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 Algebra i Analiz, 2014, Volume 26, Issue 1, Pages 40–67 (Mi aa1368)

Research Papers

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

O. Beznosovaa, A. Reznikovbc

a Department of Mathematics, Baylor University, One Bear Place \#97328, Waco, TX 76798-7328, USA
b Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
c St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka, 27, 191023, St. Petersburg, Russia

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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English version:
St. Petersburg Mathematical Journal, 2015, 26:1, 27–47

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Citation: O. Beznosova, A. Reznikov, “Sharp estimates involving $A_\infty$ and $L\log L$ constants, and their applications to PDE”, Algebra i Analiz, 26:1 (2014), 40–67; St. Petersburg Math. J., 26:1 (2015), 27–47

Citation in format AMSBIB
\Bibitem{BezRez14} \by O.~Beznosova, A.~Reznikov \paper Sharp estimates involving $A_\infty$ and $L\log L$ constants, and their applications to PDE \jour Algebra i Analiz \yr 2014 \vol 26 \issue 1 \pages 40--67 \mathnet{http://mi.mathnet.ru/aa1368} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3234812} \elib{http://elibrary.ru/item.asp?id=21826344} \transl \jour St. Petersburg Math. J. \yr 2015 \vol 26 \issue 1 \pages 27--47 \crossref{https://doi.org/10.1090/S1061-0022-2014-01329-5} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000357043200002} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84913554283} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. P. Ivanisvili, N. N. Osipov, D. M. Stolyarov, V. I. Vasyunin, P. B. Zatitskiy, “Sharp estimates of integral functionals on classes of functions with small mean oscillation”, C. R. Math. Acad. Sci. Paris, 353:12 (2015), 1081–1085
2. J. Li, J. Pipher, L. A. Ward, “Dyadic structure theorems for multiparameter function spaces”, Rev. Mat. Iberoam., 31:3 (2015), 767–797
3. Hagelstein P., Parissis I., “Weighted Solyanik Estimates for the Hardy–Littlewood Maximal Operator and Embedding of into”, J. Geom. Anal., 26:2 (2016), 924–946
4. Duoandikoetxea J., Martin-Reyes F.J., Ombrosi Sh., “On the $A_{\infty }$ conditions for general bases”, Math. Z., 282:3-4 (2016), 955–972
5. D'Onofrio L., Popoli A., Schiattarella R., “Duality for $A_\infty$ weights on the real line”, Rend. Lincei-Mat. Appl., 27:3 (2016), 287–308
6. Stolyarov D.M., Zatitskiy P.B., “Theory of locally concave functions and its applications to sharp estimates of integral functionals”, Adv. Math., 291 (2016), 228–273
7. P. Ivanisvili, D. M. Stolyarov, I V. Vasyunin, P. B. Zatitskiy, Bellman function for extremal problems in BMO II: evolution, Mem. Am. Math. Soc., 255, no. 1220, 2018, v+133 pp.
8. A. Popoli, “Sharp integrability exponents and constants for Muckenhoupt and Gehring weights as solution to a unique equation”, Ann. Acad. Sci. Fenn. Math., 43:2 (2018), 785–805
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