The proof of the nonhomogeneous 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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Once again, a proof of the nonhomogeneous 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