A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

Bruè, Elia; Calzi, Mattia; Comi, Giovanni E.; Stefani, Giorgio

doi:10.5802/crmath.300

Analyse fonctionnelle

A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II

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

Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 589-626.

Résumé

We continue the study of the space $B V^{α} (ℝ^{n})$ of functions with bounded fractional variation in $ℝ^{n}$ and of the distributional fractional Sobolev space $S^{α, p} (ℝ^{n})$ , with $p \in [1, + \infty]$ and $α \in (0, 1)$ , considered in the previous works [28, 27]. We first define the space $B V^{0} (ℝ^{n})$ and establish the identifications $B V^{0} (ℝ^{n}) = H^{1} (ℝ^{n})$ and $S^{α, p} (ℝ^{n}) = L^{α, p} (ℝ^{n})$ , where $H^{1} (ℝ^{n})$ and $L^{α, p} (ℝ^{n})$ are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient $\nabla^{α}$ strongly converges to the Riesz transform as $α \to 0^{+}$ for $H^{1} \cap W^{α, 1}$ 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 $\nabla^{α}$ as $α \to 0^{+}$ , we prove some new fractional interpolation inequalities which are stable with respect to the interpolating parameter.

Reçu le : 2020-12-11
Révisé le : 2021-11-29
Accepté le : 2021-11-30
Publié le : 2022-06-22

Zbl

DOI : 10.5802/crmath.300

Classification : 26A33, 26B30, 28A33, 47G40

Affiliations des auteurs :

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

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     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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Bruè, Elia; Calzi, Mattia; Comi, Giovanni E.; Stefani, Giorgio. A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II. Comptes Rendus. Mathématique, Tome 360 (2022) no. G6, pp. 589-626. doi : 10.5802/crmath.300. http://www.numdam.org/articles/10.5802/crmath.300/

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