Lusin approximation for functions of bounded variation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910782062592000 |
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| author | Lahti, Panu Nguyen, Khanh |
| author_facet | Lahti, Panu Nguyen, Khanh |
| contents | 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 $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07147 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lusin approximation for functions of bounded variation Lahti, Panu Nguyen, Khanh Functional Analysis 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 $Ω$. |
| title | Lusin approximation for functions of bounded variation |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2501.07147 |