Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866929314484715520 |
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| author | Hashash, Paz Poliakovsky, Arkady |
| author_facet | Hashash, Paz Poliakovsky, Arkady |
| contents | In this paper we analyse functions in Besov spaces $B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$, and functions in fractional Sobolev spaces $W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in (0,1),q\in [1,\infty)$. We prove for Besov functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$ the summability of the difference between one-sided approximate limits in power $q$, $|u^+-u^-|^q$, along the jump set $\mathcal{J}_u$ of $u$ with respect to Hausdorff measure $\mathcal{H}^{N-1}$, and establish the best bound from above on the integral $\int_{\mathcal{J}_u}|u^+-u^-|^qd\mathcal{H}^{N-1}$ in terms of Besov constants. We show for functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$ that \begin{equation} \liminf\limits_{\varepsilon \to 0^+}\fint_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for every $x$ outside of a $\mathcal{H}^{N-1}$-sigma finite set. For fractional Sobolev functions $u\in W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)$ we prove that \begin{equation} \lim_{ρ\to 0^+}\fint_{B_ρ(x)}\fint_{B_ρ(x)} |u\big(z\big)-u(y)|^qdzdy=0 \end{equation} for $\mathcal{H}^{N-rq}$ a.e. $x$, where $q\in[1,\infty)$, $r\in(0,1)$ and $rq\leq N$. We prove for $u\in W^{1,q}(\mathbb{R}^N),1<q\leq N$, that \begin{equation} \lim\limits_{\varepsilon\to 0^+}\fint_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for $\mathcal{H}^{N-q}$ a.e. $x\in \mathbb{R}^N$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_09757 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions Hashash, Paz Poliakovsky, Arkady Classical Analysis and ODEs Analysis of PDEs Functional Analysis In this paper we analyse functions in Besov spaces $B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$, and functions in fractional Sobolev spaces $W^{r,q}(\mathbb{R}^N,\mathbb{R}^d),r\in (0,1),q\in [1,\infty)$. We prove for Besov functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d)$ the summability of the difference between one-sided approximate limits in power $q$, $|u^+-u^-|^q$, along the jump set $\mathcal{J}_u$ of $u$ with respect to Hausdorff measure $\mathcal{H}^{N-1}$, and establish the best bound from above on the integral $\int_{\mathcal{J}_u}|u^+-u^-|^qd\mathcal{H}^{N-1}$ in terms of Besov constants. We show for functions $u\in B^{1/q}_{q,\infty}(\mathbb{R}^N,\mathbb{R}^d),q\in (1,\infty)$ that \begin{equation} \liminf\limits_{\varepsilon \to 0^+}\fint_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for every $x$ outside of a $\mathcal{H}^{N-1}$-sigma finite set. For fractional Sobolev functions $u\in W^{r,q}(\mathbb{R}^N,\mathbb{R}^d)$ we prove that \begin{equation} \lim_{ρ\to 0^+}\fint_{B_ρ(x)}\fint_{B_ρ(x)} |u\big(z\big)-u(y)|^qdzdy=0 \end{equation} for $\mathcal{H}^{N-rq}$ a.e. $x$, where $q\in[1,\infty)$, $r\in(0,1)$ and $rq\leq N$. We prove for $u\in W^{1,q}(\mathbb{R}^N),1<q\leq N$, that \begin{equation} \lim\limits_{\varepsilon\to 0^+}\fint_{B_{\varepsilon}(x)} |u(z)-u_{B_{\varepsilon}(x)}|^qdz=0 \end{equation} for $\mathcal{H}^{N-q}$ a.e. $x\in \mathbb{R}^N$. |
| title | Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions |
| topic | Classical Analysis and ODEs Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2304.09757 |