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


Author: A. Volberg
Original publication: Algebra i Analiz, tom 27 (2015), nomer 3.
Journal: St. Petersburg Math. J. 27 (2016), 399-413
MSC (2010): Primary 42B20
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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Additional Information

A. Volberg
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan
Email: sashavolberg@yahoo.com, volberg@math.msu.edu

DOI: https://doi.org/10.1090/spmj/1395
Keywords: Operators, dyadic shift, $T1$ theorem, nondoubling measure
Received by editor(s): November 20, 2014
Published electronically: March 30, 2016
Dedicated: To Nina Ural’tseva who taught me Mathematical Physics and how to zoom in on its beauty
Article copyright: © Copyright 2016 American Mathematical Society