Trace and extension theorems for functions of bounded variation

TL;DR

This paper characterizes the trace and extension properties of functions of bounded variation on domains with regular boundaries in doubling metric measure spaces, establishing conditions under which boundary functions can be extended into the domain.

Contribution

It proves that every $L^1$ boundary function is a trace of a BV function inside the domain and constructs a bounded linear extension from Besov spaces to BV functions.

Findings

Trace class of BV functions equals $L^1(oundary\,\Omega)$ under Poincaré inequality.

Every $L^1$ boundary function can be realized as a trace of a BV function.

Constructs a bounded linear extension from Besov spaces to BV functions.

Abstract

In this paper we show that every $L^1$-integrable function on $\partial\Omega$ can be obtained as the trace of a function of bounded variation in $\Omega$ whenever $\Omega$ is a domain with regular boundary $\partial\Omega$ in a doubling metric measure space. In particular, the trace class of $BV(\Omega)$ is $L^1(\partial\Omega)$ provided that $\Omega$ supports a 1-Poincar\'e inequality. We also construct a bounded linear extension from a Besov class of functions on $\partial\Omega$ to $BV(\Omega)$.