Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 735-753

DERIVATIVES OF INNER FUNCTIONS IN BERGMAN SPACES INDUCED BY DOUBLING WEIGHTS

Fernando Pérez-González, Jouni Rättyä and Atte Reijonen

Universidad de La Laguna, Departamento de Análisis Matemático
P.O. Box, 456, 38200 La Laguna, Tenerife, Spain; fernando.perez.gonzalez 'at' ull.es

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; jouni.rattya 'at' uef.fi

University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; atte.reijonen 'at' uef.fi

Abstract. We find a condition for the zeros of a Blaschke product B which guarantees that B' belongs to the Bergman space A_ω^p induced by a doubling weight ω, and show that this condition is also necessary if the zero-sequence of B is a finite union of separated sequences. We also give a general necessary condition for the zeros when B' ∈ A_ω^p, and offer a characterization of when the derivative of a purely atomic singular inner function belongs to A_ω^p.

2010 Mathematics Subject Classification: Primary 30J10, 30J15; Secondary 30H20.

Key words: Bergman space, Blaschke product, doubling weight, inner function.

Reference to this article: F. Pérez-González, J. Rättyä and A. Reijonen: Derivatives of inner functions in Bergman spaces induced by doubling weights. Ann. Acad. Sci. Fenn. Math. 42 (2017), 735-753.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4248