Partial differential equations/Harmonic analysis
Fractional Laplacians, extension problems and Lie groups
[Laplaciens fractionnaires, problèmes d'extension et groupes de Lie]

Présenté par : Haïm Brézis

Chamorro, Diego1 ; Jarrín, Oscar1
1 Laboratoire de mathématiques et modélisation d'Évry (LaMME), UMR 8071, Université d'Évry-Val-d'Essonne, 23, boulevard de France, 91037 Évry cedex, France
Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 517-522.

Nous généralisons aux groupes de Lie nilpotents les travaux de Caffarelli & Silvestre [1] et Stinga & Torrea [7] concernant la relation existant entre les puissances fractionnaires de l'opérateur laplacien et les solutions d'une équation aux dérivées partielles.

We generalize some results concerning the fractional powers of the Laplace operator to the setting of nilpotent Lie Groups and we study its relationship with the solutions to a partial differential equation in the spirit of the articles of Caffarelli & Silvestre [1] and Stinga & Torrea [7].

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.04.007
Chamorro, Diego 1 ; Jarrín, Oscar 1

1 Laboratoire de mathématiques et modélisation d'Évry (LaMME), UMR 8071, Université d'Évry-Val-d'Essonne, 23, boulevard de France, 91037 Évry cedex, France 
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Chamorro, Diego; Jarrín, Oscar. Fractional Laplacians, extension problems and Lie groups. Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 517-522. doi : 10.1016/j.crma.2015.04.007. http://archive.numdam.org/articles/10.1016/j.crma.2015.04.007/

[1] Caffarelli, L.; Silvestre, L. An extension problem related to the fractional Laplacian, Commun. Partial Differ. Equ., Volume 32 (2007), pp. 1245-1260

[2] Ferrari, F.; Franchi, B. Hanarck inequality for fractional sub-Laplacians in Carnot groups, Math. Z., Volume 279 (2015), pp. 435-458

[3] Frank, R.; González, M.D.M.; Monticelli, D.D.; Tan, J. An extension problem for the CR fractional Laplacian, Adv. Math., Volume 270 (2015), pp. 97-137

[4] Furioli, G.; Melzi, C.; Veneruso, A. Littlewood–Paley decomposition and Besov spaces on Lie groups of polynomial growth, Math. Nachr., Volume 279 (2006) no. 9–10, pp. 1028-1040

[5] Galé, J.E.; Miana, P.J.; Stinga, P.R. Extension problems and fractional operators: semi-groups and wave equations | arXiv

[6] Kemmppainen, M.; Sjögren, S.; Torrea, J.L. Wave extension problem for the fractional Laplacian, J. Evol. Equ., Volume 2 (2014), pp. 343-368

[7] Stinga, P.; Torrea, J. Extension problem and Harnack's inequality for some fractional operators, Commun. Partial Differ. Equ., Volume 35 (2010) no. 11, pp. 2092-2122

[8] Varopoulos, N.T.; Saloff-Coste, L.; Coulhon, T. Analysis and Geometry on Groups, Cambridge Tracts in Mathematics, vol. 100, 1992

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