Bilateral Riemann-Liouville Fractional Sobolev spaces

Abstract

We establish some notation and properties of the bilateral Riemann-Liouville fractional derivative $D^s.$ We introduce the associated Sobolev spaces of fractional order $s$, denoted by $W^{s,1}(a,b)$, and the Bounded Variation spaces of fractional order $s$, denoted by $BV^{s}(a,b)$: these spaces are studied with the aim of providing a suitable functional framework for fractional variational models in image analysis.