Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-25T22:25:29.257Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Boundary Behavior of Bounded Holomorphic Functions

Published online by Cambridge University Press:  20 November 2018

Alexander Nagel  and
Walter Rudin
Alexander Nagel
Affiliation:
University of Wisconsin, Madison, Wisconsin
Walter Rudin
Affiliation:
University of Wisconsin, Madison, Wisconsin
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 30 , Issue 3 , 01 June 1978 , pp. 583 - 592
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Burns, D. and Stout, E. L., Extending functions from submanifolds of the boundary, Duke Math. J. 43 (1976), 391–404.Google Scholar
2. Čirka, E. M., The theorems of Lindelof and Fatou in C”, Mat. Sb. 02 (134) (1973), 622-644 Math. USSR Sb. 21 (1973), 619–639).Google Scholar
3. Davie, A. M. and Øksendal, B. , Peak interpolation sets for some algebras of analytic functions, Pacific J. Math. 41 (1972), 81–87.Google Scholar
4. Henkin, G. M. and Čirka, E. M., Boundary behavior of’ hoi om orphie'functions of several complex variables, in Contemporary Problems in Mathematics, Vol. 4, Moscow, 1975 (in Russian).Google Scholar
5. Hormander, L., An introduction to complex analysis in several variables (Van Nostrand, 1966).Google Scholar
6. Hormander, L., Korányi, A., Harmonic functions on hermitian hyperbolic space, Trans. Amer. Math. Soc 135 (1969), 507–516.Google Scholar
7. Nagel, A., Smooth zero sets and interpolation sets for some algebras of holomorphic functions on strictly pseudoconvex domains, Duke Math. J. 43 (1976), 323–348.Google Scholar
8. Nagel, A., Cauchy transforms of measures, and a characterization of smooth peak interpolation sets for the ball algebra, to appear, Rocky Mountain J. Math.Google Scholar
9. Rudin, W., Peak interpolation sets of class C1, to appear, Pacific J. Math.Google Scholar
10. Stein, E. M., Boundary behavior of holomorphic functions of several complex variables (Princeton University Press, 1972).Google Scholar
11. Stein, E. M., Singular integrals and differentiability properties of functions (Princeton University Press, 1970).Google Scholar
12. Stein, E. M. and Weiss, G., Introduction to fourier analysis on euclidean spaces (Princeton University Press, 1971).Google Scholar