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 $n\in N$, 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, $A_p$ weights, Hardy spaces, BMO and VMO function classes, and connect our results with key principles and conjectures across number theory.