Calculus of Variations and Geometric Measure Theory
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A. Carbotti - G. E. Comi

A note on Riemann-Liouville fractional Sobolev spaces

created by carbotti on 22 Mar 2020
modified on 28 Oct 2020


Published Paper

Inserted: 22 mar 2020
Last Updated: 28 oct 2020

Journal: Communications on Pure and Applied Analysis
Year: 2020
Doi: 10.3934/cpaa.2020255

ArXiv: 2003.09515 PDF


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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