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://archive.numdam.org/articles/10.5802/crmath.300/

Bibliographie
Cité par

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

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

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

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

[6] Antonucci, Clara; Gobbino, Massimo; Migliorini, Matteo; Picenni, Nicola 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] Antonucci, Clara; Gobbino, Massimo; Migliorini, Matteo; Picenni, Nicola 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] Antonucci, Clara; Gobbino, Massimo; Picenni, Nicola 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] Artin, Emil The Gamma function, Athena Series. Selected Topics in Mathematics, Holt, Rinehart and Winston, 1964, vii+39 pages (translated by Michael Butler.)

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

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

[13] Bellido, José C.; Cueto, Javier; Mora-Corral, Carlos 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] Bellido, José C.; Cueto, Javier; Mora-Corral, Carlos $Γ$ -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] Bergh, Jöran; Löfström, Jörgen Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer, 1976 | DOI

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[49] Mazʼya, V.; Shaposhnikova, T. 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] Mazʼya, V.; Shaposhnikova, T. 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] Milman, Mario 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] Nguyen, Hoai-Minh $Γ$ -convergence and Sobolev norms, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 12, pp. 679-684 | DOI | MR | Zbl

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

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

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

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

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

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

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

[61] Ponce, Augusto C.; Spector, Daniel 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] Samko, Stefan G.; Kilbas, Anatoly A.; Marichev, Oleg I. Fractional integrals and derivatives, Gordon and Breach Science Publishers, 1993 | Zbl

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

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

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

[67] Shieh, Tien-Tsan; Spector, Daniel 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] Šilhavý, Miroslav 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] Spector, Daniel A noninequality for the fractional gradient, Port. Math., Volume 76 (2019) no. 2, pp. 153-168 | DOI | MR | Zbl

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

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

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

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

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

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

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

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