2012/06/05 by Hytönen, Tuomas, Nazarov, Fedor
#42B20 #42B25 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1206.0907
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 b1Q∈ Lp and b2Q∈ Lq such that 12QTb1Q∈ Lq' and 12QT^*b2Q∈ Lp', with appropriate uniformity and scaling of the norms. This is sufficient for the L2-boundedness of the Calderon-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 1Q T#b1Q and 1Q (T^*)#b2Q be uniformly in L0. The proof builds on the technique of suppressed operators from the quantitative Vitushkin conjecture due to Nazarov-Treil-Volberg.