Mateu and Orobitg proved (in Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J. 39 (1990)) that given λ > 1 and d − 1 < α ≤ d there exist constants C and N (depending on λ and α) with the following property: For any compact set K in Rd one can find a (finite) family of balls (B(xi, ri)) such that (i) K ⊂u B(xi, n), Mα denoting the α-dimensional Hausdorff content, and (iii) the dilated balls (B(xi,λri)) are an almost disjoint family with constant N. In this paper we prove that such a result is false for α ⊂ d −1. © 1993 American Mathematical Society.
|Journal||Proceedings of the American Mathematical Society|
|Publication status||Published - 1 Jan 1993|