My paper “A Schwarz lemma on the polydisk” has “officially” appeared on the Proceedings of the AMS website.
A Schwarz lemma on the polydisk
This paper can also be found here. For future reference the paper is cited as:
Greg Knese, A Schwarz lemma on the polydisk, Proc. Amer. Math. Soc., 135 (2007), 2759-2768
Abstract: This paper concerns a generalization of the infinitesimal portion of the classical Schwarz lemma inequality to the setting of the polydisk. Specifically, we give a complete description of the functions that are extremal for this inequality at every point of the polydisk: they are the transfer function of a symmetric unitary. In addition, some sufficient conditions are given for a function to be of this type.
Update: This paper has been referenced in recent preprint of J.M. Anderson, M.A. Dritschel, and J. Rovnyak. It can be found here.
Update 2: At the end of the paper we mention that we aren’t sure whether the extremal functions in the this paper always have the property that they cannot be continuously extended to the closed polydisk (except in the case of one variable). I realized recently that they indeed cannot be extended to the closed polydisk for the following reason. Rational inner functions on the polydisk which are continuous up to the boundary must be a Blaschke product of a fixed degree on every balanced disk. The extremal functions of this paper have the special property that through every point there is a balanced disk on which they are a Blaschke product of degree ONE! This is not true for every balanced disk, and therefore they cannot be extended continuously to the boundary. This all follows from the fact that continuity on the boundary forces the winding number of the holomorphic function on every balanced disk to be constant. I’m leaving out some details and being a little vague, but this is the idea.
