A note on Riemann-Liouville fractional Sobolev spaces

Abstract

Taking inspiration from a recent paper by Bergounioux, Leaci, Nardi and Tomarelli we study the Riemann-Liouville fractional Sobolev space $W_{RL,a+}^{s,p}(I)$, 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 $W_{RL,a+}^{s,1}(I)$. In addition, we define the space of functions with left Riemann-Liouville $s$-fractional bounded variation, $B{V}_{RL,a+}^s(I)$, as the set of functions $u\in L^1(I)$ such that $I_{a+}^{1-s}[u]\in BV(I)$, 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.

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