DOUBLING MEASURES AND NONQUASISYMMETRIC MAPS ON WHITNEY MODIFICATION SETS IN EUCLIDEAN SPACES

摘要

Let E be a, closed set in R~n and VV a Whitney decomposition of R~n E. Choosing one point from the interior of each cube in W 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 R~n to E ∪ F is a measure v defined on E ∪ F by v = μ on E and by v({x}) = μ(I_x) for every x ∈F, where I-x ∈ W 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 R~n. As its application, we show that there are metric spaces X, Y and a nonquasisymmetric homeo-morphism f of X onto Y such that a measure μ on X is doubling if and only if its image μ o f~(-1) is doubling on Y.