Fractional sobolev spaces and functions of bounded variation of one variable

Maïtine Bergounioux 1 , Antonio Leaci 2 , Giacomo Nardi 3 ,  and Franco Tomarelli 4
  • 1 Laboratoire MAPMO, CNRS, UMR 7349 Fédération Denis Poisson, FR 2964, Université d’Orléans, B.P. 6759 45067, Orléans, France
  • 2 Università del Salento, Dipartimento di Matematica e Fisica “Ennio De Giorgi”, I 73100, Lecce, Italy
  • 3 Institut Pasteur, Laboratoire d’Analyse d’Images Biologiques CNRS, UMR 3691, Paris, France
  • 4 Politecnico di Milano, Dipartimento di Matematica Piazza “Leonardo da Vinci”, 32 I 20133, Milano, Italy
Maïtine Bergounioux
  • Corresponding author
  • Laboratoire MAPMO, CNRS, UMR 7349 Fédération Denis Poisson, FR 2964, Université d’Orléans, B.P. 6759 45067, Orléans, France
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, Antonio Leaci
  • Università del Salento, Dipartimento di Matematica e Fisica “Ennio De Giorgi”, I 73100, Lecce, Italy
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, Giacomo Nardi and Franco Tomarelli
  • Politecnico di Milano, Dipartimento di Matematica Piazza “Leonardo da Vinci”, 32 I 20133, Milano, Italy
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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 Ws,1 for every s ∈ (0, 1) while the result remains open for BV. We study examples and address open questions.

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Fractional Calculus and Applied Analysis (FCAA) is a specialized international journal for theory and applications of an important branch of Mathematical Analysis (Calculus) where differentiations and integrations can be of arbitrary non-integer order.

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