Revista Matemática Iberoamericana


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Volume 30, Issue 1, 2014, pp. 109–131
DOI: 10.4171/RMI/771

Published online: 2014-03-23

Sobolev, Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality

Fabrice Baudoin[1] and Bumsik Kim[2]

(1) Purdue University, West Lafayette, USA
(2) Purdue University, West Lafayette, USA

By adapting some ideas of M. Ledoux ([12], [13] and [14]) to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and N. Garofalo in [3]. Our results apply in particular on all CR Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups with step two, and wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.

Keywords: Sobolev inequalities, isoperimetric inequality, Poincaré inequality, subelliptic operator

Baudoin Fabrice, Kim Bumsik: Sobolev, Poincaré, and isoperimetric inequalities for subelliptic diffusion operators satisfying a generalized curvature dimension inequality. Rev. Mat. Iberoamericana 30 (2014), 109-131. doi: 10.4171/RMI/771