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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Carleson measures and multipliers of Dirichlet-type spaces

Authors: Ron Kerman and Eric Sawyer
Journal: Trans. Amer. Math. Soc. 309 (1988), 87-98
MSC: Primary 30D55; Secondary 46E99
MathSciNet review: 957062
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Abstract: A function $ \rho $ from $ [0,\,1]$ onto itself is a Dirichlet weight if it is increasing, $ \rho '' \leqslant 0$ and $ {\lim _{x \to 0 + }}x/\rho (x) = 0$. The corresponding Dirichlet-type space, $ {D_\rho }$, consists of those bounded holomorphic functions on $ U = \{ z \in {\mathbf{C}}:\,\vert z\vert < 1\} $ such that $ \vert f'(z){\vert^2}\rho (1 - \vert z\vert)$ is integrable with respect to Lebesgue measure on $ U$. We characterize in terms of a Carleson-type maximal operator the functions in the set of pointwise multipliers of $ {D_\rho }$, $ M({D_\rho }) = \{ g:\,U \to {\mathbf{C}}:gf \in {D_\rho },\forall f \in {D_\rho }\} $.

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Article copyright: © Copyright 1988 American Mathematical Society

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