Tangential H∞-images of boundary curves

Abstract

1.Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixedn≥ 1. We define (Γ, μ) to be aFatou pair inΩ if(a) Γ is a continuous family of boundary curves γwin Ω, one ending at each w ∈ ∂Ω,(b) μ is a positive finite Borel measure on ∂Ω, and(c) the conclusion of Fatou's theorem holds with respect to Γ and μ. Let us state (a) and (c) in more detail:(a) The map(w, t) → γw(t)is continuous, from ∂Ω × [0, 1) into Ω, andfor every w in the boundary ∂Ω of Ω.(c) For everyf ∈ H∞(Ω)(the class of all bounded holomorphic functions in Ω), the limitexists a.e. [μ].