Approximations in Besov Spaces and Jump Detection of Besov Functions with Bounded Variation

Abstract

In this paper, we provide a proof that functions belonging to Besov spaces $B^{r}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$, $q\in [1,\infty)$, $r\in(0,1)$, satisfy the following formula under a certain condition: \begin{equation} \label{eq:main result in abstract} \lim_{ε\to 0^+}\frac{1}{|\lnε|}\left[u_ε\right]^q_{W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)}=N\lim_{ε\to 0^+}\int_{\mathbb{R}^N}\frac{1}{ε^N}\int_{B_ε(x)}\frac{|u(x)-u(y)|^q}{|x-y|^{rq}}dydx. \end{equation} Here, $\left[\cdot\right]_{W^{r,q}}$ represents the Gagliardo seminorm, and $u_ε$ denotes the convolution of $u$ with a mollifier $η_{(ε)}(x):=\frac{1}{ε^N}η\left(\frac{x}ε\right)$, $η\in W^{1,1}(\mathbb{R}^N),\int_{\mathbb{R}^N}η(z)dz=1$. Furthermore, we prove that every function $u$ in $BV(\mathbb{R}^N,\mathbb{R}^d)\cap B^{1/p}_{p,\infty}(\mathbb{R}^N,\mathbb{R}^d),p\in(1,\infty),$ satisfies \begin{multline} \lim_{ε\to 0^+}\frac{1}{|\lnε|}\left[u_ε\right]^q_{W^{1/q,q}(\mathbb{R}^N,\mathbb{R}^d)}=N\lim_{ε\to 0^+}\int_{\mathbb{R}^N}\frac{1}{ε^N}\int_{B_ε(x)}\frac{|u(x)-u(y)|^q}{|x-y|}dydx =\left(\int_{S^{N-1}}|z_1|~d\mathcal{H}^{N-1}(z)\right)\int_{\mathcal{J}_u} \Big|u^+(x)-u^-(x)\Big|^q d\mathcal{H}^{N-1}(x), \end{multline} for every $1<q<p$. Here $u^+,u^-$ are the one-sided approximate limits of $u$ along the jump set $\mathcal{J}_u$.