The local Tb theorem with rough test functions

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/).