A_p,E Weights, Maximal Operators, and Hardy Spaces Associated with a Family of General Sets

Suppose that E:= {E_r(x)}_{r∈I, x∈X} is a family of open subsets of a topological space X endowed with a nonnegative Borel measure μ satisfying certain basic conditions. We establish an A_E,p weights theory with respect to E and get the characterization of weighted weak type (1,1) and strong type (p,p), 1 < p ≤ ∞, for the maximal operator M_E associated with E. As applications, we introduce the weighted atomic Hardy space H¹_E,w and its dual BMO_E,w, and give a maximal function characterization of H¹_E,w. Our results generalize several well-known results.