James Pascoe: The wedge-of-the-edge theorem

An analytic function on the complex upper half plane which has a continuous real-valued extension to some neighborhood in the real line analytically continues through the neighborhood to the entire lower half plane by the Schwarz reflection principle. In several variables, an analogue of this result for a product of upper half planes is known as the edge-of-the-wedge theorem. If we restrict the range of such a function to the upper half plane, we obtain a bit more data on the analytic continuation, and, in fact, obtain analytic continuation within the real plane itself- a result called the wedge-of-the-edge theorem. However, not much is known in the way of sharp results in either case. In this talk, I will discuss the general history described above, the wedge-of-the-edge theorem in particular, and some conjectures arising by analogy with what in known for rational functions and in limiting cases.