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

Hans Triebel¹ ; Heike Winkelvoss¹

¹

Studia Mathematica, Tome 121 (1996) no. 2, pp. 149-166

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 $c_{1}r^{d} ≤ µ (B(x,r)) ≤ c_{2}r^{d}$ 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 $B^s_{p,q}(ℝ^n)$ (s ∈ ℝ, 0 p ≤ ∞, 0 q ≤ ∞) on the other hand.

Détail

DOI : 10.4064/sm-121-2-149-166

Keywords: Hausdorff dimension, Hausdorff measure, function spaces