Rearrangements in Carnot Groups
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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 ■,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_h~(1,p), then ■ is in L~p. Moreover, we found a generalization of the inequality of Pólya and Szeg? ■,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 W_h~(1,p), then ■ is in L~p. Moreover, we found a generalization of the inequality of Pólya and Szeg? ■,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 W_h~(1,p), then ■ is in L~p. Moreover, we found a generalization of the inequality of Pólya and Szeg? ■,where p ≥ 1.
引文
[1]Baernstein,A.:Symmetrization in Analysis,Cambridge University Press,Cambridge,2019
[2]Bella?che,Andre,Risler,Jean-Jacques,et al.:Sub-Riemannian Geometry,Progress in Mathematics,Vol.144,Birkhauser,Basel,1996
[3]Capogna,L.,Danielli,D.,Pauls,S.,et al.:An Introduction to the Heisenberg Group and the SubRiemannian Isoperimetric Problem,Progress in Mathematics,Vol.259,Birkhauser,Basel,2007
[4]Cohn,W.,Lam,N.,Lu,G.,et al.:The Moser-Trudinger inequality in unbounded domains of Heisenberg group and sub-elliptic equations.Nonlinear Analysis:Theory,Methods&Applications,75(12),4483-4495,(2012)
[5]Franchi,B.,Gallot,S.,Wheeden,R.:Sobolev and isoperimetric inequalities for degenerate metrics.Math.Ann.,300(4),557-571(1994)
[6]Francu,M.:Higher-order Sobolev-type embeddings on Carnot-Caratheodory spaces.Mathematische Nachrichten,290(7),1033-1052(2017)
[7]Francu,M.:Higher-order compact embeddings of function spaces on Carnot-Caratheodory spaces.Banach J.Math.Anal.,12(4),970-994(2018)
[8]Folland,G.B.,Stein,E.M.:Hardy Spaces on Homogeneous Groups,Princeton University Press,Princeton,NJ,1982
[9]Garofalo,N.,Nhieu,D.,Isoperimetric and Sobolev Inequalities for Carnot-Caratheodory Spaces and the Existence of Minimal Surfaces,Communications on Pure and Applied Mathematics,Vol.XLIX,John Wiley and Sons,pp.1081-1144,1996
[10]Lam,N.:Moser-Trudinger and Adams Type Inequalities and Their Applications,Doctoral Dissertation,Wayne State University,https://digitalcommons.wayne.edu/oa-dissertations/1016/,2015
[11]Lam,N.,Lu,G.,Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications.Advances in Mathematics,231(6),3259-3287(2012)
[12]Lam,N.,Lu,G.,Tang,H.:On nonuniformly subelliptic equations of Q-sub-Laplacian type with critical growth in the Heisenberg group.Advances in Nonlinear Studies,12(3),659-681(2012)
[13]Li,J.,Lu,G.,Zhou,M.:Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions.Calc.Var.Partial Differential Equations,57(3),Art.84,26pp.(2018)
[14]Lu,G.:Weighted poincare and Sobolev inequalities for vector fields satisfying Hormander’s condition.Rev.Matematica Iberoamericana,8(3),367-439(1992)
[15]Monti,R.,Serra Cassano,F.:Surface measure in Carnot-Caratheodory spaces.Calc.Var.Partial Differential Equations,13(3),339-376(2001)
[16]Polya,G.,Szego,G.:Isoperimetric Inequalities in Mathematicsl Physics,Ann.Math.Studies 27,Princeton University Press,Princeton,1951
[17]Schulz,F.,Vera de Serio,V.:Symmetrization with respect to a measure.Transactions of the AMS,337(1),195-210(1993)
[18]Stein,E.M.:Harmonic Analysis,Princeton University Press,Princeton,NJ,1993
[19]Tyson,J.:Sharp weighted Young’s inequalities and Moser-Trudinger inequalities on Heisenberg type groups and Grushin spaces.Potential Analysis,4(24),357-384(2006)
[2]Bella?che,Andre,Risler,Jean-Jacques,et al.:Sub-Riemannian Geometry,Progress in Mathematics,Vol.144,Birkhauser,Basel,1996
[3]Capogna,L.,Danielli,D.,Pauls,S.,et al.:An Introduction to the Heisenberg Group and the SubRiemannian Isoperimetric Problem,Progress in Mathematics,Vol.259,Birkhauser,Basel,2007
[4]Cohn,W.,Lam,N.,Lu,G.,et al.:The Moser-Trudinger inequality in unbounded domains of Heisenberg group and sub-elliptic equations.Nonlinear Analysis:Theory,Methods&Applications,75(12),4483-4495,(2012)
[5]Franchi,B.,Gallot,S.,Wheeden,R.:Sobolev and isoperimetric inequalities for degenerate metrics.Math.Ann.,300(4),557-571(1994)
[6]Francu,M.:Higher-order Sobolev-type embeddings on Carnot-Caratheodory spaces.Mathematische Nachrichten,290(7),1033-1052(2017)
[7]Francu,M.:Higher-order compact embeddings of function spaces on Carnot-Caratheodory spaces.Banach J.Math.Anal.,12(4),970-994(2018)
[8]Folland,G.B.,Stein,E.M.:Hardy Spaces on Homogeneous Groups,Princeton University Press,Princeton,NJ,1982
[9]Garofalo,N.,Nhieu,D.,Isoperimetric and Sobolev Inequalities for Carnot-Caratheodory Spaces and the Existence of Minimal Surfaces,Communications on Pure and Applied Mathematics,Vol.XLIX,John Wiley and Sons,pp.1081-1144,1996
[10]Lam,N.:Moser-Trudinger and Adams Type Inequalities and Their Applications,Doctoral Dissertation,Wayne State University,https://digitalcommons.wayne.edu/oa-dissertations/1016/,2015
[11]Lam,N.,Lu,G.,Sharp Moser-Trudinger inequality on the Heisenberg group at the critical case and applications.Advances in Mathematics,231(6),3259-3287(2012)
[12]Lam,N.,Lu,G.,Tang,H.:On nonuniformly subelliptic equations of Q-sub-Laplacian type with critical growth in the Heisenberg group.Advances in Nonlinear Studies,12(3),659-681(2012)
[13]Li,J.,Lu,G.,Zhou,M.:Concentration-compactness principle for Trudinger-Moser inequalities on Heisenberg groups and existence of ground state solutions.Calc.Var.Partial Differential Equations,57(3),Art.84,26pp.(2018)
[14]Lu,G.:Weighted poincare and Sobolev inequalities for vector fields satisfying Hormander’s condition.Rev.Matematica Iberoamericana,8(3),367-439(1992)
[15]Monti,R.,Serra Cassano,F.:Surface measure in Carnot-Caratheodory spaces.Calc.Var.Partial Differential Equations,13(3),339-376(2001)
[16]Polya,G.,Szego,G.:Isoperimetric Inequalities in Mathematicsl Physics,Ann.Math.Studies 27,Princeton University Press,Princeton,1951
[17]Schulz,F.,Vera de Serio,V.:Symmetrization with respect to a measure.Transactions of the AMS,337(1),195-210(1993)
[18]Stein,E.M.:Harmonic Analysis,Princeton University Press,Princeton,NJ,1993
[19]Tyson,J.:Sharp weighted Young’s inequalities and Moser-Trudinger inequalities on Heisenberg type groups and Grushin spaces.Potential Analysis,4(24),357-384(2006)