scholarly journals Boundary behavior of the Carathéodory, Kobayashi, and Bergman metrics on strongly pseudoconvex domains in ${\mathbf{C}}^n $ with smooth boundary

1973 ◽  
Vol 79 (4) ◽  
pp. 749-752 ◽  
Author(s):  
Ian Graham
Keyword(s):  
Smooth Boundary ◽  
Boundary Behavior ◽  
Pseudoconvex Domains ◽  
Strongly Pseudoconvex Domains ◽  
Bergman Metrics

Local Boundary Behavior of Bounded Holomorphic Functions

1978 ◽  
Vol 30 (03) ◽  
pp. 583-592 ◽  
Author(s):  
Alexander Nagel ◽  
Walter Rudin
Keyword(s):  
Smooth Boundary ◽  
Boundary Behavior ◽  
Holomorphic Functions ◽  
Area Measure ◽  
Surface Area Measure ◽  
Smooth Curves ◽  
Local Boundary ◽  
Bounded Holomorphic Function ◽  
Bounded Holomorphic ◽  
Dimensional Lebesgue Measure

Let D ⊂⊂ Cn be a bounded domain with smooth boundary ∂D, and let F be a bounded holomorphic function on D. A generalization of the classical theorem of Fatou says that the set E of points on ∂D at which F fails to have nontangential limits satisfies the condition σ (E) = 0, where a 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 certain smooth curves γ in ∂D. The condition that γ must satisfy is that its tangents avoid certain directions.

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