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Transactions of the American Mathematical Society

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

 
 

 

A commutator theorem and weighted BMO


Author: Steven Bloom
Journal: Trans. Amer. Math. Soc. 292 (1985), 103-122
MSC: Primary 42A50; Secondary 42B20, 46E40, 47B47
DOI: https://doi.org/10.1090/S0002-9947-1985-0805955-5
MathSciNet review: 805955
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Abstract: The main result of this paper is a commutator theorem: If $ \mu $ and $ \lambda $ are $ {A_p}$ weights, then the commutator $ H$, $ {M_b}$ is a bounded operator from $ {L^p}(\mu )$ into $ {L^p}(\lambda )$ if and only if $ b \in {\operatorname{BMO} _{{{(\mu {\lambda ^{ - 1}})}^{1/p}}}}$. The proof relies heavily on a weighted sharp function theorem. Along the way, several other applications of this theorem are derived, including a doubly-weighted $ {L^p}$ estimate for BMO. Finally, the commutator theorem is used to obtain vector-valued weighted norm inequalities for the Hilbert transform.


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DOI: https://doi.org/10.1090/S0002-9947-1985-0805955-5
Article copyright: © Copyright 1985 American Mathematical Society

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