Riemann–Liouville Fractional Sobolev and Bounded Variation Spaces

Antonio Leaci,Franco Tomarelli

doi:10.3390/axioms11010030

Antonio Leaci, Franco Tomarelli

Open Access

https://doi.org/10.3390/axioms11010030

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Journal: Axioms	Publication Date: Jan 14, 2022
Citations: 2	License type: CC BY 4.0

Affiliation: Politecnico di Milano, University of Salento

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.

Highlights

IntroductionAmong several different available definitions for fractional derivatives and corresponding functional spaces, this paper focuses the analysis on some classical pointwise defined or distributional fractional derivatives connected to integral-convolution operators
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
Corollaries 1 and 2 state analogous results for backward equations. This approach provides an alternative formulation of classical L1 representability; precisely, this approach leads to a straightforward extension of solvability for the Abel integral equation under conditions weaker than L1 representability, namely with data possibly belonging to BVs(a, b)

Summary

Introduction

Among several different available definitions for fractional derivatives and corresponding functional spaces, this paper focuses the analysis on some classical pointwise defined or distributional fractional derivatives connected to integral-convolution operators. We provide the definitions of the fractional Sobolev spaces Ws, and fractional bounded variation spaces BVs, 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 ([2,3,4,5,6,7,8]). Corollaries 1 and 2 state analogous results for backward equations This approach provides an alternative formulation of classical L1 representability (see [9]); precisely, this approach leads to a straightforward extension of solvability for the Abel integral equation under conditions weaker than L1 representability, namely with data possibly belonging to BVs(a, b). We thank an anonymous referee for useful remarks and pointing us to the recent article [19] containing a different approach to the Sonin–Abel equation in weighted Lebesgue spaces

Bilateral Fractional Integral and Derivative

The Bilateral Fractional Sobolev Space

Conclusions