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

Abstract

In this paper we characterize the validity of the Hardy-type inequality p,u,(0,t) q,w,(0,∞) ≤ c h 1,v,(0,∞) , where 0 < p < ∞, 0 < q ≤ +∞, u, w and v are weight functions on (0, ∞). It is pointed out that this characterization can be used to obtain new characterizations for the boundedness between weighted Lebesgue spaces for Hardy-type operators restricted to the cone of monotone functions and for the generalized Stieltjes operator. MSC: Primary 26D10; 46E20

Weighted inequalities for iterated Copson integral operators

“…The iteration of Copson and Hardy operators constitutes a related but in fact an essentially different problem. Inequalities involving such operators have been investigated for instance in [18,19,43,44].…”

Iterated Hardy-type inequalities involving suprema

“…Note that inequality (1.8) has been characterized in [14] in the case 0 < q < ∞, 1 ≤ p < ∞. We pronounce that the characterizations of inequalities (1.5) -(1.8) are important because many inequalities for classical operators can be reduced to them (for illustrations of this important fact, see, for instance, [12]). These inequalities play an important role in the theory of Morrey spaces and other topics (see [1], [2] and [3]).…”

Weighted inequalities for a class of semiadditive operators

“…− n−1 p . In the papers [1], [2] and [6] by different approaches necessary and sufficient conditions for the validity of inequalities (3.3) and (3.4) are obtained for different relations between the parameters 0 < p, q, θ ≤ ∞. Moreover, in [6] other inequalities of the type (3.3) and (3.4) are considered.…”

“…Moreover, in [6] other inequalities of the type (3.3) and (3.4) are considered. In [3] characterizations of (3.3) and (3.4) are found only for the case 1 ≤ p ≤ q < ∞ and 0 < θ < ∞ but by a method different from those in [1], [2] and [6].…”