On boundary values of holomorphic functions on balls

Abstract

It is a result of Agranovski and Valski for which Nagel and Rudin, and Stout have given alternate proofs, that if B B is the open unit ball in C n {{\mathbf {C}}^n} and if f ∈ C ( ∂ B ) f \in C(\partial B) has the property that for every complex line Λ ⊂ C n \Lambda \subset {{\mathbf {C}}^n} , f | ( Λ ∩ ∂ B ) f\left | {(\Lambda \cap \partial B)} \right . has a continuous extension to Λ ∩ B ¯ \Lambda \cap \bar B which is holomorphic in Λ ∩ B \Lambda \cap B , then f f has a continuous extension to B ¯ \bar B which is holomorphic in B B . In the paper we give an easier, more geometric proof of this result and then prove the local version of this result.