We obtain a local two weight Tb theorem with an energy side condition for higher dimensionalfractional Calderón-Zygmund operators. Our proof follows the proof for the corresponding one-dimensional Tb theorem in [41], but facing a number of new diffculties, most of which arise from the failure of Hytönen's one-dimensional two weight A2 inequality in higher dimensions. We provide a counterexample in two dimensions that shows why the analogue of Hytönen's one-dimensional result does not extend to higher dimensions. Thus, in order to obtain a local Tb theorem in higher dimensions, we use new arguments to control the diffcult 'nearby' form. We also provide refined constants for strong (p,p) inequality of the averaging Hardy operator with respect to a probability measure as well as when two measures that satisfy a special weak type inequality are involved. We obtain these results as corollaries of a more general theorem for operators with this special weak type inequality on a probability space (X,m).
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