On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to $ {L}^{1}$ vector fields

Petru Mironescu

doi:10.1016/j.crma.2010.03.019

Mathematical Analysis

On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to

L^{1}

vector fields

Presented by: Haïm Brezis

Petru Mironescu ¹

¹ Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille-Jordan, bâtiment du Doyen Jean-Braconnier, 43, boulevard du 11 novembre 1918, 69200 Villeurbanne cedex, France

Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 513-515.

Abstract (Original language)
Résumé

Bourgain and Brezis established, for maps $f \in L^{n} (T^{n})$ with zero average, the existence of a solution $Y \in W^{1, n} \cap L^{\infty}$ of (1) $div Y = f$ . Maz'ya proved that if, in addition, $f \in H^{n / 2 - 1} (T^{n})$ , then (1) can be solved in $H^{n / 2} \cap L^{\infty}$ . Their arguments are quite different. We present an elementary property of fundamental solutions of the biharmonic operator in two dimensions. This property unifies, in two dimensions, the two approaches, and implies another (apparently unrelated) estimate of Maz'ya and Shaposhnikova. We discuss higher dimensional analogs of the above results.

Bourgain and Brezis ont montré que, si $f \in L^{n} (T^{n})$ est de moyenne nulle, alors (1) $div Y = f$ a une solution $Y \in W^{1, n} \cap C^{0}$ . Maz'ya a prouvé que si, de plus, on a $f \in H^{n / 2 - 1} (T^{n})$ , alors il existe une solution de (1) dans $H^{n / 2} \cap L^{\infty}$ . Les deux preuves sont distinctes. Dans cette note, nous présentons une propriété élémentaire des solutions fondamentales de l'opérateur biharmonique en dimension deux. Cette propriété unifie, en dimension deux, les approches de Bourgain–Brezis et Maz'ya, et implique une autre estimation de Maz'ya et Shaposhnikova (apparemment non liée aux précédentes). Nous discutons des variantes de ces résultats en dimension supérieure.

Accepted: 2010-03-23
Published online: 2010-04-22

DOI: 10.1016/j.crma.2010.03.019

Author's affiliations:

Petru Mironescu ¹

¹ Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille-Jordan, bâtiment du Doyen Jean-Braconnier, 43, boulevard du 11 novembre 1918, 69200 Villeurbanne cedex, France

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     title = {On some inequalities of {Bourgain,} {Brezis,} {Maz'ya,} and {Shaposhnikova} related to $ {L}^{1}$ vector fields},
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Petru Mironescu. On some inequalities of Bourgain, Brezis, Maz'ya, and Shaposhnikova related to $ {L}^{1}$ vector fields. Comptes Rendus. Mathématique, Volume 348 (2010) no. 9-10, pp. 513-515. doi : 10.1016/j.crma.2010.03.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.03.019/

References
Cited by

[1] J. Bourgain; H. Brezis On the equation $div Y = f$ and application to control of phases, J. Amer. Math. Soc., Volume 16 (2003), pp. 393-426

[2] J. Bourgain; H. Brezis New estimates for the Laplacian, the div-curl, and related Hodge systems, C. R. Acad. Sci. Paris, Ser. I, Volume 338 (2004), pp. 539-543 (393–426)

[3] J. Bourgain; H. Brezis New estimates for elliptic equations and Hodge type systems, J. Eur. Math. Soc., Volume 9 (2007), pp. 277-315

[4] V. Maz'ya Bourgain–Brezis type inequality with explicit constants (L. De Carli; M. Milman, eds.), Interpolation Theory and Applications, Contemp. Math., vol. 445, AMS, Providence, RI, 2007, pp. 247-252

[5] V. Maz'ya Estimates for differential operators of vector analysis involving $L^{1}$ -norm, J. Eur. Math. Soc., Volume 12 (2010), pp. 221-240

[6] V. Maz'ya; T. Shaposhnikova A collection of sharp dilation invariant integral inequalities for differentiable functions (V. Maz'ya, ed.), Sobolev Spaces in Mathematics I, Int. Math. Ser. (N. Y.), vol. 8, Springer, New York, 2009, pp. 223-247

[7] E. Stein; G. Weiss Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, 1971

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