Atomic decompositions of function spaces with Muckenhoupt weights, and some relation to fractal analysis

Abstract

This paper deals with atomic decompositions in spaces of type Bsp,q (ℝn , w), Fsp,q (ℝn , w), 0 < p < ∞, 0 < q ≤ ∞, s ∈ ℝ, where the weight function w belongs to some Muckenhoupt class Ar. In particular, we consider the weight function wΓκ (x) = dist(x, Γ)κ, where Γ is some d ‐set, 0 < d < n, and κ > –(n – d). (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Entropy and approximation numbers of embeddings of function spaces with Muckenhoupt weights, II. General weights

“…For later use we recall a 'fractal' example studied in [16]. Let Γ ⊂ R n be a d-set, 0 < d < n, in the sense of [39,Def.…”

“…Recent works are due to Roudenko [29,30,13], and Bownik [1,2]. We partly rely on our approaches in [16,17].…”

Entropy and Approximation Numbers of Embeddings of Function Spaces with Muckenhoupt Weights, I

“…3, Cor.]) one can reduce the argument to the corresponding one for admissible weights, say, of type [20] implies the representability (18) (at the expense of some higher smoothness and cancellation needed for the atomic decomposition argument according to Proposition 1.12), see also [42].…”

“…Now the weight w(x) = |x| α , α > −n, may serve as a classical example. Weighted Besov and Triebel-Lizorkin spaces with Muckenhoupt weights are well known concepts, see [3-6, 15, 31, 32] and, more recently, [1,2,18]. But (the compactness of) Sobolev embeddings of such spaces were not yet studied in detail.…”

Atomic decomposition for weighted Besov and Triebel‐Lizorkin spaces

“…When the weight lies in the class A ∞ , Bownik and Ho investigated the atomic decompositions and the φ-transform in [1]. We refer to the work by Haroske and Piotrowska for the function spaces with A ∞ weights [6], which present a direct approach to atomic decompositions in these spaces. Their method depends on the celebrated paper [4].…”