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In this paper, new equivalence theorems for the boundedness of thecomposition of a quasilinear operator $T$ with the Hardy and Copson operatorsin weighted Lebesgue spaces are proved. The usefulness of the obtained resultsis illustrated in the case of weighted Hardy-type and weighted iteratedHardy-type inequalities.

New equivalence theorems for weighted inequalities involving the  composition of monotone quasilinear operators with the Hardy and Copson  operators and their applications