## Bilinear weighted Hardy inequality for nonincreasing functions.(English)Zbl 1359.26020

Summary: We characterize the validity of the bilinear Hardy inequality for nonincreasing functions
$\|f^{\ast\ast} g^{\ast\ast}\|_{L^q(w)} \leq C \|f\|_{\Lambda^{p_1}(v_1)}\|g\|_{\Lambda^{p_2}(v_2)},$
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
$\left(\int\limits_0^\infty\left(\int\limits_0^x (g^{\ast\ast}(t))^\alpha \varphi(t)\,\mathrm{d}t \right)^{\frac{\beta}{\alpha}} \psi(x)\,\mathrm{d}x \right)^{\frac1{\beta}} \leq C \left( \int\limits_0^\infty(g^\ast(x))^\gamma \omega(x) \,\mathrm{d}x \right)^{\frac1\gamma},$
$\left(\int\limits_0^\infty\left(\int\limits_x^\infty(g^{\ast\ast}(t))^\alpha \varphi(t)\,\mathrm{d}t \right)^{\frac\beta\alpha} \psi(x)\,\mathrm{d}x \right)^{\frac1\beta}\leq C \left(\int\limits_0^\infty(g^\ast(x))^\gamma \omega(x) \,\mathrm{d}x \right)^{\frac1\gamma}.$
Validity of these inequalities is characterized here for $$0<\alpha\leq\beta<\infty$$ and $$0<\gamma<\infty$$.

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators 47G10 Integral operators
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