no. 6
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

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].

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.

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

[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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