Existence and Properties of ℎ-Sets

Abstract

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.

Traces of Besov spaces on fractal $h$-sets and dichotomy results

“…There it turns out that one first needs a sound knowledge about the existence and quality of the corresponding trace spaces. Returning to the first results in that respect in [Br2], see also [Br3,Br4,Br1], we found that the approach can (and should) be extended for later applications. More precisely, for a positive continuous and non-decreasing function h : (0, 1] → R (a gauge function) with lim r→0 h(r) = 0, a non-empty compact set Γ ⊂ R n is called h-set if there exists a finite Radon measure µ in R n with supp µ = Γ and µ(B(γ, r)) ∼ h(r), r ∈ (0, 1], γ ∈ Γ, see also [Ro, Chapter 2] and [Ma,p.…”

“…60]. Here we essentially follow the presentation in [Br2,Br3,Br4,Br1], see also [Ma] for basic notions and concepts.…”

“…Note that in some later papers, cf. [Br1], instead of (2.21) and (2.22) the so-called upper and lower Boyd indices of σ are considered, given by (2.28)…”

Widths of weighted Sobolev classes with weights that are functions of distance to some h-set: some limiting cases

“…Definition 3. (see [10]). Let Γ ⊂ R d be a nonempty compact set and h ∈ H. We say that Γ is an h-set if there are a constant c * 1 and a finite countably additive measure µ on R d such that supp µ = Γ and…”

Estimates for the entropy numbers of embedding operators of function spaces on sets with tree-like structure: Some limiting cases

“…Definition 4. (see [5]). Let Γ ⊂ R d be a nonempty compact set and h ∈ H. We say that Γ is an h-set if there are a constant c * 1 and a finite countably additive measure µ on R d such that supp µ = Γ and…”