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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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