Boundary quasi-analyticity and a Phragmén–Lindelöf type theorem in classes of functions of bounded type in tubular domains

Abstract

A complete description is obtained of the Carleman classes on R n \mathbb {R}^n such that every function of bounded type in C + n \mathbb {C}^n_+ whose boundary values belong to the class under study is in fact a member of the corresponding Carleman class in C + n ∪ R n \mathbb {C}^n_+\cup \mathbb {R}^n . Also a refinement of the classical Salinas theorem is obtained, namely: under the conditions of the Salinas theorem on quasi-analyticity, instead of the assumption that a function belongs to the Carleman class in C + n ∪ R n \mathbb {C}^n_+\cup \mathbb {R}^n it suffices that its boundary values on R n \mathbb {R}^n belong to the Carleman class, and the function is of bounded type in C + n \mathbb {C}^n_+ .