A dyadic analysis approach to the problem of continuity of weighted estimates with respect to the $A_p$ characteristic

TL;DR

This paper introduces a dyadic analysis approach and Bellman function technique to establish the continuity of weighted estimates with respect to the $A_p$ characteristic, bridging the gap from dyadic to continuous settings.

Contribution

It provides a novel proof method for weighted estimate continuity, including a new transition from martingale transforms to Hilbert transforms.

Findings

Proved continuity of weighted estimates in the dyadic case.

Extended the results to the continuous setting using Bellman functions.

Established a new connection between martingale transforms and Hilbert transforms.

Abstract

This paper presents a new proof of the results regarding the continuity of weighted estimates with respect to the characteristic of the weight. Here we first prove the result in the dyadic case which is "easier" and then by the use of the Bellman function technique we pass to the continuous setting which is harder in general. To be more precise, as far as we know, this passage from the Martingale transform to the Hilbert transform described in this note is new.