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

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

doi:10.5802/crmath.300

Analyse fonctionnelle

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

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

Comptes Rendus. Mathématique, Volume 360 (2022), 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 :

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@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},
}

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JO  - Comptes Rendus. Mathématique
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PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.300
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%A Giovanni E. Comi
%A Giorgio Stefani
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%J Comptes Rendus. Mathématique
%D 2022
%P 589-626
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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/

Bibliographie
Cité par

[1] Robert A. Adams Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press Inc., 1975 | MR

[2] Angela Alberico; Andrea Cianchi; Luboš Pick; Lenka Slavíková On the limit as $s \to 0^{+}$ of fractional Orlicz-Sobolev spaces, J. Fourier Anal. Appl., Volume 26 (2020) no. 6, 80, 19 pages | MR | Zbl

[3] Luigi Ambrosio; Guido De Philippis; Luca Martinazzi Gamma-convergence of nonlocal perimeter functionals, Manuscr. Math., Volume 134 (2011) no. 3-4, pp. 377-403 | DOI | MR | Zbl

[4] Luigi Ambrosio; Nicola Fusco; Diego Pallara Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, 2000

[5] Vincenzo Ambrosio On some convergence results for fractional periodic Sobolev spaces, Opusc. Math., Volume 40 (2020) no. 1, pp. 5-20 | DOI | MR | Zbl

[6] Clara Antonucci; Massimo Gobbino; Matteo Migliorini; Nicola Picenni On the shape factor of interaction laws for a non-local approximation of the Sobolev norm and the total variation, C. R. Math. Acad. Sci. Paris, Volume 356 (2018) no. 8, pp. 859-864 | DOI | MR | Zbl

[7] Clara Antonucci; Massimo Gobbino; Matteo Migliorini; Nicola Picenni Optimal constants for a nonlocal approximation of Sobolev norms and total variation, Anal. PDE, Volume 13 (2020) no. 2, pp. 595-625 | DOI | MR | Zbl

[8] Clara Antonucci; Massimo Gobbino; Nicola Picenni On the gap between the Gamma-limit and the pointwise limit for a nonlocal approximation of the total variation, Anal. PDE, Volume 13 (2020) no. 3, pp. 627-649 | DOI | MR | Zbl

[9] Emil Artin The Gamma function, Athena Series. Selected Topics in Mathematics, Holt, Rinehart and Winston, 1964, vii+39 pages (translated by Michael Butler.)

[10] Gilles Aubert; Pierre Kornprobst Can the nonlocal characterization of Sobolev spaces by Bourgain et al. be useful for solving variational problems?, SIAM J. Numer. Anal., Volume 47 (2009) no. 2, pp. 844-860 | DOI | MR | Zbl

[11] Kaushik Bal; Kaushik Mohanta; Prosenjit Roy Bourgain-Brezis-Mironescu domains, Nonlinear Anal., Theory Methods Appl., Volume 199 (2020), 111928, 10 pages | MR | Zbl

[12] Davide Barbieri Approximations of Sobolev norms in Carnot groups, Commun. Contemp. Math., Volume 13 (2011) no. 5, pp. 765-794 | DOI | MR | Zbl

[13] José C. Bellido; Javier Cueto; Carlos Mora-Corral Fractional Piola identity and polyconvexity in fractional spaces, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 37 (2020) no. 4, pp. 955-981 | DOI | MR | Zbl

[14] José C. Bellido; Javier Cueto; Carlos Mora-Corral $Γ$ -convergence of polyconvex functionals involving $s$ -fractional gradients to their local counterparts, Calc. Var. Partial Differ. Equ., Volume 60 (2021) no. 1, 7, 29 pages | MR | Zbl

[15] Jöran Bergh; Jörgen Löfström Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer, 1976 | DOI

[16] Jean Bourgain; Haïm Brezis; Petru Mironescu Another look at Sobolev spaces, Optimal control and partial differential equations, IOS Press, 2001, pp. 439-455 | Zbl

