Аннотация:
We characterize the weighted bilinear Hardy inequality \begin{equation*} \left(\int_{0}^{\infty}( \widetilde H_2(f,g)(x))^{q}u(x)\,dx\right)^{1/q} \leqslant C\left(\int_{0}^{\infty}f^{p_{1}}(x)v_1(x)\,dx\right)^{1/p_1} \left(\int_{0}^{\infty}g^{p_{2}}(x)v_2(x)\,dx\right)^{1/p_2} \end{equation*} for all $f,g\geqslant 0$, where \begin{equation*} \widetilde H_{2}(f,g)(x)=Hf(x) \cdot H^{*}g(x) \end{equation*} is the product of the Hardy operator and its adjoint. All cases $1<p_1, p_2, q<\infty$ have been covered. We also point out that bilinear Hardy inequalities are equivalent to a pair of Hardy inequalities.
