Lusin approximation for functions of bounded variation

TL;DR

This paper establishes a Lusin approximation for BV functions in metric measure spaces with Poincaré inequality, ensuring close approximation by functions with semicontinuous properties outside small capacity sets.

Contribution

It extends Lusin approximation results to BV functions in general metric measure spaces supporting a Poincaré inequality, with additional properties like semicontinuity and maximal function continuity.

Findings

Approximation functions differ from original on sets of small capacity.

The approximations are close in BV norm within epsilon.

In Euclidean spaces, the maximal function of the approximation is continuous.

Abstract

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\Omega\subset X$, where $X=(X,d,\mu)$ is a given complete doubling metric measure space supporting a $1$-Poincar\'e inequality, then for every $\varepsilon>0$, there exist a function $f_\varepsilon$ on $\Omega$ and an open set $U_\varepsilon\subset\Omega$ such that the following properties hold true: \begin{enumerate} \item ${\rm Cap}_1(U_\varepsilon)<\varepsilon$; \item $\|f-f_\varepsilon\|_{\BV(\Omega)}< \varepsilon$; \item $f^\vee\equiv f_\varepsilon^\vee$ and $f^\wedge\equiv f_\varepsilon^\wedge$ on $\Omega\setminus U_\varepsilon$; \item $f_\varepsilon^\vee$ is upper semicontinuous on $\Omega$, and $f_\varepsilon^\wedge$ is lower semicontinuous on $\Omega$. \end{enumerate} If the space $X$ is unbounded, then such an approximating function $f_\varepsilon$ can be constructed with the additional property that the uniform limit at infinity of both $f^\vee_\varepsilon$ and $f^\wedge_\varepsilon$ is $0$. Moreover, when $X=\R^d$, we show that the non-centered maximal function of $f_\varepsilon$ is continuous in $\Omega$.