[17] Jean Bourgain; Haïm Brezis; Petru Mironescu Limiting embedding theorems for $W^{s, p}$ when $s ↑ 1$ and applications, J. Anal. Math., Volume 87 (2002), pp. 77-101 (Dedicated to the memory of Thomas H. Wolff) | DOI | MR | Zbl

[18] Jean Bourgain; Hoai-Minh Nguyen A new characterization of Sobolev spaces, C. R. Math. Acad. Sci. Paris, Volume 343 (2006) no. 2, pp. 75-80 | DOI | MR | Zbl

[19] Haïm Brezis How to recognize constant functions. A connection with Sobolev spaces, Usp. Mat. Nauk, Volume 57 (2002) no. 4(346), pp. 59-74 | MR | Zbl

[20] Haïm Brezis Functional analysis, Sobolev spaces and Partial Differential Equations, Universitext, Springer, 2011, xiv+599 pages | DOI

[21] Haïm Brezis New approximations of the total variation and filters in imaging, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 26 (2015) no. 2, pp. 223-240 | DOI | MR | Zbl

[22] Haïm Brezis; Hoai-Minh Nguyen The BBM formula revisited, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 27 (2016) no. 4, pp. 515-533 | DOI | MR | Zbl

[23] Haïm Brezis; Hoai-Minh Nguyen Two subtle convex nonlocal approximations of the BV-norm, Nonlinear Anal., Theory Methods Appl., Volume 137 (2016), pp. 222-245 | DOI | MR | Zbl

[24] Haïm Brezis; Hoai-Minh Nguyen Non-local functionals related to the total variation and connections with image processing, Ann. PDE, Volume 4 (2018) no. 1, 9, 77 pages | MR | Zbl

[25] Haïm Brezis; Hoai-Minh Nguyen Non-local, non-convex functionals converging to Sobolev norms, Nonlinear Anal., Theory Methods Appl., Volume 191 (2020), 111626, 9 pages | MR | Zbl

[26] Haïm Brezis; Jean Van Schaftingen; Po-Lam Yung A surprising formula for Sobolev norms and related topics, Proc. Natl. Acad. Sci. USA, Volume 118 (2021) no. 8, e2025254118 | DOI

[27] Giovanni E. Comi; Giorgio Stefani A distributional approach to fractional Sobolev spaces and fractional variation: Asymptotics I (2019) (to appear in Rev. Mat. Complut.) | arXiv

[28] Giovanni E. Comi; Giorgio Stefani A distributional approach to fractional Sobolev spaces and fractional variation: Existence of blow-up, J. Funct. Anal., Volume 277 (2019) no. 10, pp. 3373-3435 | DOI | MR | Zbl

[29] J. Dávila On an open question about functions of bounded variation, Calc. Var. Partial Differ. Equ., Volume 15 (2002) no. 4, pp. 519-527 | DOI | MR | Zbl

[30] Simone Di Marino; Marco Squassina New characterizations of Sobolev metric spaces, J. Funct. Anal., Volume 276 (2019) no. 6, pp. 1853-1874 | DOI | MR | Zbl

[31] Eleonora Di Nezza; Giampiero Palatucci; Enrico Valdinoci Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521-573 | DOI | MR | Zbl

[32] Oscar Dominguez; Mario Milman New Brezis-Van Schaftingen-Yung Sobolev type inequalities connected with maximal inequalities and one parameter families of operators (2020) | arXiv

[33] Lawrence C. Evans; Ronald F. Gariepy Measure theory and fine properties of functions, Textbooks in Mathematics, CRC Press, 2015 | DOI

[34] Julián Fernández Bonder; Ariel M. Salort Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., Volume 277 (2019) no. 2, pp. 333-367 | DOI | MR | Zbl

[35] Rita Ferreira; Peter Hästö; Ana Margarida Ribeiro Characterization of generalized Orlicz spaces, Commun. Contemp. Math., Volume 22 (2020) no. 2, 1850079, 25 pages | MR | Zbl

