A counterexample to a Vitali type theorem with respect to Hausdorff content

Abstract

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 \lambda > 1 and d − 1 > α ⩽ d d - 1 > \alpha \leqslant d there exist constants C C and N N (depending on λ \lambda and α \alpha ) with the following property: For any compact set K K in R d {\mathbb {R}^d} one can find a (finite) family of balls { B ( x i , r i ) } \{ B({x_i},{r_i})\} such that (i) K ⊂ ⋃ B ( x i , r i ) K \subset \bigcup {B({x_i},{r_i})} , (ii) ∑ r i α ⩽ C M α ( K ) \sum {r_i^\alpha \leqslant C{M^\alpha }(K)} , M α {M^\alpha } denoting the α \alpha -dimensional Hausdorff content, and (iii) the dilated balls { B ( x i , λ r i ) } \{ B({x_i},\lambda {r_i})\} are an almost disjoint family with constant N N . In this paper we prove that such a result is false for α ⩽ d − 1 \alpha \leqslant d - 1 .