Weighted bilinear Hardy inequalities

We characterize the weights w  , w1w1, w2w2 such that the weighted bilinear Hardy inequality(∫ab(∫axf)q(∫axg)qw(x)dx)1q⩽C(∫abfp1w1)1p1(∫abgp2w2)1p2 holds for all nonnegative functions f and g, with a positive constant C independent of f and g, for all possible values of q  , p1p1 and p2p2 with 1