On the behavior of holomorphic functions near maximum modulus sets (Q1073973)

Let B denote the open unit ball in \({\mathbb{C}}^ n\). Given a ''generalized'' maximum modulus set \(E\subset \partial B\), we produce a \(g\in \cap H^ p(B)\) which fails to have a finite radial limit at every point of E. After proving that maximum modulus sets in a certain sense exist in abundance, the above result is shown to extend a theorem of Rudin which is similar in spirit. In addition, we discuss boundary behavior along non-radial paths; in this connection several observations related to the Lindelöf principle in \({\mathbb{C}}^ n\) are made.