Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-08-06T05:41:03.222Z Has data issue: false hasContentIssue false
  • English
  • Français

A Plessner Decomposition Along Transverse Curves

Published online by Cambridge University Press:  20 November 2018

Frank Beatrous  and
Songying Li
Frank Beatrous
Affiliation:
University of Pittsburgh, Pittsburgh, Pennsylvania
Songying Li
Affiliation:
University of Pittsburgh, Pittsburgh, Pennsylvania
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.

A classical theorem of Plessner [6] asserts that any holomorphic function f on the unit disk partitions the unit circle, modulo a null set, into two disjoint pieces such that at each point of the first piece, f has a non-tangential limit, and at each point of the second piece, the cluster set of f in any Stolz angle is the entire plane. Higher dimensional versions of this result were first obtained by Calderon [2], who considered holomorphic functions on Cartesian products of half-planes. In this setting, an exact analogue of the one-dimensional result is obtained, in which the circle is replaced by the distinguished boundary, and the Stolz angles are replaced by products of cones in the coordinate half-planes. The ideas of Calderon were further developed by Rudin [8, pp. 79-83], who considered holomorphic and invariant harmonic functions in the ball of C n . In this case, the circle is replaced by the unit sphere, and the Stolz angles are replaced by the approach regions of Korányi [4].

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 5 , 01 October 1988 , pp. 1243 - 1255
Copyright
Copyright © Canadian Mathematical Society 1988

References

1. Ahem, P. and Nagel, A., Strong LP estimates for maximal functions with respect to singular measures, with applications to exceptional sets, Duke Math. J. 53 (1986), 359–393.Google Scholar
2. Calderon, A. P., The behavior of harmonic functions at the boundary, Trans. Amer. Math. Soc. 65(1950), 47–54.Google Scholar
3. Čirka, E. M., The theorems of Lindelbf and Fatou in Cn , Math. USSR Sbornik 21 (1973), 619–639.Google Scholar
4. Kor´nyi, A., The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. Math. 82 (1965), 332–350.Google Scholar
5. Nagel, A. and Rudin, W., Local boundary behavior of bounded holomorphic functions, Can. J. Math. 30 (1978), 841–865.Google Scholar
6. Plessner, A., Über das Verhalten analytischer Funktionen auf dem Rande des Definitions-bereiches, J. Reine Angew. Math. 158 (1928), 219–227.Google Scholar
7. Privalov, J., Sur une généralisation du théorème de Fatou, Rec. Math. (Math. Sbornik) 31 (1923), 232–235.Google Scholar
8. Rudin, W., Function theory in the unit ball of C n (Springer-Verlag, New York, 1980).CrossRefGoogle Scholar
9. Stein, E. M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, New Jersey, 1970).Google Scholar