The proof of the nonhomogeneous $T1$ theorem via averaging of dyadic shifts

Once again, a proof of the nonhomogeneous $T1$ theorem is given. This proof consists of three main parts: a construction of a random “dyadic” lattice as in [7,8]; an estimate of matrix coefficients of a Calderón–Zygmund operator with respect to random Haar basis if a smaller Haar support is good like in [8]; a clever averaging trick from [2,5], which involves the averaging over dyadic lattices to decompose an operator into dyadic shifts eliminating the error term that was present in the random geometric construction of [7,8]. Hence, a decomposition is established of nonhomogeneous Calderón–Zygmund operators into dyadic Haar shifts.