Inequalities for Maximal Operators Associated with a Family of General Sets

Let \({\mathbb {E}}=\{E_r(x):r>0,x\in X\}\) be a family of open subsets of a topological space X equipped with a nonnegative Borel measure \(\mu \) satisfying some basic properties. We establish sharp quantitative weighted norm inequalities for the Hardy–Littlewood maximal operator \(M_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) in terms of mixed \(A_p\)–\(A_\infty \) constants. The main ingredient to prove this result is a sharp form of a weak reverse Hölder inequality for the \(A_{\infty ,{\mathbb {E}}}\) weights. As an application of this inequality, we also provide a quantitative version of the open property for \(A_{p,{\mathbb {E}}}\) weights. Next, we prove a covering lemma in this setting and using this lemma establish the endpoint Fefferman–Stein weighted inequality for the maximal operator \(M_{{\mathbb {E}}}\). Moreover, vector-valued extensions for maximal inequalities are also obtained in this context.