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Some characterizations of the preimage of $$A_{\infty }$$ for the Hardy-Littlewood maximal operator and consequences. (English) Zbl 1444.42022
Let $$M$$ denote the non-centered Hardy-Littlewood maximal operator and $$w$$ a weight, which means that $$w$$ is a non-negative locally integrable function on $$\mathbb{R}^n$$. The classic $$A_p$$ classes for weights precisely quantify when $$M$$ maps $$L^p(w)$$ into $$L^{p,\infty}(w)$$ boundedly for $$p \geq 1$$ and have the property that $$A_p \subset A_q$$ if $$p < q$$. In this paper the author considers a weight in $$A_{\infty}$$ of a particular form, namely $$Mw \in A_{\infty}$$ and shows that this implies $$Mw \in A_1$$. Due to nesting of the $$A_p$$ classes the reverse implication is immediate so the first result of the paper is that if $$w$$ is any weight then $$Mw\in A_{\infty} \iff Mw \in A_1$$. This leads immediately to an extension of a result by Neugebauer that appeared first in [D. Cruz-Uribe, Rocky Mt. J. Math. 26, No. 2, 545–583 (1996; Zbl 0861.42007)] which gives the following characterization of weights $$w$$ such that $$Mw\in A_{\infty}$$, namely that happens if and only if there exist $$s>1$$ and $$C_0 > 0$$ such that $$(Mw^s)^{\frac{1}{s}}(x) \leq C_0 Mw(x)$$. A characterization of weights that fulfill that property was given as an open problem in [D. Cruz-Uribe and C. Pérez, J. Funct. Anal. 174, No. 1, 1–17 (2000; Zbl 0976.46018)]. This leads to further applications such as a result analogous to the characterization of $$A_1$$ weights by R. Coifman and R. Rochberg [Proc. Amer. Math. Soc. 79, 249–254 (1980; Zbl 0432.42016)] as well as improvements of some known inequalities for singular integral operators.
##### MSC:
 42B25 Maximal functions, Littlewood-Paley theory
##### Keywords:
maximal operators; $$A_\infty$$ classes
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