Year: 2026
Type: preprint
Abstract: We study composition operators acting on the weighted Bergman spaces on the bidisc, i.e. $C_Φ:A^2_β(\mathbb{D}^2)\to A^2_β(\mathbb{D}^2)$ where $Φ$ is induced by rational inner functions (RIFs) or a RIF and a smooth function (mixed case). Our approach is geometric. Our main result is a uniform criterion for all $β\in(-1,0]$ that can be summarized as follows: Boundedness of the composition operator is equivalent to transversal intersection of the level sets for non-smooth symbols, under the assumption that if any tangential intersection occurs on the singularity it must be of high order. This extends the characterization of Bayart-Kosiński to the non-smooth self maps of the bidisc. To reach our conclusions, we utilize results obtained by Anderson, Bergqvist, Bickel, Cima and Sola on Clark measures associated to RIFs and Puiseux factorizations.
Source: arXiv (Cornell University)
Author: Beslikas, Athanasios
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7155355073
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