  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 $p=\infty$ by Hofmann, Kenig, Mayboroda and Pipher.
