zbMATH — the first resource for mathematics

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.
42B25 Maximal functions, Littlewood-Paley theory
Full Text: DOI Euclid arXiv