Arbitrary finite intersections of doubling measures and applications

Abstract

We make major progress on a folkloric conjecture in analysis by constructing a measure on the real line which is doubling on all n-adic intervals for any finite list of n∈N, yet not doubling overall. In particular, we extend previous results in the area, including those of Boylan-Mills-Ward and Anderson-Hu, by using a wide array of substantially new ideas. In addition, we provide several nontrivial applications to reverse Hölder weights, Ap weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.