About boundary values in $A(\Omega)$

Abstract

Assume that Ω C ℂ n is a balanced bounded domain with a holomorphic support function (e.g. strictly pseudoconvex domain with C 2 boundary). We denote ∥f∥ 2 z := ∫ 1 0 |f(e 2πit z)| 2 dt. Let e > 0 and σ be a circular invariant Borel probability measure on ∂Ω. If g ∈ A(Ω) and h is a continuous function on ∂Ω with |g| < h on ∂Ω, then we construct nonconstant functions f 1 , f 2 ∈ A(Ω) with ∥g + f 1 ∥ z ≤ ∥h∥ z , |(g + f 2 )(z)| ≤ max |λ|=1 h(λz) for z ∈ ∂Ω and Additionally if Ω is a circular, bounded, strictly convex domain with C 2 boundary, then we give the construction of f 3 ∈ O(Ω), the holomorphic function with: ∥h ― |g + f * 3 |∥ z = 0 for all z ∈ ∂Ω, where f * denotes the radial limit of f. We also construct f 4 ∈ A(Ω) with ∥g + f 4 ∥ z = ∥h∥ z for z ∈ ∂Ω. In all cases we can make f i arbitrarily small on a given compact subset F C Ω and make it vanish to a given order at the point 0.