A New Class of General Refined Hardy-type Inequalities with Kernels

Let μ1 and μ2 be positive σ-finite measures on Ω1 and Ω2 respectively, k : Ω1 × Ω2 → R be a non-negative function, and K(x) = ∫ Ω2 k(x, y) dμ2(y), x ∈ Ω1. We state and prove a new class of refined general Hardy-type inequalities related to the weighted Lebesgue spaces L and L, where 0 < p ≤ q < ∞ or −∞ < q ≤ p < 0, convex functions and the integral operators Ak of the form Akf(x) = 1 K(x) ∫ Ω2 k(x, y)f(y) dμ2(y). We also provide a class of new sufficient conditions for a weighted modular inequality involving operator Ak to hold. As special cases of our results, we obtain refinements of the classical one-dimensional Hardy’s, Pólya–Knopp’s and Hardy–Hilbert’s inequality and of related dual inequalities, as well as a generalization and refinement of the classical Godunova’s inequality. Finally, we show that our results may be seen as generalizations of some recent results related to Riemann-Liouville’s and Weyl’s operator.