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Proceedings of the American Mathematical Society

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

 

 

Two nonequivalent conditions for weight functions


Authors: Charles Fefferman and Benjamin Muckenhoupt
Journal: Proc. Amer. Math. Soc. 45 (1974), 99-104
MSC: Primary 26A33; Secondary 42A40, 44A25
MathSciNet review: 0360952
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Abstract: A nonnegative function on the real line satisfies the condition $ {{\mathbf{A}}_\infty }$ if, given $ \varepsilon > 0$, there exists a $ \delta > 0$ such that if $ I$ is an interval, $ E \subset I$, and $ \vert E\vert < \delta \vert I\vert$, then $ \int_E {W \leq \varepsilon \int_I W } $. A nonnegative function on the real line satisfies the condition $ {\mathbf{A}}$ if for every interval $ I,\int_{2I} {W \leq C} \int_I W $, where $ 2I$ is the interval with the same center as $ I$ and twice as long, and $ C$ is independent of $ I$. An example is given of a function that satisfies $ {\mathbf{A}}$ but not $ {{\mathbf{A}}_\infty }$.


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DOI: https://doi.org/10.1090/S0002-9939-1974-0360952-X
Article copyright: © Copyright 1974 American Mathematical Society