2020
DOI: 10.4064/sm181016-5-5
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Weighted inequalities for iterated Copson integral operators

Abstract: We solve a long-standing open problem in theory of weighted inequalities concerning iterated Copson operators. We use a constructive approximation method based on a new discretization principle that is developed here. In result, we characterize all weight functions w, v, u on (0, ∞) for which there exists a constant C such that the inequalityholds for every non-negative measurable function h on (0, ∞), where p, q and m are positive parameters. We assume that p ≥ 1 because otherwise the studied inequality canno… Show more

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Cited by 9 publications
(9 citation statements)
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“…Several results were obtained recently for iterations of operators of identical type, however always under some rather unpleasant restrictions. Iteration of Copson operators was treated in [24]. Next, in [26], a three-weight inequality for the iterated Hardy operator was characterized, motivated by a specific inequality in which a weighted norm of a mean value is compared to that of the derivative of a given function.…”
Section: Introduction and The Main Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…Several results were obtained recently for iterations of operators of identical type, however always under some rather unpleasant restrictions. Iteration of Copson operators was treated in [24]. Next, in [26], a three-weight inequality for the iterated Hardy operator was characterized, motivated by a specific inequality in which a weighted norm of a mean value is compared to that of the derivative of a given function.…”
Section: Introduction and The Main Resultsmentioning
confidence: 99%
“…On the other hand, since max{A * 4 , B * 2 } < ∞ implies that max{A * 2 , B * 1 } < ∞, by using the same argument we applied in case (ii), ( 23) holds. Then, (8) combined with (23) and (24) yields that…”
Section: We Will First Show Thatmentioning
confidence: 99%
“…The special case of the dual version of (1.2) which involves iteration of the Copson operators ∞ t h is treated in [20] when p = 1, using a combination of reduction techniques and discretization. Recently, in [17], a more complicated discretization method is used to establish a characterisation of the same inequality that involves iteration of the Copson operators and is restricted to non-degenerate weights, and the case p = 1 is presented without a proof. In our approach the case p > 1 is not separated from p = 1.…”
Section: Letmentioning
confidence: 99%
“…Both of them were characterized in [20] by so-called "flipped" conditions. The "classical" conditions ensuring the validity of (2.10) was recently presented in [32].…”
Section: Background Materialsmentioning
confidence: 99%