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 W-RL,a+(s,p), for I = (a, b) for some a, b is an element of R, a < b, s is an element of (0,1) and p is an element of [1, infinity]; that is, the space of functions u is an element of L-P(I) such that the left Riemann-Liouville (1 - s)-fractional integral I-a+(1-s)[u] belongs to W-1,W-p (I). We prove that the space of functions of bounded variation BV(I) and the fractional Sobolev space W-s,W-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, BVRL,a+s (I), as the set of functions u is an element of L-1(I) such that I-a+(1-s)[u] is an element of 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.