Two-dimensional discrete Hardy operators in weighted Lebesgue spaces with mixed norms (Q2709547)

Let \((Hf)(m,n) = \sum^m_{i=1} \sum^n_{j=1} f(i,j)\), \(m,n \in\mathbb N\), be the two-dimensional (discrete) Hardy operator and \(u=\{u(m,n)\}\), \(v=\{v(m,n)\}\) weights (that is non-negative sequences). The aim of the paper is to find some particular weights \(u\) and \(v\) such that the Hardy inequality NEWLINE\[NEWLINE \Biggl(\sum^{\infty}_{n=1} \Biggl[\sum^{\infty}_{m=1} (Hf)^r(m,n) u(m,n)\Biggr]^{s/r}\Biggr)^{1/s} \leq C \Biggl(\sum^{\infty}_{n=1} \Biggl[ \sum^{\infty}_{m=1} f^p(m,n) v(m,n) \Biggr]^{q/p} \Biggr)^{1/q} \tag \(*\) NEWLINE\]NEWLINE holds for all non-negative sequences \(f=\{f(m,n)\}\) (here \(C\) is a positive constant, \(1<p\leq q < \infty\) and \(1<r\leq s<\infty)\). The author also considers the inequality \((*)\) with some variants of the operator \(H\).