Numerical heating is a serous problem in particle-in-cell (PIC) modeling of cross field diffusion. Recent work by the author has shown that for, electrostatic problems, the boundary integral treecode (BIT) has far less numerical heating than traditional PIC and that numerical heating can be nearly eliminated if regularization is added to the BIT field solver. In this work we consider the application of BIT as a sub-cell method within each PIC cell, where the boundary conditions on BIT come from the fields computed on the PIC mesh. The goal is to minimize numerical heating in PIC while allowing for mesh spacing in PIC to be much greater than a Debye length. In this work, we demonstrate a substantial reduction in numerical heating when the mesh cell is much larger than Debye length for a verily of test cases. Further, we have applied the BIT corrected PIC to the two stream instability and virtual cathode problem. In both cases we have found that the sub-cell method gives results consistent with BIT, while exhibiting vastly different temporal response times than predicted by PIC. Further, in such problems as the virtual cathode, the BIT corrected PIC is able to handle arbitrarily high densities within a mesh cell, without needing to increase the resolution of the original fixed PIC mesh. Our overall objective is to inherit the parallel capability of legacy PIC codes while providing high accuracy.