no. G6
Analyse fonctionnelle, Analyse harmonique
Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation
Van Schaftingen, Jean1 
1 Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 1041-1049

We prove Gagliardo–Nirenberg interpolation inequalities estimating the Sobolev semi-norm in terms of the bounded mean oscillation semi-norm and of a Sobolev semi-norm, with some of the Sobolev semi-norms having fractional order.

Reçu le :
Accepté le :
Publié le :
DOI : 10.5802/crmath.463
Classification : 26D10, 35A23, 42B35, 46B70, 46E35

Van Schaftingen, Jean 1

1 Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{CRMATH_2023__361_G6_1041_0,
     author = {Van Schaftingen, Jean},
     title = {Fractional {Gagliardo{\textendash}Nirenberg} interpolation inequality and bounded mean oscillation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1041--1049},
     year = {2023},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     number = {G6},
     doi = {10.5802/crmath.463},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/crmath.463/}
}
TY  - JOUR
AU  - Van Schaftingen, Jean
TI  - Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 1041
EP  - 1049
VL  - 361
IS  - G6
PB  - Académie des sciences, Paris
UR  - https://www.numdam.org/articles/10.5802/crmath.463/
DO  - 10.5802/crmath.463
LA  - en
ID  - CRMATH_2023__361_G6_1041_0
ER  - 
%0 Journal Article
%A Van Schaftingen, Jean
%T Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation
%J Comptes Rendus. Mathématique
%D 2023
%P 1041-1049
%V 361
%N G6
%I Académie des sciences, Paris
%U https://www.numdam.org/articles/10.5802/crmath.463/
%R 10.5802/crmath.463
%G en
%F CRMATH_2023__361_G6_1041_0
Van Schaftingen, Jean. Fractional Gagliardo–Nirenberg interpolation inequality and bounded mean oscillation. Comptes Rendus. Mathématique, Tome 361 (2023) no. G6, pp. 1041-1049. doi: 10.5802/crmath.463

[1] Acerbi, Emilio; Fusco, Nicola Semicontinuity problems in the calculus of variations, Arch. Ration. Mech. Anal., Volume 86 (1984) no. 2, pp. 125-145 | DOI | MR | Zbl

[2] Adams, David R.; Frazier, Michael Composition operators on potential spaces, Proc. Am. Math. Soc., Volume 114 (1992) no. 1, pp. 155-165 | DOI | MR | Zbl

[3] Bojarski, Bogdan Remarks on some geometric properties of Sobolev mappings, Functional analysis and related topics (Koshi, Shozo, ed.), World Scientific, 1991, pp. 65-76 | Zbl

[4] Brezis, Haïm; Mironescu, Petru Gagliardo–Nirenberg, composition and products in fractional Sobolev spaces, J. Evol. Equ., Volume 1 (2001) no. 4, pp. 387-404 | DOI | MR | Zbl

[5] Brezis, Haïm; Mironescu, Petru Gagliardo–Nirenberg inequalities and non-inequalities: the full story, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 35 (2018) no. 5, pp. 1355-1376 | DOI | MR | Zbl

[6] Brezis, Haïm; Mironescu, Petru Sobolev maps to the circle. From the perspective of analysis, geometry, and topology, Progress in Nonlinear Differential Equations and their Applications, 96, Birkhäuser, 2021

[7] Carleson, Lennart Two remarks on H 1 and BMO, Adv. Math., Volume 22 (1976) no. 3, pp. 269-277 | DOI | MR | Zbl

[8] Chen, Jiecheng; Zhu, Xiangrong A note on BMO and its application, J. Math. Anal. Appl., Volume 303 (2005) no. 2, pp. 696-698 | DOI | Zbl | MR

[9] Cohen, Albert Ondelettes, espaces d’interpolation et applications, Sémin. Équ. Dériv. Partielles, Volume 1999-2000 (2000), I, 14 pages | Numdam | Zbl | MR

