no. 7
Riesz transforms on connected sums
[Transformée de Riesz sur les sommes connexes]
Carron, Gilles1
1 Université de Nantes Laboratoire de Mathématiques Jean Leray (UMR 6629) 2, rue de la Houssinière B.P. 92208 44322 Nantes Cedex 3 (France)
Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2329-2343

Assume that M 0 is a complete Riemannian manifold with Ricci curvature bounded from below and that M 0 satisfies a Sobolev inequality of dimension ν>3. Let M be a complete Riemannian manifold isometric at infinity to M 0 and let p∈(ν/(ν-1),ν). The boundedness of the Riesz transform of L p (M 0 ) then implies the boundedness of the Riesz transform of L p (M)

Soit M 0 une variété riemannienne complète à courbure de Ricci bornée inférieurement et qui vérifie l’inégalité Sobolev de dimension ν>3. Si M est une variété riemannienne complète isométrique à M 0 en dehors d’un compact et si p∈(ν/(ν-1),ν) alors lorsque la transformée de Riesz est bornée sur L p (M 0 ) elle est également bornée sur L p (M).

DOI : 10.5802/aif.2334
Classification : 58J37, 58J35, 42B20
Keywords: Riesz transform, Sobolev inequalities
Mots-clés : transformée de Riesz, inégalités de Sobolev

Carron, Gilles 1

1 Université de Nantes Laboratoire de Mathématiques Jean Leray (UMR 6629) 2, rue de la Houssinière B.P. 92208 44322 Nantes Cedex 3 (France) 
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Carron, Gilles. Riesz transforms on connected sums. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2329-2343. doi: 10.5802/aif.2334

[1] Alexopoulos, G. An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math., Volume 44 (1992), pp. 691-727 | DOI | Zbl | MR

[2] Anker, J.-Ph. Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J., Volume 65 (1992) no. 2, pp. 257-297 | DOI | Zbl | MR

[3] Bakry, D. Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Lecture Notes in Math., Volume 1247 (1987), pp. 137-172 (Séminaire de Probabilités, XXI) | DOI | Zbl | MR | Numdam

[4] Carron, G. Une suite exacte en L 2 -cohomologie, Duke Math. J., Volume 95 (1998), pp. 343-372 | DOI | Zbl | MR

[5] Carron, G.; Coulhon, Th.; Hassell, A. Riesz transform for manifolds with Euclidean ends (to appear in Duke Math. Journal) | Zbl

[6] Coulhon, Th.; Dungey, N. Riesz transform and perturbation (2006) (preprint) | Zbl | MR

[7] Coulhon, Th.; Duong, X. T. Riesz transforms for 1≤p≤2, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 1151-1169 | DOI | Zbl | MR

[8] Coulhon, Th.; Duong, X. T. Riesz transform and related inequalities on non-compact Riemannian manifolds, Comm. in Pure and Appl. Math., Volume 56 (2003) no. 12, pp. 1728-1751 | DOI | Zbl | MR

[9] Coulhon, Th.; Saloff-Coste, L.; Varopoulos, N. Th. Analysis and geometry on groups, Cambridge Tracts in Mathematics, Volume 100, Cambridge University Press, Cambridge, 1992 | Zbl | MR

[10] Davies, E. B. Pointwise bounds on the space and time derivatives of heat kernels, Operator Theory, Volume 21 (1989) no. 2, pp. 367-378 | Zbl | MR

[11] Grigor’yan, A. Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana, Volume 10 (1994) no. 2, pp. 395-452 | Zbl

[12] Grigor’yan, A.; Saloff-Coste, L. Heat kernel on connected sums of Riemannian manifolds, Math. Res. Lett., Volume 6 (1999) no. 3-4, pp. 307-321 | Zbl

[13] Grigor’yan, A.; Saloff-Coste, L. Stability results for Harnack inequalities, Ann. Inst. Fourier, Volume 55 (2005) no. 3, pp. 825-890 | DOI | Zbl | Numdam

[14] Hebey, E. Optimal Sobolev inequalities on complete Riemannian manifolds with Ricci curvature bounded below and positive injectivity radius, Amer. J. Math., Volume 118 (1996) no. 2, pp. 291-300 | DOI | Zbl | MR

[15] Komatsu, H. Fractional powers of operators, Pacific J. Math., Volume 19 (1966), pp. 285-346 | Zbl | MR

[16] Li, H.-Q. La transformation de Riesz sur les variétés coniques, J. Funct. Anal., Volume 168 (1999) no. 1, pp. 145-238 | DOI | Zbl | MR

[17] Lohoué, N. Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemannienne à courbure non positive, J. Funct. Anal., Volume 61 (1985) no. 2, pp. 164-201 | DOI | Zbl | MR

[18] Varopoulos, N. Th. Hardy-Littlewood theory for semigroups, J. Funct. Anal., Volume 63 (1985) no. 2, pp. 240-260 | DOI | Zbl | MR

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