TY - JOUR

T1 - A counterexample to a vitali type theorem with respect to hausdorff content

AU - Melnikov, Mark

AU - Orobitg, Joan

PY - 1993/1/1

Y1 - 1993/1/1

N2 - 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.

AB - 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.

U2 - 10.1090/S0002-9939-1993-1137228-8

DO - 10.1090/S0002-9939-1993-1137228-8

M3 - Article

SN - 0002-9939

VL - 118

SP - 849

EP - 856

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -