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Tangential H∞-images of boundary curves

Published online by Cambridge University Press:  24 October 2008

Walter Rudin
Walter Rudin
Affiliation:
University of Wisconsin, Madison, WI 53706, U.S.A.
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Extract

1. Introduction. Suppose that Ω is a region (i.e. a connected open set) in ࠶n, for some fixed n ≥ 1. We define (Γ, μ) to be a Fatou pair in Ω if

(a) Γ is a continuous family of boundary curves γw in 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, and

for every w in the boundary ∂Ω of Ω.

(c) For every f ∈ H∞(Ω) (the class of all bounded holomorphic functions in Ω), the limit

exists a.e. [μ].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 104 , Issue 1 , July 1988 , pp. 115 - 118
Copyright
Copyright © Cambridge Philosophical Society 1988

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References

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