Angular Derivatives and Boundary Values of H(b) Spaces of Unit Ball of $\mathbb{C}^n$

TL;DR

This paper investigates deBranges-Rovnyak spaces on the unit ball of complex n-space, providing integral representations, boundary limit characterizations, and connections between Clark measures and angular derivatives.

Contribution

It introduces integral representations of functions in $H(b)$, characterizes boundary limits via angular derivatives, and links Clark measures with angular derivatives at boundary points.

Findings

Clark measure has an atom at a boundary point iff $b$ has finite angular derivative there.

Provides integral representation of $H(b)$ functions via Clark measure.

Characterizes admissible boundary limits in relation to angular derivatives.

Abstract

In this work we study deBranges-Rovnyak spaces, $H(b)$, on the unit ball of $\mathbb{C}^n$. We give an integral representation of the functions in $H(b)$ through the Clark measure on $S^n$ associated with $b$. A characterization of admissible boundary limits is given in relation with finite angular derivatives. Lastly, we examine the interplay between Clark measures and angular derivatives showing that Clark measure associated with $b$ has an atom at a boundary point if and only if $b$ has finite angular derivative at the same point.