Necessary/sufficient conditions in weighted theory

TL;DR

This paper provides a comprehensive overview of the relationships among key conditions in weighted harmonic analysis, establishing implications and counterexamples to clarify their interplay.

Contribution

It offers a nearly complete characterization of implications among fundamental weighted conditions, including new results on A_infinity weights and the C_p condition.

Findings

A_infinity weights imply the p-Pivotal condition for A_p weights.

Constructed doubling weights satisfy C_p but are not in A_infinity.

Provided a short proof of the NTV-conjecture for A_infinity weights.

Abstract

We provide an essentially complete dictionary of all implications among the basic and fundamental conditions in weighted theory such as the doubling, one weight A_p(w), A_\infty and C_p conditions as well as the two weight A_p and the "buffer" Energy and Pivotal conditions. The most notable implication is that in the case of A_\infty weights the two weight A_p condition implies the p-Pivotal condition hence giving an elegant and short proof of the known NTV-conjecture with p=2 for A_\infty weights in terms of existing T1 theory. We also provide a quite technical construction inspired by [6] proving that we can have doubling weights satisfying the C_p condition which are not in A_\infty.