A Fourier analytical characterization of the

Let ? be a closed set in R n with the Lebesgue measure j?j = 0. The rst aim of the paper is to give a Fourier analytical characterization of the Hausdorr dimension of ?. Let 0 < d < n. If there exist a Borel measure with supp ? and constants c 1 > 0 and c 2 > 0 such that c 1 r d (B(x; r)) c 2 r d for all 0 < r < 1 and all x 2 ?, where B(x; r) is a ball with centre x and radius r, then ? is called d-set. The second aim of the paper is to provide a link between the related Lebesgue spaces L p (?) , 0 < p 1, with respect to that measure on the one hand and the Fourier analytically deened Besov spaces B s p;q (R n) (s 2 R, 0 < p 1, 0 < q 1) on the other hand.