{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T11:00:43Z","timestamp":1762254043813,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T00:00:00Z","timestamp":1642118400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100003407","name":"Ministry of Education, Universities and Research","doi-asserted-by":"publisher","award":["grant number 2017BTM7SN"],"award-info":[{"award-number":["grant number 2017BTM7SN"]}],"id":[{"id":"10.13039\/501100003407","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>We establish some properties of the bilateral Riemann\u2013Liouville 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.<\/jats:p>","DOI":"10.3390\/axioms11010030","type":"journal-article","created":{"date-parts":[[2022,1,14]],"date-time":"2022-01-14T12:34:04Z","timestamp":1642163644000},"page":"30","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Riemann\u2013Liouville Fractional Sobolev and Bounded Variation Spaces"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4542-817X","authenticated-orcid":false,"given":"Antonio","family":"Leaci","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica e Fisica \u201cEnnio De Giorgi\u201d, Universit\u00e0 del Salento, 73100 Lecce, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1031-1510","authenticated-orcid":false,"given":"Franco","family":"Tomarelli","sequence":"additional","affiliation":[{"name":"Politecnico di Milano, Dipartimento di Matematica, 20133 Milan, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,1,14]]},"reference":[{"key":"ref_1","first-page":"61","article-title":"Bilateral Riemann-Liouville Fractional Sobolev spaces","volume":"41","author":"Leaci","year":"2021","journal-title":"Note Mat."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"17","DOI":"10.3934\/cpaa.2020255","article-title":"A note on Riemann-Liouville fractional Sobolev spaces","volume":"20","author":"Carbotti","year":"2021","journal-title":"Commun. 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