A decomposition problem on weakly pseudoconvex domains

Abstract

Let Ω ⊂ Cn be a bounded pseudoconvex domain with defining function ρ of class CK , 2 ≤ K ≤ ∞. Let ν ≥ 1 be a real number such that ρ = −(−ρ)1/ν is a strictly plurisubharmonic exhaustion function on Ω. For a bounded open set Ω0 ⊃ Ω we obtain for S = Ω \ Ω the following theorem. Theorem 1. Let k be a nonnegative integer and tk = [2νmax(4+3k, n−1 5 ) +1]. Then there exists a Ck map W k = (W k 1 , · · · , W k n ) : Ω × S −→ Cn with the following properties: (i) W k(·, ζ) is holomorphic for all ζ ∈ S and for (z, ζ) ∈ Ω × S one has