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Sharp estimates involving $ A_\infty$ and $ L\log L$ constants, and their applications to PDE


Authors: O. Beznosova and A. Reznikov
Original publication: Algebra i Analiz, tom 26 (2014), nomer 1.
Journal: St. Petersburg Math. J. 26 (2015), 27-47
MSC (2010): Primary 42B20
DOI: https://doi.org/10.1090/S1061-0022-2014-01329-5
Published electronically: November 21, 2014
MathSciNet review: 3234812
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Abstract | References | Similar Articles | Additional Information

Abstract: It is a well-known fact that the union $ \bigcup _{p>1} RH_p$ of the Reverse Hölder classes coincides with the union $ \bigcup _{p>1} A_p = A_\infty $ of the Muckenhoupt classes, but the $ A_\infty $ constant of the weight $ w$, which is a limit of its $ A_p$ constants, is not a natural characterization for the weight in Reverse Hölder classes. In the paper, the $ RH_1$ condition is introduced as a limiting case of the $ RH_p$ inequalities as $ p$ tends to $ 1$, and a sharp bound is found on the $ RH_1$ constant of the weight $ w$ in terms of its $ A_\infty $ constant. Also, the sharp version of the Gehring theorem is proved for the case of $ p=1$, completing the answer to the famous question of Bojarski in dimension one.

The results are illustrated by two straightforward applications to the Dirichlet problem for elliptic PDE's.

Despite the fact that the Bellman technique, which is employed to prove the main theorems, is not new, the authors believe that their results are useful and prove them in full detail.


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Additional Information

O. Beznosova
Affiliation: Department of Mathematics, Baylor University, One Bear Place #97328, Waco, Texas 76798-7328
Email: Oleksandra_Beznosova@baylor.edu

A. Reznikov
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824; St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: rezniko2@msu.edu

DOI: https://doi.org/10.1090/S1061-0022-2014-01329-5
Keywords: Muckenhoupt classes, reverse H\"older inequalities, Gehring lemma
Received by editor(s): November 10, 2012
Published electronically: November 21, 2014
Article copyright: © Copyright 2014 American Mathematical Society

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