Harmonic functions on Manifolds whose large spheres are small.
Annales Mathématiques Blaise Pascal, Tome 23 (2016) no. 2, pp. 249-261.

On étudie la croissance des fonctions harmoniques sur les variétés riemanniennes complètes dont le diamètre des grandes sphères géodésiques croît sous linéairement. Il s’agit d’une généralisation de travaux de A. Kasue. Nous obtenons aussi un résultat de continuité pour la transformée de Riesz

We study the growth of harmonic functions on complete Riemannian manifolds where the extrinsic diameter of geodesic spheres is sublinear. It is an generalization of a result of A. Kasue. Our estimates also yields a result on the boundedness of the Riesz transform.

DOI : https://doi.org/10.5802/ambp.362
Mots clés: Inégalités de Poincaré , fonctions harmoniques, transformée de Riesz.
@article{AMBP_2016__23_2_249_0,
     author = {Gilles Carron},
     title = {Harmonic functions on Manifolds whose large spheres are small.},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {249--261},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {23},
     number = {2},
     year = {2016},
     doi = {10.5802/ambp.362},
     language = {en},
     url = {ambp.centre-mersenne.org/item/AMBP_2016__23_2_249_0/}
}
Gilles Carron. Harmonic functions on Manifolds whose large spheres are small.. Annales Mathématiques Blaise Pascal, Tome 23 (2016) no. 2, pp. 249-261. doi : 10.5802/ambp.362. https://ambp.centre-mersenne.org/item/AMBP_2016__23_2_249_0/

[1] Pascal Auscher; Thierry Coulhon Riesz transform on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 4 (2005) no. 3, pp. 531-555

[2] Dominique Bakry Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Séminaire de Probabilités, XXI (Lecture Notes in Math.) Volume 1247, Springer, Berlin, 1987, pp. 137-172 | Article

[3] Peter Buser A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982) no. 2, pp. 213-230

[4] Shiu-Yuen Cheng; Shing-Tung Yau Differential equations on Riemannian manifolds and their geometric applications, Comm. Pure Appl. Math., Volume 28 (1975) no. 3, pp. 333-354 | Article

[5] Tobias H. Colding; William P. Minicozzi II Harmonic functions on manifolds, Ann. Math. (2), Volume 146 (1997) no. 3, pp. 725-747 | Article

[6] Tobias H. Colding; William P. Minicozzi II Liouville theorems for harmonic sections and applications, Comm. Pure Appl. Math., Volume 51 (1998) no. 2, pp. 113-138 | Article

[7] Carron Gilles Riesz transform on manifolds with quadratic curvature decay (2014) (https://arxiv.org/abs/1403.6278, to appear in Rev. Mat. Iberoam.)

[8] Alexander Grigor’yan; Laurent Saloff-Coste Stability results for Harnack inequalities, Ann. Inst. Fourier, Volume 55 (2005) no. 3, pp. 825-890 http://aif.cedram.org/item?id=AIF_2005__55_3_825_0 | Article

[9] Atsushi Kasue Harmonic functions with growth conditions on a manifold of asymptotically nonnegative curvature. I, Geometry and analysis on manifolds (Katata/Kyoto, 1987) (Lecture Notes in Math.) Volume 1339, Springer, Berlin, 1988, pp. 158-181 | Article

[10] Atsushi Kasue Harmonic functions of polynomial growth on complete manifolds. II, J. Math. Soc. Japan, Volume 47 (1995) no. 1, pp. 37-65 | Article

[11] Peter Li Harmonic functions of linear growth on Kähler manifolds with nonnegative Ricci curvature, Math. Res. Lett., Volume 2 (1995) no. 1, pp. 79-94 | Article

[12] John Lott; Zhongmin Shen Manifolds with quadratic curvature decay and slow volume growth, Ann. Sci. École Norm. Sup. (4), Volume 33 (2000) no. 2, pp. 275-290 | Article

[13] Christina Sormani Harmonic functions on manifolds with nonnegative Ricci curvature and linear volume growth, Pacific J. Math., Volume 192 (2000) no. 1, pp. 183-189 | Article

[14] Elias M. Stein Singular integrals and differentiability properties of functions, Princeton Mathematical Series, Volume 30, Princeton University Press, Princeton, N.J., 1970, xiv+290 pages