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 $B_{q,\infty }^r({\mathbb{R}}^N,{\mathbb{R}}^d)$, $q\in [1,\infty )$, $r\in (0,1)$, 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 $u_{\epsilon }$ denotes the convolution of $u$ with a mollifier ${\eta }_{(\epsilon )}(x):=\frac{1}{{\epsilon }^N}\eta \left(\frac{x}{\epsilon }\right)$, $\eta \in W^{1,1}({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}\eta (z)dz=1$. Furthermore, we prove that every function $u$ in $BV({\mathbb{R}}^N,{\mathbb{R}}^d)\cap B_{p,\infty }^{1/p}({\mathbb{R}}^N,{\mathbb{R}}^d),p\in (1,\infty ),$ 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 ${\mathcal{J}}_u$.