Uniform approximation of Bloch functions and the boundedness of the integration operator on H^∞

2016/04/19 by Wayne Smith, Smith, Wayne, Dmitriy Stolyarov +3
Mathematics · #42B20 #42B35 #47A30 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #F.2.2 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1604.05433

openalex publication_date 2016/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We obtain a necessary and sufficient condition for the operator of integration to be bounded on H^∞ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. This gives a full characterization of symbols of certain Volterra operators that act on bounded analytic functions in the disc if the symbol is assumed to be univalent. Without this assumption the answer is not known, and as the example at the end of the paper shows, the natural answer is definitely false.

Related

• Norm-attaining integral operators on analytic function spaces
• Volterra type operators between Bloch type spaces and weighted Banach spaces
• Integral operators mapping into the space of bounded analytic functions
• Remarks on Ritt operators, their H^∞ functional calculus and associated square function estimates
• Integral operators induced by symbols with non-negative Maclaurin coefficients mapping into H^∞