Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A commutator theorem and weighted BMO
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by Steven Bloom
Trans. Amer. Math. Soc. 292 (1985), 103-122
DOI: https://doi.org/10.1090/S0002-9947-1985-0805955-5

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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Bibliographic Information
  • © Copyright 1985 American Mathematical Society
  • 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