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 $b^1_Q\in L^p$and$b^2_Q\in L^q$suchthat$1_{2Q}Tb^1_Q\in L^{q’}$and$1_{2Q}T^b^2_Q\in L^{p’}$, with appropriate uniformity and scaling of the norms. This is sufficient for the $L^2$-boundedness of the Calderon-Zygmund operator $T$, for any $p,q\in (1,\mathrm{\infty })$, 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 $T_{\mathrm{\#}}$ and $(T^{\ast }{)}_{\mathrm{\#}}$: for the $L^2$-boundedness of $T$, it suffices that $1_Q{T}_{\mathrm{\#}}{b}_Q^1$ and $1_Q (T^)_{#}b^2_Q$beuniformlyin$L^0$. The proof builds on the technique of suppressed operators from the quantitative Vitushkin conjecture due to Nazarov-Treil-Volberg.