Inequalities of Hardy type for a class of integral operators with measures

Abstract

A b s t r a c t . Hardy type inequalities are characterized for a class of integral operators with positive Borel σ-finite measures.

Hardy-type inequalities on the cones of monotone functions

“…The full characterization of the inequality (0.18) for the operators with the Oinarov kernels in the case 1 < p, q < ∞ was obtained by D.V. Prokhorov in [52]. Here we summarize the results of [52] in Theorem 19 for the case of two measures and in Theorem 20 for the case of three measures.…”

“…Prokhorov in [52]. Here we summarize the results of [52] in Theorem 19 for the case of two measures and in Theorem 20 for the case of three measures.…”

“…Theorem 20[52]. (i) Let 1 < p ≤ q < +∞, μ, λ and ν be positive Borel σ−finite measures on [a, b], u, v, w ∈ M + , (ν a , ν s ) the Lebesgue decomposition of ν relative to λ, and dν a /dλ the Radon-Nikodym derivative of ν a with respect to λ, k a μ×λ−measurable non-negative function satisfying condition (0.20).…”

Weighted inequalities for quasilinear integral operators on the semi-axis and applications to Lorentz spaces

“…and and the constants C T and others are taken as the least possible. If q = r < ∞ these inequalities are reduced to the generalized Hardy-type inequalities which were well studied see, for instance, [2], [21], [36] with further extensions and improvements in [19], [20], [22], [23], [26], [40], [41] and others. The case q = ∞ is closely related to recently initiated studies of supremum operators [12], [13], [24], [25], [27], [29], [37].…”

Неравенство Харди с мерами, случай $0&lt;p&lt;1$

“…[1], [2] и [3]). В работах Ойнарова [4], Блума и Кермана [5], Степанова [6], Прохорова [7] рассмотрено данное неравенство для ядер, удовлетворяющих условию Ойнарова: существует константа D такая, что…”