Fractional Sobolev Spaces and Functions of Bounded Variation

Maïtine Bergounioux (1) , Antonio Leaci (2) , Giacomo Nardi (3) , Franco Tomarelli (4)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 Università del Salento [Lecce]
3 BIA - Analyse d'images biologiques - Biological Image Analysis
4 POLIMI - Politecnico di Milano [Milan]

We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space BV of functions of bounded variation, whose derivatives are not functions but measures and the space SBV, say the space of bounded variation functions whose derivative has no Cantor part. We prove that SBV is included in W^{s,1} $ for every s \in (0,1) while the result remains open for BV. We study examples and address open questions.

Mots clés: BV spaces; Fractional derivative; Riemann Liouville

Domaines: Analyse fonctionnelle [math.FA]

Maïtine Bergounioux, Antonio Leaci, Giacomo Nardi, Franco Tomarelli. Fractional Sobolev Spaces and Functions of Bounded Variation . 2016. ⟨hal-01287725v1⟩