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 cannot hold for nontrivial weights, but otherwise p, q and m are unrestricted.2010 Mathematics Subject Classification. 47G10, 47B38, 26D10.

Weighted inequalities for a superposition of the Copson operator and the Hardy operator

“…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.…”

“…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…”

Weighted inequalities involving iteration of two Hardy integral operators

“…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.…”

Norms of maximal functions between generalized and classical Lorentz spaces

“…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].…”