[36] Gerald B. Folland; Elias M. Stein Hardy spaces on homogeneous groups, Mathematical Notes, 28, Princeton University Press; University of Tokyo Press, 1982

[37] Rupert L. Frank; Robert Seiringer Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., Volume 255 (2008) no. 12, pp. 3407-3430 | DOI | MR | Zbl

[38] José García-Cuerva; José L. Rubio de Francia Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116, North-Holland, 1985

[39] Loukas Grafakos Classical Fourier analysis, Graduate Texts in Mathematics, 249, Springer, 2014

[40] Loukas Grafakos Modern Fourier analysis, Graduate Texts in Mathematics, 250, Springer, 2014

[41] John Horváth On some composition formulas, Proc. Am. Math. Soc., Volume 10 (1959), pp. 433-437 | DOI | MR | Zbl

[42] Viktor I. Kolyada; Andrei K. Lerner On limiting embeddings of Besov spaces, Stud. Math., Volume 171 (2005) no. 1, pp. 1-13 | DOI | MR | Zbl

[43] Andreas Kreuml; Olaf Mordhorst Fractional Sobolev norms and BV functions on manifolds, Nonlinear Anal., Theory Methods Appl., Volume 187 (2019), pp. 450-466 | DOI | MR | Zbl

[44] Nguyen Lam; Ali Maalaoui; Andrea Pinamonti Characterizations of anisotropic high order Sobolev spaces, Asymptotic Anal., Volume 113 (2019) no. 4, pp. 239-260 | MR | Zbl

[45] Giovanni Leoni A first course in Sobolev spaces, Graduate Studies in Mathematics, 105, American Mathematical Society, 2009

[46] Giovanni Leoni; Daniel Spector Characterization of Sobolev and $B V$ spaces, J. Funct. Anal., Volume 261 (2011) no. 10, pp. 2926-2958 | DOI | MR | Zbl

[47] Giovanni Leoni; Daniel Spector Corrigendum to “Characterization of Sobolev and $B V$ spaces” [J. Funct. Anal. 261 (10) (2011) 2926–2958], J. Funct. Anal., Volume 266 (2014) no. 2, pp. 1106-1114 | DOI | MR

[48] Ali Maalaoui; Andrea Pinamonti Interpolations and fractional Sobolev spaces in Carnot groups, Nonlinear Anal., Theory Methods Appl., Volume 179 (2019), pp. 91-104 | DOI | MR | Zbl

[49] V. Mazʼya; T. Shaposhnikova On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal., Volume 195 (2002) no. 2, pp. 230-238 | DOI | MR | Zbl

[50] V. Mazʼya; T. Shaposhnikova Erratum to: “On the Bourgain, Brezis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces” [J. Funct. Anal. 195 (2002), no. 2, 230–238], J. Funct. Anal., Volume 201 (2003) no. 1, pp. 298-300 | DOI

[51] Mario Milman Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Am. Math. Soc., Volume 357 (2005) no. 9, pp. 3425-3442 | DOI | MR | Zbl

[52] Hoai-Minh Nguyen $Γ$ -convergence and Sobolev norms, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 12, pp. 679-684 | DOI | MR | Zbl

[53] Hoai-Minh Nguyen Further characterizations of Sobolev spaces, J. Eur. Math. Soc., Volume 10 (2008) no. 1, pp. 191-229 | MR | Zbl

[54] Hoai-Minh Nguyen $Γ$ -convergence, Sobolev norms, and BV functions, Duke Math. J., Volume 157 (2011) no. 3, pp. 495-533 | MR | Zbl

[55] Hoai-Minh Nguyen; Marco Squassina On anisotropic Sobolev spaces, Commun. Contemp. Math., Volume 21 (2019) no. 1, 1850017, 13 pages | MR | Zbl

