Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions

Abstract

In this paper we analyse functions in Besov spaces $B_{q,\infty }^{1/q}({\mathbb{R}}^N,{\mathbb{R}}^d),q\in (1,\infty )$, 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 $u\in B_{q,\infty }^{1/q}({\mathbb{R}}^N,{\mathbb{R}}^d)$ 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 ${\int }_{{\mathcal{J}}_u}|u^{+}-u^{-}{|}^{q}d{\mathcal{H}}^{N-1}$ in terms of Besov constants. We show for functions $u\in B_{q,\infty }^{1/q}({\mathbb{R}}^N,{\mathbb{R}}^d),q\in (1,\infty )$ 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 $rq\le N$. We prove for $u\in W^{1,q}({\mathbb{R}}^N),1<q\le N$, 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. $x\in {\mathbb{R}}^N$.

References in corpus (1)

• On Lebesgue points of entropy solutions to the eikonal equation