  In this paper, the index groups for which the weighted Young's inequalities
hold in both continuous case and discrete case are characterized. As
applications, the index groups for the product inequalities on modulation
spaces are characterized, we also obtain the weakest conditions for the
boundedness of bilinear Fourier multipliers on modulation spaces in some sense.
For the fractional integral operator, the sharp conditions for the boundedness
of power weighted Lp-Lq estimates in both continuous case and discrete case are
obtained. By a quite different approach from others, our theorems optimize some
previous results which are committed to finding sharp conditions for some
classical convolution inequalities.
