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AbstractIn [18] and [19] we have studied compact embeddings of weighted function spaces on ℝⁿ, $id:H_q^s(w\left(x\right),ℝⁿ)\to Lₚ\left(ℝⁿ\right)$, s>0, 1 < q ≤ p< ∞, s-n/q+n/p > 0, with, for example, $w\left(x\right)=⟨x{⟩}^{\alpha }$, α > 0, or $w\left(x\right)=lo{g}^{\beta }⟨x⟩$, β > 0, and $⟨x⟩={(2+|x\left|²\right)}^{1/2}$. We have determined the behaviour of their entropy numbers eₖ(id). Now we are interested in the limiting case 1/q = 1/p + s/n. Let $w\left(x\right)=lo{g}^{\beta }⟨x⟩$, β > 0. Our results in [18] imply that id cannot be compact for any β > 0, but after replacing the target space Lₚ(ℝⁿ) by a “slightly” larger one, $Lₚ{\left(logL\right)}_{-a}\left(ℝⁿ\right)$, a > 0, the...

We study continuity envelopes of function spaces $B_{p,q}^s(ℝⁿ,w)$ and $F_{p,q}^s(ℝⁿ,w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

We study weighted function spaces of Lebesgue, Besov and Triebel-Lizorkin type where the weight function belongs to some Muckenhoupt ${}_p$ class. The singularities of functions in these spaces are characterised by means of envelope functions.

We study the existence of traces of Besov spaces on fractal h-sets Γ with a special focus on assumptions necessary for this existence; in other words, we present criteria for the non-existence of traces. In that sense our paper can be regarded as an extension of Bricchi (2004) and a continuation of Caetano (2013). Closely connected with the problem of existence of traces is the notion of dichotomy in function spaces: We can prove that-depending on the function space and the set Γ-there occurs an...

We study embeddings of spaces of Besov-Morrey type, $i{d}_{\Omega }{:}_{p₁,u₁,q₁}^{s₁}\left(\Omega \right){\hookrightarrow }_{p₂,u₂,q₂}^{s₂}\left(\Omega \right)$, where $\Omega \subset {ℝ}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $i{d}_{\Omega }$. This continues our earlier studies relating to the case of ${ℝ}^d$. Moreover, we also characterise embeddings into the scale of $L_p$ spaces or into the space of bounded continuous functions.

We study continuous embeddings of Besov spaces of type $B_{p,q}^s(ℝⁿ,w)$, where s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞, and the weight w is doubling. This approach generalises recent results about embeddings of Muckenhoupt weighted Besov spaces. Our main argument relies on appropriate atomic decomposition techniques of such weighted spaces; here we benefit from earlier results by Bownik. In addition, we discuss some other related weight classes briefly and compare corresponding results.