Tauberian conditions, Muckenhoupt weights, and differentiation properties of weighted bases

Let B be a homothecy invariant collection of convex sets in ℝⁿ. Given a measure μ, the associated weighted geometric maximal operatorM_B,μ is defined by (Formula presented) It is shown that, provided μ satisfies an appropriate doubling condition with respect to B and ν is an arbitrary locally finite measure, the maximal operator M_B,μ is bounded on Lp(ν) for sufficiently large p if and only if it satisfies a Tauberian condition of the form (Formula presented) As a consequence of this result we provide an alternative characterization of the class of Muckenhoupt weights A_∞,B for homothecy invariant Muckenhoupt bases B consisting of convex sets. Moreover, it is immediately seen that the strong maximal function M_R,μ, defined with respect to a product-doubling measure μ, is bounded on Lp(ν) for some p > 1 if and only if (Formula presented) holds for all ν-measurable sets E in ℝⁿ. In addition, we discuss applications in differentiation theory, in particular proving that a μ-weighted homothecy invariant basis of convex sets satisfying appropriate doubling and Tauberian conditions must differentiate L∞(ν).