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Derivatives of rational inner functions: geometry of singularities and integrability at the boundary
Stockholm University, Faculty of Science, Department of Mathematics.
Number of Authors: 3
2018 (English)In: Proceedings of the London Mathematical Society, ISSN 0024-6115, E-ISSN 1460-244X, Vol. 116, no 2, p. 281-329Article in journal (Refereed) Published
Abstract [en]

We analyze the singularities of rational inner functions (RIFs) on the unit bidisk and study both when these functions belong to Dirichlet-type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative Hp membership purely in terms of contact order, a measure of the rate at which the zero set of an RIF approaches the distinguished boundary of the bidisk. We also show that derivatives of RIFs with singularities fail to be in Hp for p and that higher non-tangential regularity of an RIF paradoxically reduces the Hp integrability of its derivative. We derive inclusion results for Dirichlet-type spaces from derivative inclusion for Hp. Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of RIFs fails to belong to the unweighted Dirichlet space.

Place, publisher, year, edition, pages
2018. Vol. 116, no 2, p. 281-329
National Category
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-153660DOI: 10.1112/plms.12072ISI: 000424388300003OAI: oai:DiVA.org:su-153660DiVA, id: diva2:1188265
Available from: 2018-03-07 Created: 2018-03-07 Last updated: 2018-03-07Bibliographically approved

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