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Multidimensional bilinear Hardy inequalities. (English. Russian original) Zbl 1440.26024

Dokl. Math. 100, No. 1, 374-376 (2019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 487, No. 5, 496-498 (2019).
Let \(0< q< \infty\), \(1\leq p_1, p_2 < \infty\). The following bilinear inequality is considered: \[ \displaystyle \left( \int_{{\mathbb R}^n} \left( \int_{B(|x|)} f\right)^q \left( \int_{B(|x|)} g\right)^q u(x) \ dx \right)^{\frac 1q} \leq C \left( \int_{{\mathbb R}^n} f^{p_1}v_1 \right)^{\frac{1}{p_1}} \left( \int_{{\mathbb R}^n} g^{p_2}v_2 \right)^{\frac{1}{p_2}}, \] where \(u\), \(v_1\), \(v_2\), \(f\) and \(g\) are non-negative Lebesgue-measurable functions on \(\mathbb{R}^n\), \(B(|x|)\) is a ball in \(\mathbb{R}^n\). The problem is to describe the constant \(C\) which does not depend on the functions \(f\) and \(g\) and is assumed to be least possible.
We have the following result:
If \(1< \max (p_1,p_2) < q < \infty\), then the best constant \(C\) in the above-mentioned inequality satisfies the relation \(C\approx A_1\), where \[ A_1 := \sup_{\alpha >0} \left( \int_{|x| \leq \alpha} v_1^{1-p_1'} \right)^{\frac{1}{p_1'}} \left( \int_{|x| \leq \alpha} v_2^{1-p_2'} \right)^{\frac{1}{p_2'}} \left( \int_{|x| \geq \alpha} u \right)^{\frac{1}{q}}, \] where \(p'_i=\frac{p_i}{p_i -1}\), \(i=1,2\). The relation \(A\approx B\) means that there exist constants \(c_1,c_2\) such that \( c_1A \leq B \leq c_2A\).
Results for other arrangements of parameters \(q\), \(p_1\), \(p_2\) are also given.

MSC:

26D15 Inequalities for sums, series and integrals
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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