Riesz transforms on connected sums
Annales de l'Institut Fourier, Volume 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
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)
@article{AIF_2007__57_7_2329_0,
     author = {Carron, Gilles},
     title = {Riesz transforms on connected sums},
     journal = {Annales de l'Institut Fourier},
     pages = {2329--2343},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     doi = {10.5802/aif.2334},
     mrnumber = {2394543},
     zbl = {1139.58020},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2334/}
}
TY  - JOUR
AU  - Carron, Gilles
TI  - Riesz transforms on connected sums
JO  - Annales de l'Institut Fourier
PY  - 2007
DA  - 2007///
SP  - 2329
EP  - 2343
VL  - 57
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2334/
UR  - https://www.ams.org/mathscinet-getitem?mr=2394543
UR  - https://zbmath.org/?q=an%3A1139.58020
UR  - https://doi.org/10.5802/aif.2334
DO  - 10.5802/aif.2334
LA  - en
ID  - AIF_2007__57_7_2329_0
ER  - 
%0 Journal Article
%A Carron, Gilles
%T Riesz transforms on connected sums
%J Annales de l'Institut Fourier
%D 2007
%P 2329-2343
%V 57
%N 7
%I Association des Annales de l’institut Fourier
%U https://doi.org/10.5802/aif.2334
%R 10.5802/aif.2334
%G en
%F AIF_2007__57_7_2329_0
Carron, Gilles. Riesz transforms on connected sums. Annales de l'Institut Fourier, Volume 57 (2007) no. 7, pp. 2329-2343. doi : 10.5802/aif.2334. https://aif.centre-mersenne.org/articles/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 | MR | Zbl

[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 | MR | Zbl

[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 | Numdam | MR | Zbl

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

[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) | MR | Zbl

[7] Coulhon, Th.; Duong, X. T. Riesz transforms for 1p2, Trans. Amer. Math. Soc., Volume 351 (1999), pp. 1151-1169 | DOI | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[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 | MR | Zbl

[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 | Numdam | Zbl

[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 | MR | Zbl

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

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

[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 | MR | Zbl

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

Cited by Sources: