*Accepted Paper*

**Inserted:** 22 mar 2020

**Last Updated:** 14 oct 2020

**Journal:** Communications on Pure and Applied Analysis

**Year:** 2020

**Doi:** 10.3934/cpaa.2020255

**Abstract:**

Taking inspiration from M. Bergounioux, A. Leaci, G. Nardi, and F. Tomarelli, Fractional Sobolev spaces and functions of bounded variation of one variable, Fract. Calc. Appl. Anal. 20 (2017), no. 4, 936–962, 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 $BV(I)$ and the fractional Sobolev space $W^{s, 1}(I)$ 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.

**Keywords:**
fractional Sobolev spaces, fractional calculus, fractional derivative, Riemann-Liouville fractional integral, Riemann-Liouville fractional derivative, Caputo fractional derivative, fractional $BV$ spaces

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