Geodesic
The proof of the nonhomogeneous $T1$ theorem via averaging of dyadic shifts
Algebra i analiz, Tome 27 (2015) no. 3, pp. 75-94.

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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.
Keywords: operators, dyadic shift, $T1$ theorem, nondoubling measure. 
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A. Volberg. The proof of the nonhomogeneous $T1$ theorem via averaging of dyadic shifts. Algebra i analiz, Tome 27 (2015) no. 3, pp. 75-94. http://geodesic.mathdoc.fr/item/AA_2015_27_3_a4/

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