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

Abstract

Let E be a closed set in ℝⁿ and a Whitney decomposition of ℝⁿ∖E. Choosing one point from the interior of each cube in we obtain a set F and then we say that the set E∪F is a Whitney modification of E. The Whitney modification of a measure μ on ℝⁿ to E∪F is a measure ν defined on E∪F by ν≡μ on E and by ν({x})=μ(I_x) for every x∈F, where is the cube containing the point x. We prove that a measure on E∪F is doubling if and only if it is the Whitney modification of a doubling measure on ℝⁿ. 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 μ on X is doubling if and only if its image μ○f⁻¹ is doubling on Y.