Weighted norm inequalities for generalized Fourier-type transforms and applications

We obtain necessary and sufficient conditions on weights for the generalized Fourier-type transforms to be bounded between weighted Lp-Lq spaces. As an important example, we investigate transforms with kernel of power type, as for instance the sine, Hankel, or Hα 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.