Characterisation of Embeddings in Lorentz Spaces

Abstract

Some new integral conditions characterising the embedding A p (v) <-t r q (w), 0 < p, q < oo are presented, including proofs also for the cases (i) p = oo, 0 < q < oo, (ii) q -oo, I <p < oo and (iii) p = q -oo. Only one condition is necessary for each case which means that our conditions are different from and simpler than other corresponding conditions in the literature. We even prove our results in a more general frame namely when the space T q (w) is replaced by the more general space rS(iu). In our proof we use a technique of discretisation and anti-discretisation developed by A. Gogatishvili and L. Pick, where they considered the opposite embedding.

On weighted iterated Hardy-type inequalities

“…was considered by many authors and there exist several characterizations of this inequality (see, survey paper [11], [4], [15], [10], and [27]). Using change of variables x = 1/t, we can easily obtain full characterization of the weighted inequality…”

Some new iterated Hardy-type inequalities: the case "Equation missing"

“…Let ϕ be non-decreasing and finite function on the interval I := (a, b) ⊆ R. We assign to ϕ the function λ defined on subintervals of I by Let us now recall some definitions and basic facts concerning discretization and antidiscretization which can be found in [8], [9] and [11]. Let U be an admissible function.…”

“…In this paper we characterize the validity of the inequality where 0 < p < ∞, 0 < q ≤ +∞, θ = 1, u, w and v are weight functions on (0, ∞). Note that inequality (1.1) have been considered in the case p = 1 in [4] (see also [5]), where the result is presented without proof, in the case p = ∞ in [10] and in the case θ = 1 in [11] and [22], where the special type of weight function v was considered, and, recently, in [13] in the case 0 < p < ∞, 0 < q ≤ +∞, 1 < θ ≤ ∞.…”

Редукционные Теоремы Для Весовых Интегральных Неравенств На Конусе Монотонных Функций

“…Это послужило мотивацией для широ-кого поля исследований весовых неравенств на конусах монотонных функций (см., например, [2], [5]- [14], [20], [22]- [37], [41]- [45], [47], [53], [55], [62], [66]- [68], [73], [76], [78], [88]- [91], [94], [99], [103], [104], [110], [111], [112], [117] и др. ).…”