Tyson型集及Borel函数的图的拟对称极小性  

Quasisymmetric Minimality of Sets of Tyson Type and Graphs of Borel Functions

作　　者：党云贵 文胜友 Yun Gui DANG;Sheng You WEN(Faculty of Mathematics and Statistics,Hubei University,Hubei Key Laboratory of Applied Mathematics,Wuhan 430062,P.R.China;Department of Mathematics,Liiliang University,Liiliang 033000,P.R.China)

机构地区：[1]湖北大学数学与统计学学院应用数学湖北省重点实验室,武汉430062 [2]吕梁学院数学系,吕梁033000

出　　处：《数学学报（中文版）》2020年第6期621-628,共8页Acta Mathematica Sinica：Chinese Series

基　　金：国家自然科学基金资助项目(11871200,11671189);山西省高等学校科技创新项目(2019L0963)。

摘　　要：本文将欧氏空间Rd中形如[0,1]×Z的集称为Tyson型集,其中d>1,Z■R^d-1、已知当Z是R^d-1中的紧集时,Tyson型集是拟对称极小集.本文改进了这个结果,证明了当Z是R^d-1中的Borel集时,Tyson型集仍是拟对称极小集.作为应用,我们证明了Tyson型集三个形变版本的拟对称极小性,其中一个结果是:设Z是R^d-1中的任一Borel集,h:Z→R^1是Borel函数,满足dimH({h≠0}∩Z)=dimH Z,则h的图G(h)是拟对称极小集,其中h的图G(h)定义为G(h)={(z,y):z∈Z,y∈[0,h(z)]}.In this paper,sets of the form[0,1]×Z in Rd will be called sets of Tysontype,where d>1 and Z■R^d-1.It is known that every set E of Tyson type isquasisymmetrically minimal,provided that Z is compact in R^d-1.We prove that thisresult holds by only assuming that Z is Borel in R^d-1.As applications,we prove thatthe quasisymmetric minimality of three variants of sets of Tyson type.One of theresults is that the graph G(h)of h is quasisymmetrically minimal,provided that Zis a Borel set in R^d-1 and h:Z→R^1 is a Borel function satisfying the condition dimH({h≠0}∩Z)=dimH Z,where G(h)={(z,y):z∈Z,y∈[0,h(z)]}.

关 键 词：Tyson型集 HAUSDORFF维数 拟对称极小集 

分 类 号：P174.12[天文地球—天文学]