Two-dimensional discrete Hardy inequalities (Q1586348)

The two-dimensional discrete Hardy operator is defined by \[ ({\mathbf H}{\mathcal F})(n_1,n_2):=\sum_{i_1=1}^{n_1} \sum_{i_2=1}^{n_2} {\mathcal F} (i_1,i_2), \] where \({\mathcal F}(i_1,i_2)\) denotes a nonnegative double sequence. The author, among others, gives a necessary condition for \(H:\ell_v^p \to \ell^q_u,\) i.e., derives sufficient conditions on nonnegative double sequences \({\mathcal U}(n_1,n_2)\) and \({\mathcal V}(n_1,n_2)\) for which there is a constant \(C>0\) such that \[ \left(\sum_{n_1=1}^\infty \sum_{n_2=1}^\infty\bigl({\mathbf H}{\mathcal F}\bigr)^q(n_1,n_2) {\mathcal U} (n_1,n_2)\right)^{1/q} \leq C\left(\sum_{n_1=1}^\infty\sum_{n_2=1}^\infty {\mathcal F}^p\bigl(n_1,n_2\bigr) {\mathcal V}\bigl(n_1,n_2\bigr)\right)^{1/p} \] holds for all double sequences \({\mathcal F} (\cdot,\cdot)\geq 0\) and \(1<p\leq q<\infty.\)