Geometric-arithmetic averaging of dyadic weights

TL;DR

This paper introduces a new averaging method to construct A_p and reverse Holder weights from dyadic families, enhancing the understanding of weighted inequalities in analysis.

Contribution

It presents a novel averaging process that constructs A_p and RH_p weights from dyadic weights, extending to polydiscs and linking to BMO.

Findings

Averaging process constructs A_p weights from dyadic A_p weights.

Method also constructs RH_p weights from dyadic RH_p weights.

Extension of the process to polydiscs broadens its applicability.

Abstract

The theory of (Muckenhoupt) weights arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing A_p weights from a measurably varying family of dyadic A_p weights. This averaging process is suggested by the relationship between the A_p weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Holder (RH_p) conditions from families of dyadic RH_p weights, and extends to the polydisc as well.