On the embedding of $A_1$ into $A_\infty$
HTML articles powered by AMS MathViewer
- by Guillermo Rey PDF
- Proc. Amer. Math. Soc. 144 (2016), 4455-4470 Request permission
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.References
- R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
- C. Domingo-Salazar, M. T. Lacey, and G. Rey, Borderline weak-type estimates for singular integrals and square functions, Bulletin of the London Mathematical Society, Dec. 2015.
- Tuomas Hytönen, Carlos Pérez, and Ezequiel Rela, Sharp reverse Hölder property for $A_\infty$ weights on spaces of homogeneous type, J. Funct. Anal. 263 (2012), no. 12, 3883–3899. MR 2990061, DOI 10.1016/j.jfa.2012.09.013
- Antonios D. Melas, A sharp $L^p$ inequality for dyadic $A_1$ weights in $\Bbb R^n$, Bull. London Math. Soc. 37 (2005), no. 6, 919–926. MR 2186725, DOI 10.1112/S0024609305004765
- F. Nazarov, A. Reznikov, V. Vasyunin, and A. Volberg, A Bellman function counterexample to the $A_1$ conjecture: The blow-up of the weak norm estimates of weighted singular operators, ArXiv e-prints, June 2015.
- F. Nazarov, S. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909–928. MR 1685781, DOI 10.1090/S0894-0347-99-00310-0
- Adam Osȩkowski, Sharp inequalities for dyadic $A_1$ weights, Arch. Math. (Basel) 101 (2013), no. 2, 181–190. MR 3089774, DOI 10.1007/s00013-013-0537-9
- Guillermo Rey and Alexander Reznikov, Extremizers and sharp weak-type estimates for positive dyadic shifts, Adv. Math. 254 (2014), 664–681. MR 3161110, DOI 10.1016/j.aim.2013.12.030
- Leonid Slavin, Alexander Stokolos, and Vasily Vasyunin, Monge-Ampère equations and Bellman functions: the dyadic maximal operator, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 585–588 (English, with English and French summaries). MR 2412802, DOI 10.1016/j.crma.2008.03.003
- V. I. Vasyunin, The exact constant in the inverse Hölder inequality for Muckenhoupt weights, Algebra i Analiz 15 (2003), no. 1, 73–117 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 1, 49–79. MR 1979718, DOI 10.1090/S1061-0022-03-00802-1
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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 4455-4470
- MSC (2010): Primary 42B35; Secondary 46E30
- DOI: https://doi.org/10.1090/proc/13087
- MathSciNet review: 3531194