## 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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