Heisenberg群上内插的L^∞范数估计  

L^∞ Norm Estimation of Interpolations on the Heisenberg Group

作　　者：陈平 Chen Ping(School of Mathematics and Information Technology,Jiangsu Second Normal University,Nanjing 210013,China)

机构地区：[1]江苏第二师范学院数学与信息技术学院,江苏南京210013

出　　处：《南京师大学报（自然科学版）》2020年第2期6-9,共4页Journal of Nanjing Normal University(Natural Science Edition)

基　　金：国家自然科学基金青年项目(11601193);2017年度江苏省高校“青蓝工程”优秀青年骨干教师培养对象计划项目;江苏省高校优秀中青年教师和校长境外研修项目.

摘　　要：本文研究了Heisenberg群(Hn,d,L^2n+1)上费用函数为Ф(d(x,y))时最优计划γ的内插μt.其中Ф为严格凸函数.内插的本质就是一种测度.我们证明了该内插μt关于Lebsegue测度L^2n+1是绝对连续的,同时也对μt的L^∞范数进行了估计.此外,利用这一估计结果,我们还对Heisenberg群上的变分逼近问题解的内插(e^toS)#γγε进行了估计.本文的证明主要利用Heisenberg群上的L^2n+1测度收缩性质以及最优运输理论中的循环单调性以及Ф的严格凸性.In this paper,we discuss the interpolation μt of optimal transport plan γ in the Heisenberg group(Hn,d,L^2n+1)with the cost function Ф(d(x,y))where Ф is a strictly convex function. An interpolation is actually a kind of measure. We show that the interpolation μt is absolutely continuous with respect to the Lebsegue measure L^2n+1. We also give a L^∞ norm estimation on μt. Furthermore,as a corollary of the above estimation result,we also estimate the interpolation (e^toS)#γγε of solutions of the variational approximation problem in the Heisenberg group. The main methods we used include the L^2n+1 measure contraction property of the Heisenberg group,the cyclically monotonicity in the optimal transport theory and the strictly convexity of Ф.

关 键 词：HEISENBERG群 内插 L^∞范数估计 

分 类 号：O186.14[理学—数学]