  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 which integrates over some right
neighbourhood of zero is equivalent to the boundedness of the same operator on
the cone of positive non-increasing functions. It is well known that an
analogous inequality for integration away from zero is not true. However, as we
show in this paper, the equivalence of the restricted inequality for the
non-restricted one is still true for certain class of kernel-type operators,
regardless of the measure of the integration domain.
