On Weighted Bilinear Inequalities with Mixed Hardy Operators

We characterize the weighted bilinear Hardy inequality $$(∫₀∞( H_2(f,g)(x))qu(x)\,dx)1/q ≤ C(∫₀∞f^{p₁}(x)v_1(x)\,dx)1/p_1 (∫₀∞g^{p₂}(x)v_2(x)\,dx)1/p_2$$ 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 $$1<p_1, p_2, q<∞$$ have been covered. We also point out that bilinear Hardy inequalities are equivalent to a pair of Hardy inequalities.