The Blocking Technique, Weighted Mean Operators and Hardy’s Inequality

Abstract

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Discretization and anti-discretization of rearrangement-invariant norms

“…Our approach is based on discretization and (more importantly) antidiscretization methods combined with the blocking technique from [17]. Let us outline our approach and the structure of the paper.…”

Boundedness of Hardy-type operators with a kernel: integral weighted conditions for the case $$0<q<1\le p<\infty $$ 0 < q < 1 ≤ p < ∞

“…The results are presented in Theorems 8, 9 and Corollaries 10, 11. The proofs are based on the well-known method of dyadic discretization (or blocking technique, see [8] for a basic introduction into this method). The particular variant of the technique employed here is essentially the same as the one used in [11].…”

“…Notice that, in B 7 , the term corresponding to n = 0 is indeed omitted, since for any t ∈ μ it holds that t t μ w(x)U q (x, t) dx μ U q (t μ , t) and the right-hand side is thus already represented by the 0-th term in B 8 . Let us note that in what follows, expressions such as sup x∈ (y,∞] ϕ(x) appear even where the argument ϕ(x) is undefined for x = ∞.…”

Интегральные Операторы Харди-Стеклова. Часть I

“…Неравенство Харди для последовательностей характеризова-но в работах Г. Беннета [13] К.-Г. Гроссе-Эрдмана [14] и М. Л. Гольдмана [15]. В частности, в [14; теорема 9.2] указан следующий результат.…”