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

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.