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

Let Γ be a closed set in Rn with the Lebesgue measure |Γ| = 0. The first aim of the paper is to give a Fourier analytical characterization of the Hausdorff dimension of Γ. Let 0 < d < n. If there exist a Borel measure μ with supp μ ⊂ Γ and constants c1 > 0 and c2 > 0 such that c1r ≤ μ(B(x, r)) ≤ c2r 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 dset. The second aim of the paper is to provide a link between the related Lebesgue spaces Lp(Γ) , 0 < p ≤ ∞, with respect to that measure μ on the one hand and the Fourier analytically defined Besov spaces Bs p,q(R) (s ∈ R, 0 < p ≤ ∞, 0 < q ≤ ∞) on the other hand.