Bilinear weighted Hardy inequality for nonincreasing functions

Abstract

We characterize the validity of the bilinear Hardy inequality for nonincreasing functions \[ \|f^{**} g^{**}\|_{L^q(w)} \le C \|f\|_{\Lambda^{p_1}(v_1)}\|g\|_{\Lambda^{p_2}(v_2)}, \] in terms of the weights $v_1$, $v_2$, $w$, covering the complete range of exponents $p_1,p_2,q\in (0,\infty]$. The problem is solved by reducing it into the iterated Hardy-type inequalities \begin{align*} \left( \int\limits_0^\infty \biggl( \int\limits_0^x (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}, \\ \left( \int\limits_0^\infty \biggl( \int\limits_x^\infty (g^{**}(t))^\alpha \varphi(t)\,\mathrm{d}t \biggr)^\frac{\beta}{\alpha} \psi(x)\,\mathrm{d}x \right)^\frac{1}{\beta} & \le C \biggl( \int\limits_0^\infty (g^*(x))^\gamma \omega(x) \,\mathrm{d}x \biggr)^\frac{1}{\gamma}. \end{align*} Validity of these inequalities is characterized here for $0\lt\alpha\le\beta\lt\infty$ and $0\lt\gamma\lt\infty$.