Let Ω be a domain in Cn. It is known that a holomorphic function on Ω behaves better in complex tangential directions. When Ω is of finite type, the best possible improvement is quantified at each point by the distance to the boundary in the complex tangential directions (see the papers on the geometry of finite type domains of Catlin, Nagel-Stein and Wainger for precise definition). We show that this improvement is characteristic: for a holomorphic function, a regularity in complex tangential directions implies the corresponding regularity in all directions. We give a pointwise inequality in both directions between the gradients and the complex tangential gradients. We characterize Besov, Sobolev and Lipschitz spaces of holomorphic functions defined on Ω by the behavior of complex tangential derivatives.
@article{urn:eudml:doc:41152,
title = {Behavior of holomorphic functions in complex tangential directions in a domain of finite type in Cn.},
journal = {Publicacions Matem\`atiques},
volume = {36},
year = {1992},
pages = {251-292},
zbl = {0758.32004},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41152}
}
Grellier, Sandrine. Behavior of holomorphic functions in complex tangential directions in a domain of finite type in Cn.. Publicacions Matemàtiques, Tome 36 (1992) pp. 251-292. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41152/