The local Tb theorem with rough test functions

Abstract

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 $b_Q^1\in L^p$ and $b_Q^2\in L^q$ such that $1_{2Q}T{b}_Q^1\in L^{q^{\prime }}$ and $1_{2Q}{T}^{\ast }{b}_Q^2\in L^{p^{\prime }}$, with appropriate uniformity and scaling of the norms. This is sufficient for the $L^2$-boundedness of the Calderon-Zygmund operator $T$, for any $p,q\in (1,\infty )$, 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 $(T^{\ast }{)}_{\#}$: for the $L^2$-boundedness of $T$, it suffices that $1_Q{T}_{\#}{b}_Q^1$ and $1_Q(T^{\ast }{)}_{\#}{b}_Q^2$ be uniformly in $L^0$. The proof builds on the technique of suppressed operators from the quantitative Vitushkin conjecture due to Nazarov-Treil-Volberg.

arXiv commentV2: 24 pages, incorporates referee comments, accepted for publication in Adv Math