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A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces. (English) Zbl 0864.46020

Summary: Let \(\Gamma\) be a closed set in \(\mathbb{R}^n\) with Lebesgue measure \(|\Gamma|=0\). The first aim of the paper is to give a Fourier analytical characterization of the Hausdorff dimension of \(\Gamma\).
Let \(0<d<n\). If there exist a Borel measure \(\mu\) with \(\text{supp }\mu\subset\Gamma\) and constants \(c_1>0\) and \(c_2>0\) such that \(c_1r^d\leq\mu(B(x,r))\leq c_2r^d\) for all \(0<r<1\) and all \(x\in\Gamma\), where \(B(x,r)\) is a ball with centre \(x\) and radius \(r\), then \(\Gamma\) is called a \(d\)-set. The second aim of the paper is to provide a link between the related Lebesgue spaces \(L_p(\Gamma)\), \(0<p\leq\infty\), with respect to that measure \(\mu\) on the one hand and the Fourier analytically defined Besov spaces \(B^s_{p,q}(\mathbb{R}^n)\) (\(s\in\mathbb{R}\), \(0<p\leq\infty\), \(0<q\leq\infty\)) on the other hand.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
28A78 Hausdorff and packing measures
28A80 Fractals
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