  Local Tb theorems with Lp type testing conditions, which are not scale
invariant, have been studied widely in the case of the Lebesgue measure. In the
non-homogeneous world local Tb theorems have only been proved assuming scale
invariant ($L^{\infty}$ or BMO) testing conditions. In this paper, for the
first time, we overcome these obstacles in the non-homogeneous world, and prove
a non-homogeneous local Tb theorem with L2 type testing conditions. This paper
is in the setting of the vertical and conical square functions defined using
general measures and kernels. The proof uses various recent innovations
including a Whitney averaging formula and the insertion of a Calderon-Zygmund
stopping data of a fixed function in to the construction of the twisted
martingale difference operators.
