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

Let Γ be a closed set in ${\displaystyle {ℝ}^n}$ 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 ${\displaystyle c_1>0}$ and ${\displaystyle c_2>0}$ such that ${\displaystyle c_1{r}^d\le \mu \left(B(x,r)\right)\le c_2{r}^d}$ 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 ${\displaystyle L_p(\Gamma )}$, 0 < p ≤ ∞, with respect to that measure µ on the hand and the Fourier analytically defined Besov spaces ${\displaystyle B^s\_\{p,q\}\left({ℝ}^n\right)}$ (s ∈ ℝ, 0 < p ≤ ∞, 0 < q ≤ ∞) on the other hand.