  We obtain necessary and sufficient conditions on weights for the generalized
Fourier-type transforms to be bounded between weighted $L^p-L^q$ spaces. As an
important example, we investigate transforms with kernel of power type, as for
instance the sine, Hankel or $\mathscr{H}_\alpha$ transforms. The obtained
necessary and sufficient conditions are given in terms of weights, but not in
terms of their decreasing rearrangements, as in several previous
investigations.
