Reduction principle for a certain class of kernel-type operators

The classical Hardy-Littlewood inequality asserts that the integral of a product of two functions is always majorized by that of their non-increasing rearrangements. One of the pivotal applications of this result is the fact that the boundedness of an integral operator acting near zero is equivalent to the boundedness of the same operator restricted to the cone of positive non-increasing functions. It is well known that an analogous inequality for integration away from zero is not true. We will show in this paper that, nevertheless, the equivalence of the two inequalities is still preserved for certain rather general class of kernel-type operators under a mild restriction and regardless of the measure of the underlying integration domain.