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Global Poincaré Inequalities on the Heisenberg Group and Applications
作者姓名:Yu  Xin  DONG  Guo  Zhen  LU  Li  Jing  SUN
作者单位:[1]Department of Mathematics, Fudan University, Shanghai 200433, P. R. China [2]Department of Mathematics, Beijing Normal University, Beijing 100875, P. R. China [3]Department of Mathematics, Wayne State University, Detroit, MI 48202, USA
基金项目:The first author is supported-by Zhongdian grant of NSFC; The second author is supported by a global grant at Wayne State University and by NSF of USA
摘    要:Let f be in the localized nonisotropic Sobolev space Wloc^1,p (H^n) on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n + 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n |△H^n f|^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞(H^n) under the norm of (∫H^n |f| Qp/Q-p)^Q-p/Qp + (∫ H^n |△H^n f|^p)^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L(2Q/Q-2) Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.

关 键 词:Poincaré  不等式  海森堡群  应用  Sobolev不等式  Sobolev空间
修稿时间:2004-05-25

Global Poincaré Inequalities on the Heisenberg Group and Applications
Yu Xin DONG Guo Zhen LU Li Jing SUN.Global Poincaré Inequalities on the Heisenberg Group and Applications[J].Acta Mathematica Sinica,2007,23(4):35-744.
Abstract:
Keywords:Heisenberg group  Sobolev inequalities  Poincaré  inequalities  best constants
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