Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces

Abstract

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.

Symmetrized fractional total variation for signal and image analysis

“…As we are dealing with Riemann-Liouville fractional calculus we recall the definitions and the main results useful in the following ( [33]). For additional details concerning a bilateral approach we refer to [4,25,26].…”

“…We recall the definition of the Riemann-Liouville fractional integrals and derivatives for L 1 -functions, which, like their bilateral versions (Riesz potentials and related distributional derivatives) [25], can be represented by convolutions. Definition 2.1 Let u ∈ L 1 (0, 1).…”

“…Next, we define the Riemann-Liouville fractional derivative as Definition 2.2 (Distributional Riemann-Liouville fractional derivative) Let u ∈ L 1 (0, 1) and 0 < s < 1. Then, referring to [25] and [26], the left, right, and bilateral Riemann-Liouville derivatives of u at x ∈ (0, 1) are defined by…”

“…We emphasize that, referring to the trivial extension of u (u = 0 a.e. on R \ (0, 1)), all integrals in (2.1)-(2.5) are convolutions, though we consider their values restricted to the open set (0, 1) (see [25,26]):…”

“…We investigate such a variational model in the framework of Riemann-Liouville fractional derivatives. We show that there exists a unique solution that belongs to the space of Fractional Bounded Variation space BV s as defined in [26], see also [4,21,25]. Both 1D and 2D cases are considered, respectively, in Sects.…”