2023/11/21 by Peter Kristel, Eric Schippers, Kristel, Peter +3
Mathematics · #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Geometric and Algebraic Topology #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Representation Theory (math.RT) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2311.12972
openalex publication_date 2023/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To a conformal map f from the disk \mathbbD into the complex plane onto a domain with rectifiable Ahlfors-regular boundary, we associate a new kind of Grunsky operator on the Hardy space of the unit disk. This is analogous to the classical Grunsky operator, which itself can be viewed as an operator on Bergman or Dirichlet space. We show that the pull-back of the Smirnov space of the complement of f(\mathbbD) by f is the graph of the Grunsky operator. We also characterize those domains with rectifiable Ahlfors-regular boundaries such that the Grunsky operator is Hilbert-Schmidt. In particular, we show that if the Grunsky operator is Hilbert-Schmidt, then f(\mathbbD) is a Weil-Petersson quasidisk. The formulations of the results and proofs make essential use of a geometric treatment of Smirnov space as a space of half-order differentials.