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Discrete bilinear Hardy inequalities. (English. Russian original) Zbl 1444.26031

Dokl. Math. 100, No. 3, 554-557 (2019); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 489, No. 5, 445-448 (2019).
The problem of the characterization of the bilinear discrete Hardy inequality is solved.
Let \( 0 < p_1, p_2, s < \infty \), \(u=\{ u_n\}\), \(v=\{v_n\}\) and \(w=\{w_n\}\) be fixed sequences of nonnegative numbers. Let \(a=\{a_n\}\) and \( b=\{b_n\}\) be arbitrary sequences of nonnegative numbers.
The symbol \(\approx\) is used as follows: \(A\approx B\) iff there exist constants \(c_1,c_2\) depending only on the parameters \(p_1\), \(p_2\) and \(s\) such that \(c_2 A \leq B \leq c_1 A\). If \(p\not= 1\), \(0<p<\infty\), the notation \(p':=\frac{p}{p-1}\) is used.
The bilinear discrete Hardy inequality has the following form: \[ \left[ \sum_{n=1}^{\infty} \left( \sum_{i=1}^{n} a_i \right)^s \left( \sum_{i=1}^{n} b_i \right)^s u_n \right]^{\frac 1s} \leq C \left( \sum_{n=1}^{\infty} a_n^{p_1} v_n \right)^{\frac{1}{p_1}} \left( \sum_{n=1}^{\infty} b_n^{p_2} w_n \right)^{\frac{1}{p_2}}, \quad \quad \quad (1) \] where the constant \(C\) does not depend on \(a\) or \(b\) and \(C\) is the best possible.
The following theorem gives information about the constant \(C\) when the parameter \(s\) is greater than \(p_1\) and \(p_2\).
Theorem 1. Inequality (1) holds with the best constant \(C\) if and only if \(A_i< \infty\) where \(C\approx A_i\), \(i=1,2,3,4\), and \(A_i\) is described as following:
(i) If \(1<\min (p_1,p_2) \leq \max (p_1,p_2) \leq s <\infty\), then \[ A_1:= \sup_{n\in \mathbb{N}} \left( \sum_{i=n}^{\infty} u_i \right)^{\frac{1}{s}} \left( \sum_{i=1}^{n} v_i^{1-p_1'} \right)^{\frac{1}{p_1'}} \left( \sum_{i=1}^{n} w_i^{1-p_2'} \right)^{\frac{1}{p_2'}}. \]
(ii) If \( 0< p_1 \leq 1 < p_2 \leq s <\infty\), then \[ A_2:=\sup_{n\in \mathbb{N}} \left( \sum_{i=n}^{\infty} u_i \right)^{\frac{1}{s}} \left( \sum_{i=1}^{n} w_i^{1-p_2'} \right)^{\frac{1}{p_2'}} \sup_{1\leq k\leq n} v_k^{\frac{1}{p_1}}. \]
(iii) If \( 0 < p_2 \leq 1 < p_1 \leq s \), then \[ A_3:=\sup_{n\in \mathbb{N}} \left( \sum_{i=n}^{\infty} u_i \right)^{\frac{1}{s}} \left( \sum_{i=1}^{n} v_i^{1-p_1'} \right)^{\frac{1}{p_1'}} \sup_{1\leq k\leq n} w_k^{\frac{1}{p_2}}. \]
(iv) If \(0<\max (p_1,p_2) \leq \min (1,s) <\infty\), then \[ A_4:= \sup_{n\in \mathbb{N}} v_n^{\frac{1}{p_1}} \left( \sum_{j=n}^{\infty} u_j \right)^{\frac{1}{s}} \sup_{1\leq k\leq n} w_k^{\frac{1}{p_2}} + \sup_{n\in \mathbb{N}} w_n^{\frac{1}{p_2}} \left( \sum_{j=n}^{\infty} u_j \right)^{\frac{1}{s}} \sup_{1\leq k\leq n} v_k^{\frac{1}{p_1}}. \] In this paper, other cases for the parameters \(p_1,p_2\) and \(s\) are also solved. Precisely, the constant \(C\) is given for the following domains:
\(\mathrm{II}_1\)
\( 1 < \min (p_1,p_2) \leq s < \max (p_1,p_2) <\infty\)
\(\mathrm{II}_2\)
\( 0 < \min (p_1,p_2) \leq \min (1,s) \leq 1 < \max (p_1,p_2) < \infty\)
\(\mathrm{II}_3\)
\(0<\min (p_1,p_2) \leq s < \max (p_1,p_2) \leq 1\)
\(\mathrm{III}_1\)
\( 0 < s <\min (p_1,p_2) \), \( \min (p_1,p_2) > 1 \)
\(\mathrm{III}_2\)
\( 0 < s < \min (p_1,p_2) \leq 1 < \max (p_1,p_2) <\infty\)
\(\mathrm{III}_3\)
\( 0 < s < \min (p_1,p_2) \leq \max (p_1,p_2) \leq 1\).

MSC:

26D15 Inequalities for sums, series and integrals
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References:

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