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作者（2019）在《Rearrangements in Carnot Groups》一文中研究指出：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.
Abstract
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.
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