Boundary behavior of functions in the de Branges--Rovnyak spaces

TL;DR

This paper investigates the boundary behavior of functions in de Branges--Rovnyak spaces, providing criteria for radial limits of derivatives and conditions for boundary continuity implying analyticity, with implications for Bernstein's inequality.

Contribution

It generalizes Ahern-Clark's criterion for radial limits and links boundary continuity to analyticity within de Branges--Rovnyak spaces.

Findings

Established a criterion for the existence of radial limits of derivatives.

Showed that boundary continuity of all functions implies their analyticity on that boundary.

Applied results to questions related to Bernstein's inequality.

Abstract

This paper deals with the boundary behavior of functions in the de Branges--Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges--Rovnyak spaces. This criterion generalizes a result of Ahern-Clark. Then we prove that the continuity of all functions in a de Branges--Rovnyak space on an open arc $I$ of the boundary is enough to ensure the analyticity of these functions on $I$. We use this property in a question related to Bernstein's inequality.