Fine properties of Besov functions $B_{q,\infty }^r$ in metric spaces

Abstract

Let $X$ be a metric space and $\mu$ an $s$-regular Ahlfors measure. Let $Y$ be a metric space. We prove that for Besov functions $u\in B_{q,\infty }^r(X,\mu ;Y)$, every point is a {\it general average Lebesgue point} of $u$ outside a $\sigma$-finite set with respect to the Hausdorff measure ${\mathcal{H}}^{s-rq}$. The proof is based on density-type estimates involving Hausdorff measure. In addition, we prove that for functions $u$ in the fractional Sobolev space $W^{r,q}(X,\mu ;Y)$, almost every point with respect to ${\mathcal{H}}^{s-rq}$ is an {\it average Lebesgue point} of $u$. Finally, if $Y$ is also complete, we prove that for $u\in B_{q,\infty }^r(X,\mu ;Y)$, almost every point is a {\it Lebesgue point} outside a set of Hausdorff dimension at most $s-rq$.