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Composition operators on Hardy spaces on Lavrentiev domains

Authors: Eva A. Gallardo-Gutiérrez, María J. González and Artur Nicolau
Journal: Trans. Amer. Math. Soc. 360 (2008), 395-410
MSC (2000): Primary 47B38, 30C85
Published electronically: July 23, 2007
MathSciNet review: 2342009
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Abstract: For any simply connected domain $ \Omega$, we prove that a Littlewood type inequality is necessary for boundedness of composition operators on $ \mathcal{H}^p(\Omega)$, $ 1\leq p<\infty$, whenever the symbols are finitely-valent. Moreover, the corresponding ``little-oh'' condition is also necessary for the compactness. Nevertheless, it is shown that such an inequality is not sufficient for characterizing bounded composition operators even induced by univalent symbols. Furthermore, such inequality is no longer necessary if we drop the extra assumption on the symbol of being finitely-valent. In particular, this solves a question posed by Shapiro and Smith (2003). Finally, we show a striking link between the geometry of the underlying domain $ \Omega$ and the symbol inducing the composition operator in $ \mathcal{H}^p(\Omega)$, and in this sense, we relate both facts characterizing bounded and compact composition operators whenever $ \Omega$ is a Lavrentiev domain.

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Additional Information

Eva A. Gallardo-Gutiérrez
Affiliation: Departamento de Matemáticas, Universidad de Zaragoza, Plaza San Francisco s/n, 50009 Zaragoza, Spain

María J. González
Affiliation: Departamento de Matemáticas, Universidad de Cádiz, Apartado 40, 11510 Puerto Real (Cádiz), Spain

Artur Nicolau
Affiliation: Departamento de Matemáticas, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain

Keywords: Composition operator, Lavrentiev domain, Nevanlinna counting function
Received by editor(s): April 28, 2005
Received by editor(s) in revised form: February 21, 2006
Published electronically: July 23, 2007
Additional Notes: The first author was partially supported by Plan Nacional I+D grant no. MTM2006-06431 and Gobierno de Aragón ref. DGA E-64. The second and third authors were partially supported by Plan Nacional I+D grant no. MTM2005-00544 and 2005SGR00774.
Article copyright: © Copyright 2007 American Mathematical Society

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