Tailored Besov spaces and h‐sets

Abstract

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

Homogeneity, non-smooth atoms and Besov spaces of generalised smoothness on quasi-metric spaces

“…Let us recall the definition and some properties of h-sets in R n studied in [3]. As we have already said, we will rely on what is known about this kind of sets in R n to extend this theory to more general spaces.…”

“…The above definition was given in [3,Chapter 3], where Bricchi showed that the definition makes sense and that, if we apply it to σ = (0), we get…”

“…These indices were used by Bricchi [3] to deal with Besov spaces with generalised smoothness. The indices s(σ) and s(σ) [respectively s(σ) and s(σ)] may not coincide.…”

“…If N j = 2 j , j ∈ N 0 , we recover the Besov spaces of generalised smoothness studied by Bricchi [3] and we write B σ p,q (R n ).…”

“…We say in this case that Γ = (Γ, n , µ) is an h-set. In [3] and [5] Bricchi characterised the class of functions h involved. There he also introduced and studied Besov spaces of generalised smoothness on these fractal sets.…”