Characterization of weighted inequalities for superpositions of integral and supremal operators

Abstract

In this paper, we provide a characterization of the boundedness of positive sublinear operators that are superposition of three operators: Copson, Hardy, and Tandori (supremal) operators defined on the halfaxis from a weighted Lebesgue space L_p(v) to another weighted Lebesgue space L₁(w), where v and w are weight functions on the half-axis (0,∞), and 1 < p < ∞. Our characterization is entirely different from existing results in the literature. The motivation for investigating such inequalities stems from the problem of finding a minimal rearrangement invariant space that contains the cones of non-increasing rearrangement of the functions represented by generalized fractional maximal function acting on functions from weighted Lorentz function spaces. More specifically, by obtaining two-sided estimates for the best constant in the corresponding inequality, we derive a characterization of the associate space of minimal rearrangement invariant spaces containing cones of non-increasing rearrangement of generalized fractional maximal function. To achieve this goal, we are using discretization and anti-discretization methods. In particular, we extend existing discretization techniques to handle the operators formed by iterating the Copson, Hardy, and Tandori operators. We first establish a discrete characterization in Theorem 3. Then, applying anti-discretization techniques, we derive a continuous characterization in Theorem 1.