A two weight local $Tb$ theorem for $n$-dimensional fractional singular integrals

Abstract

We obtain a local two weight $Tb$ theorem with an energy side condition for higher dimensional fractional Calder\'{o}n-Zygmund operators. The proof follows the general outline of the proof for the corresponding one-dimensional $Tb$ theorem in [SaShUr12], but encountering a number of new challenges, including several arising from the failure in higher dimensions of T. Hyt\"{o}nen's one-dimensional two weight $A_{2}$ inequality [Hyt].