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

Volberg, A.

doi:10.1090/spmj/1395

The proof of the nonhomogeneous $T1$ theorem via averaging of dyadic shifts
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by A. Volberg

St. Petersburg Math. J. 27 (2016), 399-413

DOI: https://doi.org/10.1090/spmj/1395

Published electronically: March 30, 2016

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Abstract:

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 two earlier papers by Nazarov, Treil, and Volberg, dated back to 2003 and 1997; an estimate for matrix coefficients of a Carderón–Zygmund operator with respect to random Haar basis if a smaller Haar support is good like in the paper of 1997 mentioned above; a clever averaging trick used by Hytönen, Peres, Treil, and Volberg in two papers of 2012 and 2014, 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 employed in the papers of 2003 and 1997 mentioned above. Hence, a decomposition is established of nonhomogeneous Calderón–Zygmund operators into dyadic Haar shifts.

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