Lusin approximation for functions of bounded variation
Panu Lahti, Khanh Nguyen
TL;DR
This work establishes a Lusin-type approximation for functions of bounded variation on open sets of complete doubling metric measure spaces supporting a $1$-Poincaré inequality. For any BV function $f$ and any $\varepsilon>0$, the authors construct a BV-approximation $f_\varepsilon$ on $\Omega$ and a small-capacity set $U_\varepsilon$ so that $f_\varepsilon$ matches the upper and lower approximate limits of $f$ outside $U_\varepsilon$ and enjoys upper semicontinuity of $f_\varepsilon^\vee$ and lower semicontinuity of $f_\varepsilon^\wedge$ there; moreover, $f_\varepsilon$ approximates $f$ in BV-norm within $\varepsilon$ and the exceptional set has Cap$_1$-measure less than $\varepsilon$. The paper also shows that on unbounded spaces one can enforce vanishing limits at infinity for these semicontinuous representatives, and in Euclidean spaces the non-centered maximal function of $f_\varepsilon$ is continuous. A metric-space counterexample demonstrates limitations in generality for maximal-function regularity, while the Euclidean case confirms stronger regularity for the Lusin-approximant. The work advances BV analysis on metric spaces by providing a robust Lusin approximation with global semicontinuity properties and boundary-truncation techniques, with implications for fine properties and maximal operator regularity.
Abstract
We prove a Lusin approximation of functions of bounded variation. If $f$ is a function of bounded variation on an open set $Ω\subset X$, where $X=(X,d,μ)$ is a given complete doubling metric measure space supporting a $1$-Poincaré inequality, then for every $\varepsilon>0$, there exist a function $f_\varepsilon$ on $Ω$ and an open set $U_\varepsilon\subsetΩ$ such that the following properties hold true: \begin{enumerate} \item ${\rm Cap}_1(U_\varepsilon)<\varepsilon$; \item $\|f-f_\varepsilon\|_{\BV(Ω)}< \varepsilon$; \item $f^\vee\equiv f_\varepsilon^\vee$ and $f^\wedge\equiv f_\varepsilon^\wedge$ on $Ω\setminus U_\varepsilon$; \item $f_\varepsilon^\vee$ is upper semicontinuous on $Ω$, and $f_\varepsilon^\wedge$ is lower semicontinuous on $Ω$. \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 $Ω$.