A wedge‐of‐the‐edge theorem: analytic continuation of multivariable Pick functions in and around the boundary

Abstract

In 1956, quantum physicist N. Bogoliubov discovered the edge-of-the-wedge theorem, a theorem used to analytically continue a function through the boundary of a domain under certain conditions. We discuss an analogous phenomenon, a wedge-of-the-edge theorem, for the boundary values of Pick functions, functions from the poly upper half plane into the half plane. We show that Pick functions which have a continuous real-valued extension to a union of two hypercubes with a certain orientation in R d have good analytic continuation properties. Furthermore, we establish bounds on the behavior of this analytic continuation, which makes normal families arguments accessible on the boundary for Pick functions in several variables. Moreover, we obtain a Hartog's phenomenon type result for locally inner functions.