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

Entropy numbers of embedding operators of weighted Sobolev spaces with weights that are functions of distance from some h-set

“…For each subtree D ⊂ A we denote Ω[D] = ∪ ξ∈V(D) Ω[ξ]. It was proved in [40,44,45] that for any j 0 j min , i 0 ∈ Ĩj 0 and for any vertex η j 0 ,i 0 there exists a linear continuous operator P η j 0 ,i 0 : L q,v (Ω) → P r−1 (Ω) such that for any subtree D with minimal vertex η j 0 ,i 0 and for any function f ∈ W r p,g (Ω)…”

“…In other cases we apply Lemma 11 or Corollary 1. In [42, p. 50], [40] and [45] the number k * * = k * * (Z * ) ∈ N is defined and the functions {ψ t,j } j∈Jt ∈ C ∞ (R d ) are constructed with the following properties:…”

“…In this paper we obtain order estimates for entropy numbers of embedding operator of weighted Sobolev spaces on a John domain into weighted Lebesgue space. Estimates for n-widths of such embeddings were recently obtained in [42,45]. Definition 1.…”