Reproducing kernel functions and boundary limits in model spaces

Yongjiang Duan, Yufei Li and Javad Mashreghi

Vol. 337 (2025), No. 2, 225–244

DOI: 10.2140/pjm.2025.337.225

Abstract

We investigate the boundary behavior of functions in model spaces, emphasizing their approach towards boundary points through arbitrary but fixed regions. We generalize classical results to encompass any approach region, thereby resolving an open question about reproducing kernel functions. The study connects the geometric nature of approach regions to the analytic properties of functions in the model space as they approach the unit circle. Key conditions are established to fully characterize boundedness, weak convergence, and the existence of limits within the approach region for functions in these spaces. Furthermore, precise criteria for norm convergence of reproducing kernel functions are provided, offering deeper insights into how boundary geometry influences analytic behavior. These results extend classical boundary value problems to an ultimately broader framework, highlighting the interplay between geometry and analysis.

Keywords

model space, boundary behavior, inner function

Mathematical Subject Classification

Primary: 30J05, 46E22

Milestones

Received: 22 January 2025

Revised: 9 May 2025

Accepted: 10 May 2025

Published: 3 June 2025