Estimates for norms of two‐weighted summation operators on a tree under some restrictions on weights

Abstract

Phone: +7 495 939 5632 V j (ξ ) := V T j (ξ ) := {ξ ξ : ρ T (ξ, ξ ) = j}.Given ξ ∈ V(T ), we denote by T ξ = (T ξ , ξ) the subtree in T with the vertex setDefinition 1.1 Let W ⊂ V(T ). We say that G ⊂ T is a maximal subgraph on the set of vertices W if V(G) = W and if any two vertices ξ , ξ ∈ W that are adjacent in T are also adjacent in G.Given two subtrees T 1 ⊂ T and T 2 ⊂ T , we denote by T 1 \T 2 the maximal subgraph on the set of verticesLet G be a subgraph in (T , ξ 0 ). Denote by V max (G) and V min (G) the set of maximal and minimal vertices in G, respectively.Let G be a disjoint union of trees (T j , ξ j ), 1 j k. Then the partial order on each tree T j induces the partial order on G. Let u, w : V(G) → [0, ∞) be weight functions. Define the summation operator S u,w,G by

Estimates for n-widths of two-weighted summation operators on trees

“…The lower estimate for S p,q A,u,w was obtained in [1,Lemma 3.3]. In addition, the following result was proved (see [1, Lemma 3.1]).…”

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

“…From (121) it follows that we can apply Theorem 3.6 in [38] and estimate S p,q Ât,j ,u,w (σ) from above. For sufficiently small σ = σ(Z 0 , t) > 0 we get 145) and ( 146) we obtain Assumptions 1, 2, 3.…”

“…From Theorem F in [36], Theorem 3.6 in [38] and Assumption C it follows that if ξ * ∈ V A j−j min (ξ 0 ), then in the case κ > θ q S p,q D,u,w…”

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

“…Let D be a subtree in A. Denote by S p,q D,u,w the operator norm of S u,w,D : l p (D) → l q (D). Applying the results of [40], [46], we get that for j 2 and for any i ∈ I j we have S p,q Aη j,i ,u,w ≍ Z C(j), where C(j) is defined as follows.…”