Geodesic
A Plessner Decomposition Along Transverse Curves
Canadian journal of mathematics, Tome 40 (1988) no. 5, pp. 1243-1255

Voir la notice de l'article provenant de la source Cambridge University Press

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]. 
Beatrous, Frank; Li, Songying. A Plessner Decomposition Along Transverse Curves. Canadian journal of mathematics, Tome 40 (1988) no. 5, pp. 1243-1255. doi: 10.4153/CJM-1988-053-1
@article{10_4153_CJM_1988_053_1,
     author = {Beatrous, Frank and Li, Songying},
     title = {A {Plessner} {Decomposition} {Along} {Transverse} {Curves}},
     journal = {Canadian journal of mathematics},
     pages = {1243--1255},
     year = {1988},
     volume = {40},
     number = {5},
     doi = {10.4153/CJM-1988-053-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-053-1/}
}
TY  - JOUR
AU  - Beatrous, Frank
AU  - Li, Songying
TI  - A Plessner Decomposition Along Transverse Curves
JO  - Canadian journal of mathematics
PY  - 1988
SP  - 1243
EP  - 1255
VL  - 40
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-053-1/
DO  - 10.4153/CJM-1988-053-1
ID  - 10_4153_CJM_1988_053_1
ER  - 
%0 Journal Article
%A Beatrous, Frank
%A Li, Songying
%T A Plessner Decomposition Along Transverse Curves
%J Canadian journal of mathematics
%D 1988
%P 1243-1255
%V 40
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-053-1/
%R 10.4153/CJM-1988-053-1
%F 10_4153_CJM_1988_053_1

[1] 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] 2. Calderon, A. P., The behavior of harmonic functions at the boundary, Trans. Amer. Math. Soc. 65(1950), 47–54. Google Scholar

[3] 3. Čirka, E. M., The theorems of Lindelbf and Fatou in Cn , Math. USSR Sbornik 21 (1973), 619–639. Google Scholar

[4] 4. Kor´nyi, A., The Poisson integral for generalized half-planes and bounded symmetric domains, Ann. Math. 82 (1965), 332–350. Google Scholar

[5] 5. Nagel, A. and Rudin, W., Local boundary behavior of bounded holomorphic functions, Can. J. Math. 30 (1978), 841–865. Google Scholar

[6] 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] 7. Privalov, J., Sur une généralisation du théorème de Fatou, Rec. Math. (Math. Sbornik) 31 (1923), 232–235. Google Scholar

[8] 8. Rudin, W., Function theory in the unit ball of n (Springer-Verlag, New York, 1980). Google Scholar | DOI

[9] 9. Stein, E. M., Singular integrals and differentiability properties of functions (Princeton University Press, Princeton, New Jersey, 1970). Google Scholar

Cité par Sources :