We characterize the weighted bilinear Hardy inequality (₀^ (H₂ (f, g) (x) ) ^qu (x) \, dx) ^1/q C (₀^f^p₁ (x) v₁ (x) \, dx) ^1/p₁ (₀^g^p₂ (x) v₂ (x) \, dx) ^1/p₂ for all f, g 0, where H₂ (f, g) (x) =Hf (x) H^*g (x) is the product of the Hardy operator and its adjoint. All cases 1<p₁, p₂, q< have been covered. We also point out that bilinear Hardy inequalities are equivalent to a pair of Hardy inequalities.
Mohanty et al. (Mon,) studied this question.