Weighted norm inequalities for integral transforms with splitting kernels

Given an integral transform on the positive real line, we say that its kernel is
splitting if it satisfies upper pointwise estimates given by products of two
functions, each of them taken in a different variable. We discuss necessary and/or
sufficient conditions weighted norm inequalities involving these transforms to
hold. Sharpness of the results is directly related to the sharpness of the upper
estimates for the kernels that we find.