Angular derivatives and boundary values of H(b) spaces of unit ball of ℂⁿ

Abstract

In this work, deBranges–Rovnyak spaces, $H\left(b\right)$, on the unit ball of ${\mathbb{C}}^n$ are studied. An integral representation of the functions in $H\left(b\right)$ through the Clark measure on $S^n$ associated with b is given and a characterization of admissible boundary limits is given in relation with finite angular derivatives. Lastly, the interplay between Clark measures and angular derivatives is examined and it is obtained 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.

COMMUNICATED BY:

• H. Boas

Keywords:

• deBranges–Rovnyak spaces
• angular derivatives
• Clark measures
• admissible boundary limits

AMS SUBJECT CLASSIFICATIONS:

• 32A37 (primary)
• 32A35
• 32A40 (secondary)

Acknowledgments

I would like to thank the anonymous referee for his/her valuable comments and suggestions which improved the representation of this study greatly.

Disclosure statement

No potential conflict of interest was reported by the author.