A reduction theorem for supremum operators

We show that the two-weight Hardy inequality restricted to nonincreasing functions, namely ∫ 0 ∞ ∫ 0 t f * ( s ) d s q w ( t ) d t 1 / q ≲ ∫ 0 ∞ f * ( t ) p v ( t ) d t 1 / p , where 0 < p ⩽ 1 and 0 < q < ∞ , is equivalent to slightly different inequalities. Consequently, we can reduce this inequality to a pair of unrestricted inequalities (a reduction theorem). As an application, we prove an analogous assertion for a three-weight inequality involving a supremum operator, namely ∫ 0 ∞ sup t ⩽ s < ∞ u ( s ) f * * ( s ) q w ( t ) d t 1 / q ≲ ∫ 0 ∞ f * ( t ) p v ( t ) d t 1 / p , in which the weight u is assumed to be continuous on ( 0 , ∞ ) . This result in turn enables us to establish necessary and sufficient conditions on the weights ( u , v , w ) for which this inequality holds.