Tangential boundary behavior in Hilbert spaces of analytic functions

TL;DR

This paper extends boundary behavior analysis in Hilbert spaces of analytic functions, generalizing classical results to higher derivatives and broader approach regions, with implications for angular derivatives and boundary regularity.

Contribution

It generalizes recent boundary behavior results to higher order derivatives in $H(b)$ spaces and broader approach regions, providing a new proof and deeper understanding.

Findings

Generalizes boundary behavior results to higher derivatives in $H(b)$ spaces

Shows non-tangential approach regions are optimal for angular derivatives

Provides a new self-contained proof of the generalized result

Abstract

Sarason's Hilbert space version of Carath\'eodory-Julia Theorem connects the non-tangential boundary behavior of functions in de Branges-Rovnyak space $H(b)$ with the existence of angular derivatives in the sense of Carath\'eodory for $b$, an analytic self-mapping of the unit disk. In this article, we continue the study of higher order extensions of this result that deal with derivatives of functions in $H(b)$, and we consider notions of approach regions more general than the non-tangential ones. Our main result generalizes the recent work of Duan-Li-Mashreghi on boundary behavior in model spaces to $H(b)$-spaces and to higher order derivatives, and we give a new self-contained proof of that result. It also generalizes earlier radial results of Fricain-Mashreghi. In relation to existence of angular derivatives, we show that in the classical Carath\'eodory-Julia Theorem one cannot replace the non-tangential approach region by any essentially larger region.