  Taking inspiration from a recent paper by Bergounioux, Leaci, Nardi and
Tomarelli we study the Riemann-Liouville fractional Sobolev space $W^{s,
p}_{RL, a+}(I)$, for $I = (a, b)$ for some $a, b \in \mathbb{R}, a < b$, $s \in
(0, 1)$ and $p \in [1, \infty]$; that is, the space of functions $u \in
L^{p}(I)$ such that the left Riemann-Liouville $(1 - s)$-fractional integral
$I_{a+}^{1 - s}[u]$ belongs to $W^{1, p}(I)$. We prove that the space of
functions of bounded variation and the fractional Sobolev space, $BV(I)$ and
$W^{s, 1}(I)$, continuously embed into $W^{s, 1}_{RL, a+}(I)$. In addition, we
define the space of functions with left Riemann-Liouville $s$-fractional
bounded variation, $BV^{s}_{RL,a+}(I)$, as the set of functions $u \in
L^{1}(I)$ such that $I^{1 - s}_{a+}[u] \in BV(I)$, and we analyze some fine
properties of these functions. Finally, we prove some fractional Sobolev-type
embedding results and we analyze the case of higher order Riemann-Liouville
fractional derivatives.
