Partial Differential Equations
Inequalities related to liftings and applications
Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 957-962.

We present two inequalities for liftings of smooth maps from the torus Td into the unit circle S1. Using these inequalities, we answer a question of J. Bourgain, H. Brezis, and P. Mironescu in [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Comm. Pure Appl. Math. 58 (2005) 529–551] and establish an estimate of liftings in the spirit of R.R. Coifman and Y. Meyer [R.R. Coifman, Y. Meyer, Une généralisation du théorème de Calderon sur l'intégrale de Cauchy, in: Fourier Analysis, in: Proc. Sem., El Escorial, vol. 1, Asoc. Mat. Espa nola, Madrid, 1980, pp. 87–116].

Nous présentons deux inégalités pour des relèvements des applications régulières du tore Td dans le cercle unité S1. Ces inégalités nous permettent de répondre à une question de J. Bourgain, H. Brezis, et P. Mironescu dans [J. Bourgain, H. Brezis, P. Mironescu, Lifting, degree, and distributional Jacobian revisited, Comm. Pure Appl. Math. 58 (2005) 529–551] et d'établir une estimation des relèvements dans l'esprit de R.R. Coifman et Y. Meyer [R.R. Coifman, Y. Meyer, Une généralisation du théorème de Calderon sur l'intégrale de Cauchy, in : Fourier Analysis, in : Proc. Sem., El Escorial, vol. 1, Asoc. Mat. Espa nola, Madrid, 1980, pp. 87–116].

Accepted:
Published online:
DOI: 10.1016/j.crma.2008.07.026
Nguyen, Hoai-Minh 1

1 Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
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Nguyen, Hoai-Minh. Inequalities related to liftings and applications. Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 957-962. doi : 10.1016/j.crma.2008.07.026. http://www.numdam.org/articles/10.1016/j.crma.2008.07.026/

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[6] Coifman, R.R.; Meyer, Y. Une généralisation du théorème de Calderon sur l'intégrale de Cauchy, Fourier Analysis, Proc. Sem., El Escorial, vol. 1, Asoc. Mat. Espa nola, Madrid, 1980, pp. 87-116

[7] Mironescu, P. Sobolev maps on manifolds: degree, approximation, lifting, Perspectives in Nonlinear Partial Differential Equations, Contemp. Math., vol. 446, Amer. Math. Soc., Providence, RI, 2007, pp. 413-436 (In honor of Ham Brezis)

[8] Mironescu, P. Lifting default for S1-valued maps, C. R. Acad. Sci. Paris, Ser. I, Volume 346 (2008) | DOI

[9] Nguyen, H.-M. Optimal constant in a new estimate for the degree, J. d'Analyse Math., Volume 101 (2007), pp. 367-395

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