Boundary values via reproducing kernels: The Julia-Carath\'eodory theorem

TL;DR

This paper generalizes the Julia-Carathéodory theorem using reproducing kernels, introducing new boundary concepts and composition factors, applicable to various spaces and leading to improved classical results.

Contribution

It introduces a novel boundary framework and generalizes key theorems to arbitrary sets using reproducing kernels and composition factors.

Findings

Generalized Julia-Carathéodory theorem to arbitrary sets

Proved Julia's lemma in an abstract setting

Improved classical results for weighted Dirichlet and Besov spaces

Abstract

Given a reproducing kernel $k$ on a nonempty set $X$, we define the reproductive boundary of $X$ with respect to $k$. Furthermore, we generalize the well known nontangential and horocyclic approach regions of the unit circle to this new kind of boundary. We also introduce the concept of a composition factor of $k$, an abstract analogue of analytic selfmaps of the unit disk. Using these notions, we obtain a far reaching generalization of the Julia-Carath\'eodory theorem, stated on an arbitrary set. We also prove Julia's lemma in the abstract setting and give sufficient conditions for the convergence of iterates of some selfmaps. As an application we improve the classical theorem on the unit disk for contractive multipliers of standard weighted Dirichlet spaces, as well as Besov spaces on the unit ball. Many examples and questions are provided for these novel objects of study.