In this paper we define Besov-type spaces with generalised smoothness on a rather vast class of (isotropic) irregular sets of fractal type in the Euclidean space (h-sets). As a special case we shall obtain the definition of Besov spaces with the usual scalar-index of regularity. To deal with this problem we rely on the one hand on the measure-geometric theory we have developed for h-sets and, on the other, we rely on some known results for Besov spaces with generalised smoothness in R n and on some advanced techniques concerning these function spaces (local means and atoms), which have been recently developed in full generality

We describe the growth envelope of Besov and Triebel-Lizorkin spaces B σ pq (R n) and F σ pq (R n) with generalized smoothness, i.e. instead of the usual scalar regularity index σ ∈ R we consider now the more general case of a sequence σ = {σ j } j∈N 0. We take under consideration the range of the parameters σ, p, q which, in analogy to the classical terminology, we call sub-critical.

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In this note we shall consider the following problem: which conditions should satisfy a function ℎ : (0, 1) → ℝ in order to guarantee the existence of a (regular) measure μ in with compact support and for some positive constants 𝑐2, and 𝑐2 independent of γ ∈ Γ and 𝑟 ∈ (0,1)? The theory of self-similar fractals provides outstanding examples of sets fulfilling (♡) with ℎ(𝑟) = 𝑟𝑑, 0 ≤ 𝑑 ≤ 𝑛, and a suitable measure μ. Analogously, we shall rely on some recent techniques for the construction of pseudo self-similar fractals in order to deal with our more general task.

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