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

TL;DR

This paper establishes new results on the approximation and boundary behavior of solutions to elliptic equations using bounded variation functions and Carleson measures, extending classical theorems to broader contexts.

Contribution

It introduces a novel approach to trace and extension maps for functions of bounded variation related to elliptic PDEs, generalizing previous BMO and $p= fty$ results.

Findings

Proves continuity and surjectivity of the trace map onto $L_p$.

Establishes $L_p$ Carleson approximability for elliptic solutions.

Extends classical results to non-smooth divergence form 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.