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.
Revised:
Accepted:
Published online:
DOI: 10.5802/crmath.300
Elia Bruè 1; Mattia Calzi 2; Giovanni E. Comi 3; Giorgio Stefani 4
@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/
[1] Sobolev spaces, Pure and Applied Mathematics, 65, Academic Press Inc., 1975 | MR
[2] On the limit as of fractional Orlicz-Sobolev spaces, J. Fourier Anal. Appl., Volume 26 (2020) no. 6, 80, 19 pages | MR | Zbl
[3] Gamma-convergence of nonlocal perimeter functionals, Manuscr. Math., Volume 134 (2011) no. 3-4, pp. 377-403 | DOI | MR | Zbl
[4] Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, Clarendon Press, 2000
[5] On some convergence results for fractional periodic Sobolev spaces, Opusc. Math., Volume 40 (2020) no. 1, pp. 5-20 | DOI | MR | Zbl
[6] 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] 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] 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] The Gamma function, Athena Series. Selected Topics in Mathematics, Holt, Rinehart and Winston, 1964, vii+39 pages (translated by Michael Butler.)
[10] 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] Bourgain-Brezis-Mironescu domains, Nonlinear Anal., Theory Methods Appl., Volume 199 (2020), 111928, 10 pages | MR | Zbl
[12] Approximations of Sobolev norms in Carnot groups, Commun. Contemp. Math., Volume 13 (2011) no. 5, pp. 765-794 | DOI | MR | Zbl
[13] 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] -convergence of polyconvex functionals involving -fractional gradients to their local counterparts, Calc. Var. Partial Differ. Equ., Volume 60 (2021) no. 1, 7, 29 pages | MR | Zbl
[15] Interpolation spaces. An introduction, Grundlehren der Mathematischen Wissenschaften, 223, Springer, 1976 | DOI
[16] Another look at Sobolev spaces, Optimal control and partial differential equations, IOS Press, 2001, pp. 439-455 | Zbl
[17] Limiting embedding theorems for when and applications, J. Anal. Math., Volume 87 (2002), pp. 77-101 (Dedicated to the memory of Thomas H. Wolff) | DOI | MR | Zbl
[18] A new characterization of Sobolev spaces, C. R. Math. Acad. Sci. Paris, Volume 343 (2006) no. 2, pp. 75-80 | DOI | MR | Zbl
[19] 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] Functional analysis, Sobolev spaces and Partial Differential Equations, Universitext, Springer, 2011, xiv+599 pages | DOI
[21] 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] 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] Two subtle convex nonlocal approximations of the BV-norm, Nonlinear Anal., Theory Methods Appl., Volume 137 (2016), pp. 222-245 | DOI | MR | Zbl
[24] 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] Non-local, non-convex functionals converging to Sobolev norms, Nonlinear Anal., Theory Methods Appl., Volume 191 (2020), 111626, 9 pages | MR | Zbl
[26] A surprising formula for Sobolev norms and related topics, Proc. Natl. Acad. Sci. USA, Volume 118 (2021) no. 8, e2025254118 | DOI
[27] A distributional approach to fractional Sobolev spaces and fractional variation: Asymptotics I (2019) (to appear in Rev. Mat. Complut.) | arXiv
[28] 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] 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] New characterizations of Sobolev metric spaces, J. Funct. Anal., Volume 276 (2019) no. 6, pp. 1853-1874 | DOI | MR | Zbl
[31] Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., Volume 136 (2012) no. 5, pp. 521-573 | DOI | MR | Zbl
[32] New Brezis-Van Schaftingen-Yung Sobolev type inequalities connected with maximal inequalities and one parameter families of operators (2020) | arXiv
[33] Measure theory and fine properties of functions, Textbooks in Mathematics, CRC Press, 2015 | DOI
[34] Fractional order Orlicz-Sobolev spaces, J. Funct. Anal., Volume 277 (2019) no. 2, pp. 333-367 | DOI | MR | Zbl
[35] Characterization of generalized Orlicz spaces, Commun. Contemp. Math., Volume 22 (2020) no. 2, 1850079, 25 pages | MR | Zbl
[36] Hardy spaces on homogeneous groups, Mathematical Notes, 28, Princeton University Press; University of Tokyo Press, 1982
[37] Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal., Volume 255 (2008) no. 12, pp. 3407-3430 | DOI | MR | Zbl
