  We prove a version of the local Tb Theorem assuming that the accretive
functions b_Q and T b_Q are locally L ^{p} integrable, for any 1< p < \infty .
This improves a recent result of Hytonen-Nazarov. The proof strategy relies
upon the their strategy, with additional techniques concerning twisted
martingale differences and the use of random dyadic grids in the local Tb
setting.
