  Citation: Leaci, A.; Tomarelli, F. Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces. Axioms 2022, 11, 30. https://doi.org/10.3390/ axioms11010030 Academic Editors: Natália Martins, Ricardo Almeida, Moulay Rchid Sidi Ammi and Cristiana João Soares da Silva Received: 27 November 2021 Accepted: 11 January 2022 Published: 14 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). axioms Article Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces † Antonio Leaci 1, * and Franco Tomarelli 2 1 Dipartimento di Matematica e Fisica “Ennio De Giorgi”, Università del Salento, 73100 Lecce, Italy 2 Politecnico di Milano, Dipartimento di Matematica, 20133 Milan, Italy; franco.tomarelli@polimi.it * Correspondence: antonio.leaci@unisalento.it † Dedicated to Delfim F. M. Torres on the Occasion of His 50th Birthday. Abstract: We establish some properties of the bilateral Riemann–Liouville fractional derivative D s . We set the notation, and study the associated Sobolev spaces of fractional order s, denoted by W s,1 (a, b), and the fractional bounded variation spaces of fractional order s, denoted by BV s (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. Keywords: fractional derivatives; distributional derivatives; Sobolev spaces; bounded variation functions; embeddings; compactness; calculus of variations; Abel equation 1. Introduction Among several different available definitions for fractional derivatives and corre- sponding functional spaces, this paper focuses the analysis on some classical pointwise defined or distributional fractional derivatives connected to integral-convolution operators. Precisely, we refer to bilateral definitions of Riemann–Liouville fractional derivatives and related Sobolev and bounded variation spaces that we introduced in [1]: here, we show some compactness and embedding properties of these spaces. First, we recall the classical Riemann–Liouville left and right fractional derivatives (d/dx) s + and (d/dx) s − and introduce the distributional Riemann–Liouville left and right fractional derivatives D s + , D s − together with their bilateral even and odd versions, respec- tively D s e , D s o , all of them defined for non-integer orders s,0 < s < 1 (see Definition 4). Second, we provide the definitions of the fractional Sobolev spaces W s,1 and fractional bounded variation spaces BV s , associated to these bilateral derivatives (see Definitions 9 and 10). These function spaces are studied here (see Theorem 6, Examples 2–5, 6 and 8) in comparison with their non-bilateral counterpart ([2–8]). The spaces W s,1 and BV s turn out to be the natural setting for data of Abel integral equations in order to make them well-posed problems in the distributional framework too: see Propositions 2 and 3 showing that if f ∈ BV s ( a, b) with −∞ < a ≤ b ≤ +∞, then the distributional Abel integral equation I s a+ [u]= f admits a unique solution and provides an explicit resolvent formula. Corollaries 1 and 2 state analogous results for backward equations. This approach provides an alternative formulation of classical L 1 representability (see [9]); precisely, this approach leads to a straightforward extension of solvability for the Abel integral equation under conditions weaker than L 1 representability, namely with data possibly belonging to BV s ( a, b). Basic properties of the functional spaces introduced in present article (weak compact- ness property stated by Theorems 3 and 11 together with comparison embeddings and strict embeddings stated in Theorems 6 and 8 and by (92) and (93)), namely BV( a, b) ⊂  =  σ∈(0,1) W σ,1 ( a, b) ⊂  = W s,1 ( a, b) ⊂  = BV s + ( a, b) ∀s ∈ (0, 1), Axioms 2022, 11, 30. https://doi.org/10.3390/axioms11010030 https://www.mdpi.com/journal/axioms