A three-weighted Hardy-type inequality on the cone of quasimonotone functions

From the text (translated from the Russian): "Hardy-type inequalities play a large role in mathematical analysis and its applications, as well as in the theory of differential equations. Sharp estimates for the norms of Hardy-type operators on the set of all nonnegative measurable functions are well known. However, in the theory of function spaces it is often necessary to use such inequalities on a set of functions with additional (quasi) monotonicity conditions, in which case the classical Hardy inequalities lose their accuracy. Examples of functions with two monotonicity conditions are second nonincreasing permutations of functions and moduli of continuity in the theory of function spaces, and also the Peetre K-functional in interpolation theory. Sharp two-weighted Hardy-type inequalities have been obtained only recently. In the present paper we study three-weighted Hardy-type inequalities on the set of functions with the monotonicity properties with respect to two given functions."