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

Abstract

We prove continuity and surjectivity of the trace map onto $L_p$, 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.

This is an extended version of an earlier arXiv preprint (not to be published) "Approximate and exact extensions of Lebesgue boundary functions", which also includes Lp epsilon-approximability results for divergence form elliptic equations. To this third version we have also added a section on applications to estimates of harmonic measure