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

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 $Y$.