The ρ-quasiconcave Functions and Weighted Inequalities

On weighted iterated Hardy-type inequalities

“…The usual Stieltjes transform is obtained on putting U(x) ≡ x. In the case U(x) ≡ x λ , λ > 0, the boundedness of the operator S between weighted L s and L q spaces, namely inequality [13] (see also [14]) (when 1 < s < ∞, 0 < q ≤ ∞), where the result is presented without proof. This problem also was considered in [16] and [20,21], where completely different approach was used, based on the so called "gluing lemma" (see also [17]).…”

Norms of maximal functions between generalized and classical Lorentz spaces

“…Note that inequality (2.8) have been considered in the case m = 1 in [6] (see also [14]), where the result was presented without proof, and in the case p = 1 in [15] and [56], where the special type of weight function v was considered. Recall that the inequality has been completely characterized in [18] and [19] in the case 0 < m < ∞, 0 < q ≤ ∞, 1 ≤ p < ∞ by using discretization and anti-discretization methods.…”

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

“…Next, we state a necessary lemma which is also of independent interest. 4) and the best constant in inequality (3.1) satisfies…”

“…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 ≤ ∞.…”