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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: \[ \frac {w(E)}{w(Q)} \leq \biggl ( \frac {|E|}{|Q|} \biggr )^\epsilon , \] 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
MR Author ID: 1050866
ORCID: 0000-0001-8112-7262
Email: reyguill@math.msu.edu

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