Illinois Journal of Mathematics

Doubling measures and nonquasisymmetric maps on Whitney modification sets in Euclidean spaces

Xiaohua Wang, Shengyou Wen, and Zhixiong Wen

Full-text: Open access

Abstract

Let $E$ be a closed set in $\mathbb{R}^n$ and $\mathcal{W}$ a Whitney decomposition of $\mathbb{R}^n\setminus E$. Choosing one point from the interior of each cube in $\mathcal{W}$ we obtain a set $F$ and then we say that the set $E\cup F$ is a Whitney modification of $E$. The Whitney modification of a measure $\mu$ on $\mathbb{R}^n$ to $E\cup F$ is a measure $\nu$ defined on $E\cup F$ by $\nu\equiv\mu$ on $E$ and by $\nu(\{x\})=\mu(I_x)$ for every $x\in F$, where $I_x\in\mathcal{W}$ is the cube containing the point $x$. We prove that a measure on $E\cup F$ is doubling if and only if it is the Whitney modification of a doubling measure on $\mathbb{R}^n$. As its application, we show that there are metric spaces $X,Y$ and a nonquasisymmetric homeomorphism $f$ of $X$ onto $Y$ such that a measure $\mu$ on $X$ is doubling if and only if its image $\mu\circ f^{-1}$ is doubling on $Y$.

Article information

Source
Illinois J. Math., Volume 52, Number 4 (2008), 1291-1300.

Dates
First available in Project Euclid: 18 November 2009

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1258554363

Digital Object Identifier
doi:10.1215/ijm/1258554363

Mathematical Reviews number (MathSciNet)
MR2595768

Zentralblatt MATH identifier
1193.28011

Subjects
Primary: 28A80: Fractals [See also 37Fxx]

Citation

Wang, Xiaohua; Wen, Shengyou; Wen, Zhixiong. Doubling measures and nonquasisymmetric maps on Whitney modification sets in Euclidean spaces. Illinois J. Math. 52 (2008), no. 4, 1291--1300. doi:10.1215/ijm/1258554363. https://projecteuclid.org/euclid.ijm/1258554363


Export citation

References

  • J. Heinonen, Lectures on analysis on metric spaces, Springer-Verlag, New York, 2001.
  • R. Kaufman and J. M. Wu, Two problems on doubling measures, Rev. Math. Iberoam. 11 (1995), 527–545.
  • T. Laakso, Plane with $A_{\infty}$-weighted metric not bilipschitz embeddable to $\mathbb{R}^n$, Bull. Lond. Math. Soc. 34 (2002), 667–676.
  • J. Luukkainen and E. Saksman, Every complete doubling metric space carries a doubling measure, Proc. Amer. Math. Soc. 126 (1998), 531–534.
  • E. Saksman, Remarks on the nonexistence of doubling measures, Ann. Acad. Sci. Fenn. Math. 24 (1999), 155–163.
  • S. Semmes, On the nonexistence of bilipschitz parameterizations and geometric problems about $A_\infty$-weights, Rev. Math. Iberoam. 12 (1996), 337–410.
  • A. L. Vol'berg and S. V. Konyagin, On measures with the doubling condition, Math. USSR-Izv. 30 (1988), 629–638.