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Riemann-Liouville Fractional Sobolev and Bounded Variation Spaces

We establish some properties of the bilateral Riemann–Liouville fractional derivative Ds.  We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by Ws,1(a,b), and the fractional bounded variation spaces of fractional order s, denoted by BVs(a,b). Examples, embeddings and compactness properties related to these spaces are addressed, aiming to set a functional framework suitable for fractional variational models for image analysis.