The proof of non-homogeneous T1 theorem via averaging of dyadic shifts

TL;DR

This paper presents a new proof of the non-homogeneous T1 theorem by constructing a random dyadic lattice, estimating matrix coefficients, and employing an averaging trick to decompose operators into dyadic shifts, improving upon previous methods.

Contribution

It introduces a novel proof approach for the non-homogeneous T1 theorem using random dyadic lattices and an averaging technique to simplify the decomposition of Calderón-Zygmund operators.

Findings

Successful construction of a random dyadic lattice.

Effective estimation of matrix coefficients with good Haar support.

Elimination of error terms through averaging over dyadic lattices.

Abstract

We give again a proof of non-homogeneous T1 theorem. Our proof consists of three main parts: a construction of a random dyadic lattice; an estimate of matrix coefficients of a Calder\'on--Zygmund operator with respect to random Haar basis if a smaller Haar support is good; a clever averaging trick from Hyt\"onen's papers which uses the averaging over dyadic lattices to decompose operator into dyadic shifts eliminating the error term that was present in the previous proofs by Nazarov--Treil--Volberg.