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

TL;DR

This paper investigates the Lebesgue point properties of Besov and fractional Sobolev functions in metric spaces with Ahlfors regular measures, establishing conditions under which points are Lebesgue points outside small sets.

Contribution

It proves that Besov functions have Lebesgue points outside sets of controlled Hausdorff measure and extends these results to fractional Sobolev spaces in metric spaces.

Findings

Every point is a general average Lebesgue point outside a $\sigma$-finite set for Besov functions.

Almost every point is an average Lebesgue point for functions in fractional Sobolev spaces.

For complete $Y$, almost every point is a Lebesgue point outside a set of Hausdorff dimension at most $s - rq$.

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 $u \in B^r_{q,\infty}(X,\mu;Y)$, 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 $u \in B^r_{q,\infty}(X,\mu;Y)$, almost every point is a {\it Lebesgue point} outside a set of Hausdorff dimension at most $s - rq$.