Uniform Regularity in a Wedge and Regularity of Traces of CR Functions

Abstract

We discuss in Sect. 1 the property of regularity at the boundary of separately holomorphic functions along families of discs and apply, in Sect. 2, to two situations. First, let $\mathcal{W}$ be a wedge of ℂⁿ with C ^ω, generic edge ℰ: a holomorphic function f on $\mathcal{W}$ has always a generalized (hyperfunction) boundary value bv(f) on ℰ, and this coincides with the collection of the boundary values along the discs which have C ^ω transversal intersection with ℰ. Thus Sect. 1 can be applied and yields the uniform continuity at ℰ of f when bv(f) is (separately) continuous. When $\mathcal{W}$ is only smooth, an additional property, the temperateness of f at ℰ, characterizes the existence of boundary value bv(f) as a distribution on ℰ. If bv(f) is continuous, this operation is consistent with taking limits along discs (Theorem 2.8). By Sect. 1, this yields again the uniform continuity at ℰ of tempered holomorphic functions with continuous bv. This is the theorem by Rosay (Trans. Am. Math. Soc. 297(1):63–72, 1986), in whose original proof the method of “slicing” by discs is not used.

As related literature we mention, among others, Sato et al. (Lecture Notes in Mathematics, vol. 287, pp. 265–529, Springer, Berlin, 1973), Komatsu (J. Fac. Sci., Univ. Tokyo Sect. IA, Math. 19:201–214, 1972), Hörmander (Grundlehren der mathematischen Wissenschaften, vol. 256, Springer-Verlag, Berlin, 1984), Cordaro and Treves (Annals of Mathematics Studies, vol. 136, Princeton University Press, Princeton, 1994), Baouendi et al. (Princeton Mathematical Series, Princeton University Press, Princeton, 1999) and Berhanu and Hounie (Math. Z. 255:161–175, 2007).