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A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II
Comptes Rendus. Mathématique, Volume 360 (2022), pp. 589-626.

We continue the study of the space BV α ( n ) of functions with bounded fractional variation in  n and of the distributional fractional Sobolev space S α,p ( n ), with p[1,+] and α(0,1), considered in the previous works [28, 27]. We first define the space BV 0 ( n ) and establish the identifications BV 0 ( n )=H 1 ( n ) and S α,p ( n )=L α,p ( n ), where H 1 ( n ) and L α,p ( n ) are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient α strongly converges to the Riesz transform as α0 + for H 1 W α,1 and S α,p functions. We also study the convergence of the L 1 -norm of the α-rescaled fractional gradient of W α,1 functions. To achieve the strong limiting behavior of  α as α0 + , we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

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DOI : 10.5802/crmath.300
Classification : 26A33, 26B30, 28A33, 47G40

Elia Bruè 1 ; Mattia Calzi 2 ; Giovanni E. Comi 3 ; Giorgio Stefani 4

1 School of Mathematics, Institute for Advanced Study, 1 Einstein Dr., Princeton NJ 05840, USA
2 Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133 Milano, Italy
3 Universität Hamburg, Fakultät für Mathematik, Informatik und Naturwissenschaften, Fachbereich Mathematik, Bundesstraße 55, 20146 Hamburg, Germany
4 Department Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits
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     author = {Elia Bru\`e and Mattia Calzi and Giovanni E. Comi and Giorgio Stefani},
     title = {A distributional approach to fractional {Sobolev} spaces and fractional variation: asymptotics {II}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {589--626},
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     year = {2022},
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Elia Bruè; Mattia Calzi; Giovanni E. Comi; Giorgio Stefani. A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 589-626. doi : 10.5802/crmath.300. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.300/

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