], Coman [5], and Li [11]. Recently, Ha and the second author gave a general related result to a Hölder data under the hypothesis that Ω satisfies an f -property (see Definition 2.1 below).…”