Bounded variation approximation of L_p dyadic martingales and solutions to elliptic equations

We prove continuity and surjectivity of the trace map onto $L_p$, from a space of functions of locally bounded variation, defined by the Carleson functional. The extension map is constructed through a stopping time argument. This extends earlier work by Varopoulos in the BMO case, related to the Corona theorem. We also prove $L_p$ Carleson approximability results for solutions to elliptic non-smooth divergence form equations, which generalize results in the case $p=\mathrm{\infty }$ by Hofmann, Kenig, Mayboroda and Pipher.