mathcal {A}_{p, {mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that ({mathbb {E}}:={E_r(x)}_{rin {mathcal {I}}, xin X}) is a family of open subsets of a topological space (X) endowed with a nonnegative Borel measure (mu ) satisfying certain basic conditions. We establish an (mathcal {A}_{{mathbb {E}}, p}) weights theory with respect to ({mathbb {E}}) and get the characterization of weighted weak type (1,1) and strong type ((p,p)) , (1 , for the maximal operator ({mathcal {M}}_{{mathbb {E}}}) associated with ({mathbb {E}}) . As applications, we introduce the weighted atomic Hardy space (H^1_{{mathbb {E}}, w}) and its dual (BMO_{{mathbb {E}},w}) , and give a maximal function characterization of (H^1_{{mathbb {E}},w}) . Our results generalize several well-known results.