Beschreibung:
<jats:p>Let <jats:italic>D</jats:italic> ⊂⊂ <jats:italic>C<jats:sup>n</jats:sup>
</jats:italic> be a bounded domain with smooth boundary
∂<jats:italic>D,</jats:italic> and let <jats:italic>F be</jats:italic> a
bounded holomorphic function on <jats:italic>D.</jats:italic> A generalization of
the classical theorem of Fatou says that the set <jats:italic>E</jats:italic> of
points on ∂<jats:italic>D</jats:italic> at which <jats:italic>F</jats:italic>
fails to have nontangential limits satisfies the condition σ
<jats:italic>(E)</jats:italic> = 0, where <jats:italic>a</jats:italic> denotes
surface area measure. We show in the present paper that this result remains true
when σ is replaced by 1-dimensional Lebesgue measure on
<jats:italic>certain</jats:italic> smooth curves γ in ∂D. The condition that γ
must satisfy is that its tangents avoid certain directions.</jats:p>