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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 $b^1_Q\inL^p$ and $b^2_Q\in L^q$ such that $1_{2Q}Tb^1_Q\in L^{q'}$ and$1_{2Q}T^*b^2_Q\in L^{p'}$, with appropriate uniformity and scaling of thenorms. This is sufficient for the $L^2$-boundedness of the Calderon-Zygmundoperator $T$, for any $p,q\in(1,\infty)$, a result previously unknown forsimultaneously small values of $p$ and $q$. We obtain this as a corollary of alocal $Tb$ theorem for the maximal truncations $T_{\#}$ and $(T^*)_{\#}$: forthe $L^2$-boundedness of $T$, it suffices that $1_Q T_{\#}b^1_Q$ and $1_Q(T^*)_{\#}b^2_Q$ be uniformly in $L^0$. The proof builds on the technique ofsuppressed operators from the quantitative Vitushkin conjecture due toNazarov-Treil-Volberg.

The local Tb theorem with rough test functions