The local Tb theorem with rough test functions

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http://hdl.handle.net/10138/319154

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Hytonen , T & Nazarov , F 2020 , ' The local Tb theorem with rough test functions ' , Advances in Mathematics , vol. 372 , 107306 . https://doi.org/10.1016/j.aim.2020.107306

Title: The local Tb theorem with rough test functions
Author: Hytonen, Tuomas; Nazarov, Fedor
Contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2020-10-07
Language: eng
Number of pages: 36
Belongs to series: Advances in Mathematics
ISSN: 0001-8708
URI: http://hdl.handle.net/10138/319154
Abstract: We prove a local Tbtheorem under close to minimal (up to certain 'buffering') integrability assumptions, conjectured by S. Hofmann (El Escorial, 2008): Every cube is assumed to support two non-degenerate functions b(Q)(1) is an element of L-p and b(Q)(2) is an element of L-q such that 1(2Q)Tb(Q)(1) is an element of L-q' and 1(2Q)T*b(Q)(2) is an element of L-p', with appropriate uniformity and scaling of the norms. This is sufficient for the L-2-boundedness of the Calderon-Zygmund operator T, for any p, q is an element of(1, infinity), a result previously unknown for simultaneously small values of pand q. We obtain this as a corollary of a local Tbtheorem for the maximal truncations T-# and (T*)(#): for the L-2-boundedness of T, it suffices that 1(Q)T#b(Q)(1) and 1Q(T*)# b(Q)(2) be uniformly in L-0. The proof builds on the technique of suppressed operators from the quantitative Vitushkin conjecture due to Nazarov-Treil-Volberg. (C) 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Subject: Singular integral
Calderon-Zygmund operator
Boundedness criterion
Accretive system
Stopping time
111 Mathematics
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