On the embedding of 𝐴₁ into 𝐴_{∞}

Rey, Guillermo

doi:10.1090/proc/13087

On the embedding of 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 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

is any dyadic cube and

is any subset of

. This embedding yields a sharp reverse-Hölder inequality as an easy corollary.

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