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Composition operators between Hardy and weighted Bergman spaces on convex domains in $ {\bf C}\sp N$


Authors: Barbara D. MacCluer and Peter R. Mercer
Journal: Proc. Amer. Math. Soc. 123 (1995), 2093-2102
MSC: Primary 47B38; Secondary 32A35, 46E15
DOI: https://doi.org/10.1090/S0002-9939-1995-1254846-9
MathSciNet review: 1254846
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Abstract: Suppose $ \Omega $ is a bounded, strongly convex domain in $ {{\mathbf{C}}^N}$ with smooth boundary and $ \phi :\Omega \to \Omega $ is an arbitrary holomorphic map. While in general the composition operator $ {C_\phi }$ need not map the Hardy space $ {H^p}(\Omega )$ into itself when $ N > 1$, our main theorem shows that $ {C_\phi }$ does map $ {H^p}(\Omega )$ boundedly into a certain weighted Bergman space on $ \Omega $, where the weight function depends on the dimension N. We also consider properties of $ {C_\phi }$ on $ {H^p}(\Omega )$ when $ \phi (\Omega )$ is contained in an approach region in $ \Omega $.


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DOI: https://doi.org/10.1090/S0002-9939-1995-1254846-9
Article copyright: © Copyright 1995 American Mathematical Society

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