Inequalities for maximal operators associated with a family of general sets (Q6595830)

Let \(X\) be a topological space equipped with a nonnegative Borel measure \(\mu\). For each \(\in X\) and \(r > 0\), let \(E_r(x)\) be a nonempty, bounded open subset of \(X\) containing \(x\) and let \(\mathbb E = \{E_r(x) : 0 < r < \infty, x\in X\}\). Assume that the family \(\mathbb E\) and the measure \(\mu\) satisfy the following conditions: (A) \(\cup_{r>0} E_r(x) = X\); (B) \(\cap_{r>0} E_r(x) = \{x\}\); (C) \(E_r(x)\subset E_s(x)\) if \(0 < r \le s\); (D) for all \(x \in X\) and \(r > 0\), we have \(0 <\mu(E_r(x))< \infty\), and \(\mu\) satisfies a doubling condition, i.e., there exists a constant \(C_\mu > 1\) such that \(\mu(E_{2r}(x))\le C_\mu \mu(E_r(x))\) for all \(x\in X\) and \(E_r(x)\in \mathbb E\); (E) for each open set \(U\) and \(r > 0\), the function \(x \to \mu(E_r(x)\cap U)\) is continuous; (F) there exists a constant \(\theta>1\) such that for all \(E_r(x)\in\mathbb E\), \(y\in E_r(x)\) implies \(E_r(x)\subset E_{\theta r}(y)\) and \(E_r(y)\subset E_{\theta r}(x)\); (G) the mapping \(r \to\mu(E_r(x))\) is continuous for each \(x\in X\). \N\NThe authors 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, they also provide a quantitative version of the open property for \(A_{p,\mathbb E}\) weights. They furthermore 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. \N\NThe family \(\mathbb E\) is introduced in [\textit{Y. Ding} et al., J. Fourier Anal. Appl. 20, No. 3, 608--667 (2014; Zbl 1305.42019)]. The authors' results extend the known classical cases, including the usual Euclidean case and the case of a space of homogeneous type.