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Acta Mathematica Sinica, English Series

, Volume 35, Issue 7, pp 1115–1127 | Cite as

Rearrangements in Carnot Groups

  • Juan J. ManfrediEmail author
  • Virginia N. Vera de SerioEmail author
Article
  • 12 Downloads

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 ‖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 \(W_{\rm{h}}^{1,p}\), then \({{{D_{\rm{h}}}{u^ \star }(x)} \over {\left| {{D_{\rm{h}}}(\left\| x \right\|)} \right|}}\) is in Lp. Moreover, we found a generalization of the inequality of Pólya and Szegö
$$\int {{{{{\left| {{D_{\rm{h}}}{u^ \star }} \right|}^p}} \over {{{\left| {{D_{\rm{h}}}(\left\| x \right\|)} \right|}^p}}}\;} dx \le C\;\int {{{\left| {{D_{\rm{h}}}u} \right|}^p}\;dx,} $$
where p ≥ 1.

Keywords

Symmetrization rearrangements Carnot groups 

MR(2010) Subject Classification

30C65 42B31 

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Notes

Acknowledgements

Part of this work was done while the first author visited the Universidad Nacional de Cuyo as a FOMEC visiting professor during the Summers of 1998 and 2000. He wishes to express his appreciation for the kind hospitality and the nice working atmosphere.

We did not submit this manuscript for publication because we were trying to proof the sharp version of Theorem 5; that is Cper = 1, which to the best of our knowledge remains open as of today. Nevertheless, the preprint was circulated among the community. We received several requests from colleagues, and it has been quoted several times in the literature. See for example the book [3], the dissertation [10], and the papers [4, 6, 11–13, 19], and [7]. In addition, the main results have not been published elsewhere. This is why we decided to submit this manuscript to publication. We are very thankful to Acta Mathematica Sinica, English Series for considering this manuscript.

We are thankful to the referees for the careful reading and thoughtful suggestions.

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Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PittsburghPittsburghUSA
  2. 2.Facultad de Ciencias EconómicasUniversidad Nacional de CuyoMendozaArgentina

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