The local Tb theorem with rough test functions

We prove a local Tb theorem 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 bQ1∈Lp and bQ2∈Lq such that 12QTbQ1∈Lq′ and 12QT⁎bQ2∈Lp′, with appropriate uniformity and scaling of the norms. This is sufficient for the L2-boundedness of the Calderón–Zygmund operator T, for any p,q∈(1,∞), a result previously unknown for simultaneously small values of p and q. We obtain this as a corollary of a local Tb theorem for the maximal truncations T# and (T⁎)#: for the L2-boundedness of T, it suffices that 1QT#bQ1 and 1Q(T⁎)#bQ2 be uniformly in L0. The proof builds on the technique of suppressed operators from the quantitative Vitushkin conjecture due to Nazarov–Treil–Volberg.