Weighted bilinear Hardy inequalities

We characterize the weights w, $w_1$, $w_2$ such that the weighted bilinear Hardy inequality$\left(\underset{a}{\overset{b}{\int }}\right(\underset{a}{\overset{x}{\int }}f{)}^q(\underset{a}{\overset{x}{\int }}g{)}^{q}w(x)dx{)}^{\frac{1}{q}}?C(\underset{a}{\overset{b}{\int }}{f}^{p_1}{w}_1{)}^{\frac{1}{p_1}}(\underset{a}{\overset{b}{\int }}{g}^{p_2}{w}_2{)}^{\frac{1}{p_2}}$ holds for all nonnegative functions f and g, with a positive constant C independent of f and g, for all possible values of q, $p_1$ and $p_2$ with $1<q,p_1,p_2<\infty$. We also characterize the good weights for the weighted bilinear n-dimensional Hardy inequality to hold.