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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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