[10] Cohen, Albert; Dahmen, Wolfgang; Daubechies, Ingrid; DeVore, Ronald Harmonic analysis of the space BV, Rev. Mat. Iberoam., Volume 19 (2003) no. 1, pp. 235-263 | DOI | Zbl | MR

[11] Cohen, Albert; Meyer, Yves; Oru, Frédérique Improved Sobolev embedding theorem, Sémin. Équ. Dériv. Partielles, Volume 1997-1998 (1998), XVI, 16 pages | Numdam | Zbl

[12] Dao, Nguyen Anh Gagliardo–Nirenberg type inequalities using fractional Sobolev spaces and Besov spaces (2022) | arXiv

[13] Fefferman, Charles; Stein, Elias M. H p spaces of several variables, Acta Math., Volume 129 (1972) no. 3-4, pp. 137-193 | DOI | Zbl

[14] Fiorenza, Alberto; Formica, Maria Rosaria; Roskovec, Tomáš G.; Soudský, Filip Detailed proof of classical Gagliardo-Nirenberg interpolation inequality with historical remarks, Z. Anal. Anwend., Volume 40 (2021) no. 2, pp. 217-236 | DOI | Zbl | MR

[15] Gagliardo, Emilio Ulteriori proprietà di alcune classi di funzioni in più variabili, Ric. Mat., Volume 8 (1959), pp. 24-51 | Zbl

[16] Hajłasz, Piotr Sobolev spaces on an arbitrary metric space, Potential Anal., Volume 5 (1996) no. 4, pp. 403-415 | DOI | MR | Zbl

[17] Jabin, Pierre-Emmanuel Differential equations with singular fields, J. Math. Pures Appl., Volume 94 (2010) no. 6, pp. 597-621 | DOI | MR | Zbl

[18] John, Fritz; Nirenberg, Louis On functions of bounded mean oscillation, Commun. Pure Appl. Math., Volume 14 (1961), pp. 415-426 | DOI | MR | Zbl

[19] Kałamajska, Agnieszka Pointwise multiplicative inequalities and Nirenberg type estimates in weighted Sobolev spaces, Stud. Math., Volume 108 (1994) no. 3, pp. 275-290 | DOI | MR | Zbl

[20] Kozono, Hideo; Wadade, Hidemitsu Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO, Math. Z., Volume 259 (2008) no. 4, pp. 935-950 | DOI | MR | Zbl

[21] Liu, Fon Che A Luzin type property of Sobolev functions, Indiana Univ. Math. J., Volume 26 (1977) no. 4, pp. 645-651 | DOI | MR | Zbl

[22] Lokharu, E. È. Gagliardo–Nirenberg inequality for maximal functions that measure smoothness, Zap. Nauchn. Semin. (POMI), Volume 389 (2011), pp. 143-161 | Zbl

[23] Mazʼya, Vladimir; Shaposhnikova, Tatyana On pointwise interpolation inequalities for derivatives, Math. Bohem., Volume 124 (1999) no. 2-3, pp. 131-148 | MR | Zbl

[24] Meyer, Yves; Rivière, Tristan A partial regularity result for a class of stationary Yang–Mills fields in high dimension, Rev. Mat. Iberoam., Volume 19 (2003) no. 1, pp. 195-219 | DOI | MR | Zbl

[25] Miyazaki, Yoichi A short proof of the Gagliardo–Nirenberg inequality with BMO term, Proc. Am. Math. Soc., Volume 148 (2020) no. 10, pp. 4257-4261 | DOI | MR | Zbl

[26] Nirenberg, Louis On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 13 (1959), pp. 115-162 | MR | Zbl

[27] Stein, Elias M. Singular integrals and differentiability properties of functions, Princeton Mathematical Series, 30, Princeton University Press, 1970, xiv+290 pages

[28] Strzelecki, Pawel Gagliardo–Nirenberg inequalities with a BMO term, Bull. Lond. Math. Soc., Volume 38 (2006) no. 2, pp. 294-300 | DOI | MR | Zbl

Cité par Sources :