A two weight local $Tb$ theorem for the Hilbert transform

Abstract

We obtain a two weight local $Tb$ theorem for any elliptic and gradient elliptic fractional singular integral operator $T^{\alpha }$ on the real line $\mathbb{R}$, and any pair of locally finite positive Borel measures $(\sigma ,\omega )$ on $\mathbb{R}$. The Hilbert transform is included in the case $\alpha =0$, and is bounded from $L^2(\sigma )$ to $L^2(\omega )$ if and only if the Muckenhoupt and energy conditions hold, as well as $b_Q$ and $b_Q^{\ast }$ testing conditions over intervals $Q$, where the families $\{b_Q\}$ and $\{b_Q^{\ast }\}$ are $p$-weakly accretive for some $p>2$. A number of new ideas are needed to accommodate weak goodness, including a new method for handling the stubborn nearby form, and an additional corona construction to deal with the stopping form. In a sense, this theorem improves the $T1$ theorem obtained by the authors and M. Lacey.