We continue the study of the space of functions with bounded fractional variation in and of the distributional fractional Sobolev space , with and , considered in the previous works [28, 27]. We first define the space and establish the identifications and , where and are the (real) Hardy space and the Bessel potential space, respectively. We then prove that the fractional gradient strongly converges to the Riesz transform as for and functions. We also study the convergence of the -norm of the -rescaled fractional gradient of functions. To achieve the strong limiting behavior of as , 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
Elia Bruè 1 ; Mattia Calzi 2 ; Giovanni E. Comi 3 ; Giorgio Stefani 4
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@article{CRMATH_2022__360_G6_589_0, 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}, publisher = {Acad\'emie des sciences, Paris}, volume = {360}, year = {2022}, doi = {10.5802/crmath.300}, zbl = {07547261}, language = {en}, }
TY - JOUR AU - Elia Bruè AU - Mattia Calzi AU - Giovanni E. Comi AU - Giorgio Stefani TI - A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II JO - Comptes Rendus. Mathématique PY - 2022 SP - 589 EP - 626 VL - 360 PB - Académie des sciences, Paris DO - 10.5802/crmath.300 LA - en ID - CRMATH_2022__360_G6_589_0 ER -
%0 Journal Article %A Elia Bruè %A Mattia Calzi %A Giovanni E. Comi %A Giorgio Stefani %T A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics II %J Comptes Rendus. Mathématique %D 2022 %P 589-626 %V 360 %I Académie des sciences, Paris %R 10.5802/crmath.300 %G en %F CRMATH_2022__360_G6_589_0
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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