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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

A rearranged good $ \lambda$ inequality


Authors: Richard J. Bagby and Douglas S. Kurtz
Journal: Trans. Amer. Math. Soc. 293 (1986), 71-81
MSC: Primary 42B25
MathSciNet review: 814913
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Abstract: Let $ Tf$ be a maximal Calderón-Zygmund singular integral, $ Mf$ the Hardy-Littlewood maximal function, and $ w$ an $ {A_\infty }$ weight. We replace the ``good $ \lambda$'' inequality

$\displaystyle w\left( {\{ x:\,Tf(x) > 2\lambda \,{\text{and}}\,Mf(x) \leq \vare... ...a \} } \right) \leq C(\varepsilon )w\left( {\{ x:\,Tf(x) > \lambda \} } \right)$

by the rearrangement inequality

$\displaystyle (Tf)_w^ \ast (t) \leq C(Mf)_w^ \ast (t/2) + (Tf)_w^ \ast (2t)$

and show that it gives better estimates for $ Tf$. In particular, we obtain best possible weighted $ {L^p}$ bounds, previously unknown exponential integrability estimates, and simplified derivations of known unweighted estimates for $ {(Tf)^ \ast }$.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1986-0814913-7
PII: S 0002-9947(1986)0814913-7
Keywords: Singular integral operator, maximal function, weight, rearrangement
Article copyright: © Copyright 1986 American Mathematical Society