[38] Weighted norm inequalities and related topics, North-Holland Mathematics Studies, 116, North-Holland, 1985
[39] Classical Fourier analysis, Graduate Texts in Mathematics, 249, Springer, 2014
[40] Modern Fourier analysis, Graduate Texts in Mathematics, 250, Springer, 2014
[41] On some composition formulas, Proc. Am. Math. Soc., Volume 10 (1959), pp. 433-437 | DOI | MR | Zbl
[42] On limiting embeddings of Besov spaces, Stud. Math., Volume 171 (2005) no. 1, pp. 1-13 | DOI | MR | Zbl
[43] Fractional Sobolev norms and BV functions on manifolds, Nonlinear Anal., Theory Methods Appl., Volume 187 (2019), pp. 450-466 | DOI | MR | Zbl
[44] Characterizations of anisotropic high order Sobolev spaces, Asymptotic Anal., Volume 113 (2019) no. 4, pp. 239-260 | MR | Zbl
[45] A first course in Sobolev spaces, Graduate Studies in Mathematics, 105, American Mathematical Society, 2009
[46] Characterization of Sobolev and spaces, J. Funct. Anal., Volume 261 (2011) no. 10, pp. 2926-2958 | DOI | MR | Zbl
[47] Corrigendum to “Characterization of Sobolev and spaces” [J. Funct. Anal. 261 (10) (2011) 2926–2958], J. Funct. Anal., Volume 266 (2014) no. 2, pp. 1106-1114 | DOI | MR
[48] Interpolations and fractional Sobolev spaces in Carnot groups, Nonlinear Anal., Theory Methods Appl., Volume 179 (2019), pp. 91-104 | DOI | MR | Zbl
[49] 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] 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] 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] -convergence and Sobolev norms, C. R. Math. Acad. Sci. Paris, Volume 345 (2007) no. 12, pp. 679-684 | DOI | MR | Zbl
[53] Further characterizations of Sobolev spaces, J. Eur. Math. Soc., Volume 10 (2008) no. 1, pp. 191-229 | MR | Zbl
[54] -convergence, Sobolev norms, and BV functions, Duke Math. J., Volume 157 (2011) no. 3, pp. 495-533 | MR | Zbl
[55] On anisotropic Sobolev spaces, Commun. Contemp. Math., Volume 21 (2019) no. 1, 1850017, 13 pages | MR | Zbl
[56] The Mazʼya-Shaposhnikova limit in the magnetic setting, J. Math. Anal. Appl., Volume 449 (2017) no. 2, pp. 1152-1159 | DOI | MR | Zbl
[57] Magnetic BV-functions and the Bourgain-Brezis-Mironescu formula, Adv. Calc. Var., Volume 12 (2019) no. 3, pp. 225-252 | DOI | MR | Zbl
[58] An estimate in the spirit of Poincaré’s inequality, J. Eur. Math. Soc., Volume 6 (2004) no. 1, pp. 1-15 | DOI | Zbl
[59] A new approach to Sobolev spaces and connections to -convergence, Calc. Var. Partial Differ. Equ., Volume 19 (2004) no. 3, pp. 229-255 | DOI | MR | Zbl
[60] Elliptic PDEs, measures and capacities, EMS Tracts in Mathematics, 23, European Mathematical Society, 2016 | DOI
[61] 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] Fractional integrals and derivatives, Gordon and Breach Science Publishers, 1993 | Zbl
[63] theory for fractional gradient PDE with 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] Regularity for a fractional -Laplace equation, Commun. Contemp. Math., Volume 20 (2018) no. 1, 1750003, 6 pages | MR | Zbl
[65] An -type estimate for Riesz potentials, Rev. Mat. Iberoam., Volume 33 (2017) no. 1, pp. 291-303 | DOI | MR | Zbl
[66] On a new class of fractional partial differential equations, Adv. Calc. Var., Volume 8 (2015) no. 4, pp. 321-336 | MR | Zbl
[67] 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] 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] A noninequality for the fractional gradient, Port. Math., Volume 76 (2019) no. 2, pp. 153-168 | DOI | MR | Zbl
[70] An optimal Sobolev embedding for , J. Funct. Anal., Volume 279 (2020) no. 3, 108559, 26 pages | MR | Zbl
[71] Bourgain-Brézis-Mironescu formula for magnetic operators, C. R. Math. Acad. Sci. Paris, Volume 354 (2016) no. 8, pp. 825-831 | DOI | Zbl
[72] Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton University Press, 1970
[73] Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993
[74] Sobolev spaces, Colloq. Math., Volume 60/61 (1990) no. 1, pp. 129-139 | DOI | Zbl
[75] Estimates on translations and Taylor expansions in fractional Sobolev spaces, Nonlinear Anal., Theory Methods Appl., Volume 200 (2020), 111995, 12 pages | MR | Zbl
[76] Limits of Besov norms, Arch. Math., Volume 96 (2011) no. 2, pp. 169-175 | DOI | MR | Zbl
Cited by Sources:
Comments - Policy