Spaces of generalised smoothness have been considered by several mathematicians within different approaches. We refer to Gol'dman (using modulus of continuity, cf. [Gol76]), Kalyabin and Lizorkin (approximation theory, cf. [KL87]), Merucci, Cobos and Fernandez (interpolation theory, cf. [Mer84], [CF88]) among others. A survey has been given in [KL87]. More historical references can be found in [Leo98a]. Our approach is similar to that in [Leo98a], that is, we use the point of view of Fourier analysis and, moreover, consider the more general context of quasi-Banach spaces. The interest of Leopold in [Leo98a] was in using spaces of generalised smoothness of Besov type to handle embedding properties in delicate limiting situations. Our study was strongly motivated by the articles [ET98] and [ET99]. There, Edmunds and Triebel used spaces of generalised smoothness of Besov type when studying the behaviour of eigenvalues in problems which correspond to the vibration of a drum, the whole mass of which is concentrated on a fractal subset of the drum. In order to explain the relationship between fractals and function spaces we need some previous considerations. The fractals considered by Edmunds and Triebel in the above papers are (isotropic) perturbed d-sets, called (d, Ψ)-sets. Let Γ be a non-empty closed subset of R n , 0 < d < n and Ψ a positive monotone function on the interval (0, 1] with c 1 Ψ (2 −j) ≤ Ψ (2 −2j) ≤ c 2 Ψ (2 −j), j ∈ N 0 , (0.1) for some positive constants c 1 and c 2. Then Γ is called a (d, Ψ)-set if there is a Radon measure µ with supp µ = Γ and two positive constants c 1 and c 2 such that c 1 r d Ψ (r) ≤ µ(B(γ, r)) ≤ c 2 r d Ψ (r) (0.2) for any ball B(γ, r) centred at γ ∈ Γ of radius r ∈ (0, 1). If, additionally, Ψ is decreasing with lim r→0 Ψ (r) = ∞, and (0.2) holds for d = n, then Γ is called an (n, Ψ)-set. Let Ω be a bounded C ∞ domain in R n and let −∆ be the Dirichlet Laplacian in Ω. According to Theorem 2.28 and Corollary 2.30 of [ET99], the operator B = (−∆) −1 • tr Γ (0.3) is a compact self-adjoint non-negative operator inW 1 2 (Ω), where Γ ⊂ Ω is a (d, Ψ)-set with n − 2 < d ≤ n and tr Γ is closely related to the trace tr Γ ofW 1 2 (Ω) on Γ. Moreover, the positive eigenvalues µ k of B, ordered so that µ k+1 ≤ µ k , k ∈ N, and repeated according to their algebraic multiplicity, can be estimated as follows: c 1 k −1 (kΨ (k −1)) (n−2)/d ≤ µ k ≤ c 2 k −1 (kΨ (k −1)) (n−2)/d , k ∈ N, (0.4) for some positive constants c 1 and c 2 .
