A note on Riemann-Liouville fractional Sobolev spaces

Taking inspiration from a recent paper by Bergounioux et al., we study the Riemann-Liouville fractional Sobolev space \begin{document}$ W^{s, p}_{RL, a+}(I) $\end{document}, for id="M2">\begin{document}$ I = (a, b) $\end{document} some id="M3">\begin{document}$ a, b \in \mathbb{R}, < id="M4">\begin{document}$ s (0, 1) and id="M5">\begin{document}$ p [1, \infty] $\end{document}; that is, of functions id="M6">\begin{document}$ u L^{p}(I) such left id="M7">\begin{document}$ (1 - s) $\end{document}-fractional integral id="M8">\begin{document}$ I_{a+}^{1 s}[u] belongs to id="M9">\begin{document}$ W^{1, p}(I) $\end{document}. We prove bounded variation id="M10">\begin{document}$ BV(I) id="M11">\begin{document}$ 1}(I) continuously embed into id="M12">\begin{document}$ 1}_{RL, In addition, define with id="M13">\begin{document}$ variation, id="M14">\begin{document}$ BV^{s}_{RL,a+}(I) as set id="M15">\begin{document}$ L^{1}(I) id="M16">\begin{document}$ I^{1 s}_{a+}[u] analyze fine properties these functions. Finally, Sobolev-type embedding results case higher order derivatives.