Rearrangements in Carnot Groups  

作　　者：Juan J. MANFREDI Virginia N. VERA DE SERIO 

机构地区：[1]Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA [2]Facultad de Ciencias Econdmicas, Universidad Nacional de Cuyo, 5500 Mendoza, Argentina

出　　处：《Acta Mathematica Sinica,English Series》2019年第7期1115-1127,共13页数学学报（英文版）

基　　金：supported in part by NSF(Grant No.DMS-9970687);SECTyP-UNCuyo,Argentina(Res.3853/16-R)

摘　　要：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.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 ■,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 Wh1,p, then ■ is in Lp. Moreover, we found a generalization of the inequality of Pólya and Szeg? ■,where p ≥ 1.

关 键 词：SYMMETRIZATION REARRANGEMENTS Carnot GROUPS 

分 类 号：O1[理学—数学]