A Structure Theorem on Intersections of General Doubling Measures and Its Applications

TL;DR

This paper constructs measures on the real line that are doubling in p-adic and q-adic contexts for distinct primes but not globally doubling, revealing new insights into harmonic analysis and weight classes.

Contribution

It introduces explicit measures that are p-adic and q-adic doubling but not globally doubling, connecting dyadic analysis with number theory.

Findings

Constructed measures that are p-adic and q-adic doubling but not doubling.

Applied these measures to reverse H"older and Muckenhoupt A_p classes.

Revealed interplay between geometric and number theoretic properties.

Abstract

We unite two themes in dyadic analysis and number theory by studying an analogue of the failure of the Hasse principle in harmonic analysis. Explicitly, we construct an explicit family of measures on the real line that are $p$-adic and $q$-adic doubling for any distinct primes $p$ and $q$, yet not doubling, and we apply these results to show analogous statements about the reverse H\"older and Muckenhoupt $A_p$ classes of weights. The proofs involve a delicate interplay among several geometric and number theoretic properties.