Infinite intersections of doubling measures, weights, and function classes

TL;DR

This paper investigates whether intersecting all prime p-adic versions of measures and weights recovers the original objects, providing new insights and generalizations that answer longstanding questions in harmonic analysis.

Contribution

It generalizes a number-theoretic construction to show that the intersection of all prime p-adic versions does recover the full object, resolving open questions.

Findings

Intersection of all prime p-adic versions recovers the original object

Generalization of a number-theoretic construction to harmonic analysis

Answers longstanding questions about measure and weight intersections

Abstract

A series of longstanding questions in harmonic analysis ask if the intersection of all prime ``$p$-adic versions" of an object, such as a doubling measure, or a Muckenhoupt or reverse H\"older weight, recovers the full object. Investigation into these questions was reinvigorated in 2019 by work of Boylan-Mills-Ward, culminating in showing that this recovery fails for a finite intersection in work of Anderson-Bellah-Markman-Pollard-Zeitlin. Via generalizing a new number-theoretic construction therein, we answer these questions.