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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A factorization theorem for the derivative
of a function in $H^{p}$


Author: William S. Cohn
Journal: Proc. Amer. Math. Soc. 127 (1999), 509-517
MSC (1991): Primary 32A35
MathSciNet review: 1605936
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Abstract: We show that a function $G$ is the derivative of a function $f$ in the Hardy space $H^{p}$ of the unit disk $D$ for $0<p<\infty $ if and only if $G=F\Phi ^{\prime }$ where $F\in H^{p}$ and $\Phi \in BMOA$. Here, $F$ can be chosen to be non-vanishing, $||\Phi ||_{BMOA} \le 1$, and $||F||_{H^{p}} \le C||f||_{H^{p}}$. As an application, we characterize positive measures $\mu $ on the unit disk such that the operator $L_{\mu }g(\zeta )=\int _{D} g(z) {\frac{d\mu (z) }{(1-\zeta \bar {z})^{2}}}$ is bounded from the tent space $T^{p}_{\infty }$ to $H^{p}$, where ${\frac{1}{2}} <p<\infty $.


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Additional Information

William S. Cohn
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48070

DOI: https://doi.org/10.1090/S0002-9939-99-04870-4
Received by editor(s): May 28, 1997
Communicated by: Theodore W. Gamelin
Article copyright: © Copyright 1999 American Mathematical Society