A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces

Triebel, Hans; Winkelvoss, Heike

Studia Mathematica, Tome 119 (1996), p. 149-166 / Harvested from The Polish Digital Mathematics Library

Résumé

Let Γ be a closed set in $ℝ^{n}$ with Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure µ with supp µ ⊂ Γ and constants $c_{1} > 0$ and $c_{2} > 0$ such that $c_{1} r^{d} \leq µ (B (x, r)) \leq c_{2} r^{d}$ for all 0 < r < 1 and all x ∈ Γ, where B(x,r) is a ball with centre x and radius r, then Γ is called a d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces $L_{p} (Γ)$ , 0 < p ≤ ∞, with respect to that measure µ on the hand and the Fourier analytically defined Besov spaces $B_{p, q}^{s} (ℝ^{n})$ (s ∈ ℝ, 0 < p ≤ ∞, 0 < q ≤ ∞) on the other hand.

Publié le : 1996-01-01

Zbl 0864.46020

EUDML-ID : urn:eudml:doc:216348

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     title = {A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces},
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     volume = {119},
     year = {1996},
     pages = {149-166},
     zbl = {0864.46020},
     language = {en},
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Triebel, Hans; Winkelvoss, Heike. A Fourier analytical characterization of the Hausdorff dimension of a closed set and of related Lebesgue spaces. Studia Mathematica, Tome 119 (1996) pp. 149-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv121i2p149bwm/

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