Lusin approximation for functions of bounded variation

Abstract

We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $\hat{I}\text{c◯}\subset X$, where $X=(X,d,\hat{I}¼)$ is a given complete doubling metric measure space supporting a $1$-Poincaré inequality, then for every $\epsilon >0$, there exist a function $f_{\epsilon }$ on $\hat{I}\text{c◯}$ and an open set $U_{\epsilon }\subset \hat{I}\text{c◯}$ such that the following properties hold true: \begin{enumerate} \item ${\mathrm{Cap}}_1(U_{\epsilon })<\epsilon$; \item $\|f-f_\varepsilon\|_{\BV(Ω)}< \varepsilon$; \item $f^{\vee }\equiv f_{\epsilon }^{\vee }$ and $f^{\wedge }\equiv f_{\epsilon }^{\wedge }$ on $\hat{I}\text{c◯}\setminus U_{\epsilon }$; \item $f_{\epsilon }^{\vee }$ is upper semicontinuous on $\hat{I}\text{c◯}$, and $f_{\epsilon }^{\wedge }$ is lower semicontinuous on $\hat{I}\text{c◯}$. \end{enumerate} If the space $X$ is unbounded, then such an approximating function $f_{\epsilon }$ can be constructed with the additional property that the uniform limit at infinity of both $f_{\epsilon }^{\vee }$ and $f_{\epsilon }^{\wedge }$ is $0$. Moreover, when $X={\mathbb{R}}^d$, we show that the non-centered maximal function of $f_{\epsilon }$ is continuous in $\hat{I}\text{c◯}$.