Fractional Sobolev Spaces and Functions of Bounded Variation

We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space $BV$ of functions of bounded variation, whose derivatives are not functions but measures and the space $SBV$, say the space of bounded variation functions whose derivative has no Cantor part. We prove that $SBV$ is included in $W^{s,1}$ for every $s \in (0,1)$whiletheresultremainsopenfor$BV$. We study examples and address open questions.