In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br , or equivalently with respe...In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br , or equivalently with respect to a gauge‖x‖, and prove basic regularity properties of this construction. If u is a bounded nonnegative real function with compact support, we denote by u*its rearrangement. Then, the radial function u* is of bounded variation. In addition, if u is continuous then u* is continuous, and if u belongs to the horizontal Sobolev space W 1,ph , then Dhu*(x)/Dh( ‖x‖ )| is in Lp. Moreover, we found a generalization of the inequality of P(o)lya and Szeg(o) ∫|Dhu*|p/Dh(‖x‖)|pdx≤C ∫|Dhu|pdx,where p ≥ 1.展开更多
In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carn...In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction （projection） of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and （intrinsic） regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.展开更多
基金supported in part by NSF(Grant No.DMS-9970687)SECTyP-UNCuyo,Argentina(Res.3853/16-R)
文摘In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls Br , or equivalently with respect to a gauge‖x‖, and prove basic regularity properties of this construction. If u is a bounded nonnegative real function with compact support, we denote by u*its rearrangement. Then, the radial function u* is of bounded variation. In addition, if u is continuous then u* is continuous, and if u belongs to the horizontal Sobolev space W 1,ph , then Dhu*(x)/Dh( ‖x‖ )| is in Lp. Moreover, we found a generalization of the inequality of P(o)lya and Szeg(o) ∫|Dhu*|p/Dh(‖x‖)|pdx≤C ∫|Dhu|pdx,where p ≥ 1.
基金supported by the National Natural Science Foundation of China(No.10471063)
文摘In this paper we give a geometric interpretation of the notion of the horizontal mean curvature which is introduced by Danielli Garofalo-Nhieu and Pauls who recently introduced sub- Riemannian minimal surfaces in Carnot groups. This will be done by introducing a natural nonholonomic connection which is the restriction （projection） of the natural Riemannian connection on the horizontal bundle. For this nonholonomic connection and （intrinsic） regular hypersurfaces we introduce the notions of the horizontal second fundamental form and the horizontal shape operator. It turns out that the horizontal mean curvature is the trace of the horizontal shape operator.
基金Supported by National Natural Science Foundation of China(11001130)Fundamental Research Funds for the Central Universities(30917011335)Scientific Research Innovation Project of Jiangsu Province(KYCX17-0327)。
