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

Abstract

We prove continuity and surjectivity of the trace map onto $L_p({\mathbb{R}}^n)$, 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.