Sharp weighted convolution inequalities and some applications

The index groups for which weighted Young inequalities hold in both the continuous case and discrete case are characterized. As applications, the index groups for the product inequalities on modulation spaces are characterized, and we also obtain the weakest conditions for the boundedness of bilinear Fourier multipliers on modulation spaces. For the fractional integral operator, sharp conditions for the power weighted L-p - L-q estimates in both the continuous and discrete cases are obtained. By a novel unified approach, we complete some previous results on sharp conditions for some classical inequalities.