Calculus of Variations and Geometric Measure Theory

M. Bergounioux - A. Leaci - G. Nardi - F. Tomarelli

Fractional Sobolev Spaces and Functions of Bounded Variation of One Variable

created by leaci on 06 Sep 2017


Published Paper

Inserted: 6 sep 2017
Last Updated: 6 sep 2017

Journal: Fract. Calc. Appl. Anal.
Volume: 20
Number: 4
Pages: 936-962
Year: 2017
Doi: 10.1515/fca-2017-0049

ArXiv: 1603.05033v3 PDF

There is a minor change in the title with respect to the arxiv preprint.

Links: Journal site


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.

Keywords: Sobolev spaces, fractional calculus, bounded variation functions, Riemann-Liouville derivative, Marchaud derivative