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Fractional Sobolev Spaces and Functions of Bounded Variation of One Variable

Abstract : 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.
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Maïtine Bergounioux, Antonio Leaci, Giacomo Nardi, Franco Tomarelli. Fractional Sobolev Spaces and Functions of Bounded Variation of One Variable. Fractional Calculus & Applied Analysis, 2017, ⟨10.1515/fca-2017-0049⟩. ⟨hal-01287725v2⟩

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