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On the embedding of $ A_1$ into $ A_\infty$


Author: Guillermo Rey
Journal: Proc. Amer. Math. Soc. 144 (2016), 4455-4470
MSC (2010): Primary 42B35; Secondary 46E30
DOI: https://doi.org/10.1090/proc/13087
Published electronically: April 25, 2016
MathSciNet review: 3531194
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Abstract: We give a quantitative embedding of the Muckenhoupt class $ A_1$ into $ A_\infty $. In particular, we show how $ \epsilon $ depends on $ [w]_{A_1}$ in the inequality which characterizes $ A_\infty $ weights:

$\displaystyle \frac {w(E)}{w(Q)} \leq \biggl ( \frac {\vert E\vert}{\vert Q\vert} \biggr )^\epsilon , $

where $ Q$ is any dyadic cube and $ E$ is any subset of $ Q$. This embedding yields a sharp reverse-Hölder inequality as an easy corollary.

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Additional Information

Guillermo Rey
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: reyguill@math.msu.edu

DOI: https://doi.org/10.1090/proc/13087
Received by editor(s): April 27, 2015
Received by editor(s) in revised form: January 2, 2016
Published electronically: April 25, 2016
Communicated by: Svitlana Mayboroda
Article copyright: © Copyright 2016 American Mathematical Society

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