[56] Andrea Pinamonti; Marco Squassina; Eugenio Vecchi The Mazʼya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., Volume 449 (2017) no. 2, pp. 1152-1159 | DOI | MR | Zbl

[57] Andrea Pinamonti; Marco Squassina; Eugenio Vecchi Magnetic BV-functions and the Bourgain-Brezis-Mironescu formula, Adv. Calc. Var., Volume 12 (2019) no. 3, pp. 225-252 | DOI | MR | Zbl

[58] Augusto C. Ponce An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc., Volume 6 (2004) no. 1, pp. 1-15 | DOI | Zbl

[59] Augusto C. Ponce A new approach to Sobolev spaces and connections to $Γ$ -convergence, Calc. Var. Partial Differ. Equ., Volume 19 (2004) no. 3, pp. 229-255 | DOI | MR | Zbl

[60] Augusto C. Ponce Elliptic PDEs, measures and capacities, EMS Tracts in Mathematics, 23, European Mathematical Society, 2016 | DOI

[61] Augusto C. Ponce; Daniel Spector A note on the fractional perimeter and interpolation, C. R. Math. Acad. Sci. Paris, Volume 355 (2017) no. 9, pp. 960-965 | DOI | MR | Zbl

[62] Stefan G. Samko; Anatoly A. Kilbas; Oleg I. Marichev Fractional integrals and derivatives, Gordon and Breach Science Publishers, 1993 | Zbl

[63] Armin Schikorra; Tien-Tsan Shieh; Daniel Spector $L^{p}$ theory for fractional gradient PDE with $V M O$ coefficients, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl., Volume 26 (2015) no. 4, pp. 433-443 | DOI | MR | Zbl

[64] Armin Schikorra; Tien-Tsan Shieh; Daniel Spector Regularity for a fractional $p$ -Laplace equation, Commun. Contemp. Math., Volume 20 (2018) no. 1, 1750003, 6 pages | MR | Zbl

[65] Armin Schikorra; Daniel Spector; Jean Van Schaftingen An $L^{1}$ -type estimate for Riesz potentials, Rev. Mat. Iberoam., Volume 33 (2017) no. 1, pp. 291-303 | DOI | MR | Zbl

[66] Tien-Tsan Shieh; Daniel Spector On a new class of fractional partial differential equations, Adv. Calc. Var., Volume 8 (2015) no. 4, pp. 321-336 | MR | Zbl

[67] Tien-Tsan Shieh; Daniel Spector On a new class of fractional partial differential equations II, Adv. Calc. Var., Volume 11 (2018) no. 3, pp. 289-307 | DOI | MR | Zbl

[68] Miroslav Šilhavý Fractional vector analysis based on invariance requirements (Critique of coordinate approaches), M. Continuum Mech. Thermodyn., Volume 32 (2020) no. 1, pp. 207-228 | DOI | MR | Zbl

[69] Daniel Spector A noninequality for the fractional gradient, Port. Math., Volume 76 (2019) no. 2, pp. 153-168 | DOI | MR | Zbl

[70] Daniel Spector An optimal Sobolev embedding for $L^{1}$ , J. Funct. Anal., Volume 279 (2020) no. 3, 108559, 26 pages | MR | Zbl

[71] Marco Squassina; Bruno Volzone Bourgain-Brézis-Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 8, pp. 825-831 | DOI | Zbl

[72] Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton University Press, 1970

[73] Elias M. Stein Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993

[74] Robert S. Strichartz $H^{p}$ Sobolev spaces, Colloq. Math., Volume 60/61 (1990) no. 1, pp. 129-139 | DOI | Zbl

[75] Félix del Teso; David Gómez-Castro; Juan Luis Vázquez Estimates on translations and Taylor expansions in fractional Sobolev spaces, Nonlinear Anal., Theory Methods Appl., Volume 200 (2020), 111995, 12 pages | MR | Zbl

[76] Hans Triebel Limits of Besov norms, Arch. Math., Volume 96 (2011) no. 2, pp. 169-175 | DOI | MR | Zbl

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