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

Abstract

In this paper, estimates for norms of weighted summation operators (discrete Hardy-type operators) on a tree are obtained for 1 < p < q < ∞ and for arbitrary weights and trees.Given ξ ∈ V(T ), we denote by T ξ = (T ξ , ξ) the subtree in T with the vertex setLet W ⊂ V(T ). We say that G ⊂ T is a maximal subgraph on the vertex set W if V(G) = W and if any two vertices ξ ′ , ξ ′′ ∈ W that are adjacent in T are also adjacent in G. Given ξ, ξ ′ ∈ V(T ), ξ ξ ′ , we denote by [ξ, ξ ′ ] the maximal subgraph on the vertex set {η ∈ V(T ) : ξ η ξ ′ }.