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

Melnikov, Mark; Orobitg, Joan

doi:10.1090/S0002-9939-1993-1137228-8

A counterexample to a Vitali type theorem with respect to Hausdorff content
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by Mark Melnikov and Joan Orobitg PDF

Proc. Amer. Math. Soc. 118 (1993), 849-856 Request permission

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

References

Joan Mateu and Joan Orobitg, Lipschitz approximation by harmonic functions and some applications to spectral synthesis, Indiana Univ. Math. J. 39 (1990), no. 3, 703–736. MR 1078735, DOI 10.1512/iumj.1990.39.39035
Joan Mateu and Joan Verdera, BMO harmonic approximation in the plane and spectral synthesis for Hardy-Sobolev spaces, Rev. Mat. Iberoamericana 4 (1988), no. 2, 291–318. MR 1028743, DOI 10.4171/RMI/75