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

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.

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

DOI: 10.5802/ambp.362
Keywords: Poincaré inequality, harmonic function, Riesz transform
Gilles Carron 1

1 Laboratoire de Mathématiques Jean Leray (UMR 6629), Université de Nantes, 2, rue de la Houssinière, B.P. 92208, 44322 Nantes Cedex 3, France
@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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.362/}
}
TY  - JOUR
AU  - Gilles Carron
TI  - Harmonic functions on Manifolds whose large spheres are small.
JO  - Annales mathématiques Blaise Pascal
PY  - 2016
DA  - 2016///
SP  - 249
EP  - 261
VL  - 23
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.362/
UR  - https://doi.org/10.5802/ambp.362
DO  - 10.5802/ambp.362
LA  - en
ID  - AMBP_2016__23_2_249_0
ER  - 
%0 Journal Article
%A Gilles Carron
%T Harmonic functions on Manifolds whose large spheres are small.
%J Annales mathématiques Blaise Pascal
%D 2016
%P 249-261
%V 23
%N 2
%I Annales mathématiques Blaise Pascal
%U https://doi.org/10.5802/ambp.362
%R 10.5802/ambp.362
%G en
%F AMBP_2016__23_2_249_0
Gilles Carron. Harmonic functions on Manifolds whose large spheres are small.. Annales mathématiques Blaise Pascal, Volume 23 (2016) no. 2, pp. 249-261. doi : 10.5802/ambp.362. https://ambp.centre-mersenne.org/articles/10.5802/ambp.362/

[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 | DOI

[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 | DOI

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

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

[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 | DOI

[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 | DOI

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

[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 | DOI

[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 | DOI

[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 | DOI

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

Cited by Sources: