2024/03/31 by Bartosz Malman, Malman, Bartosz, Daniel Seco +1 · 1 citation
Mathematics · #30H45 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2404.00736
openalex publication_date 2024/03/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study conditions for containment of a given space X of analytic functions on the unit disk \mathbbD in the de Branges-Rovnyak space H(b). We deal with the non-extreme case in which b admits a Pythagorean mate a, and derive a multiplier boundedness criterion on the function ϕ= b/a which implies the containment X ⊂ H(b). With our criterion, we are able to characterize the containment of the Hardy space Hp inside H(b), for p ∈ [2, ∞]. The end-point cases have previously been considered by Sarason, and we show that in his result, stating that ϕ∈ H2 is equivalent to H^∞ ⊂ H(b), one can in fact replace H^∞ by BMOA. We establish various other containment results, and study in particular the case of the Dirichlet space D, containment of which is characterized by a Carleson measure condition. In this context, we show that matters are not as simple as in the case of the Hardy spaces, and we carefully work out an example.