Weighted inequalities for discrete iterated kernel operators

We develop a new method that enables us to solve the open problem of characterizing discrete inequalities for kernel operators involving suprema. More precisely, we establish necessary and sufficient conditions under which there exists a positive constant $C$ such that \begin{equation*} \Bigg(\sum_{n\in$\mathbb{Z}$}\Bigg(\sum_{i=-\infty}^n {U}(i,n) a_i\Bigg)^{q} {w}n\Bigg)^{\frac{1}{q}} \le C \Bigg(\sum{n\in$\mathbb{Z}$}a_n^p {v}n\Bigg)^{\frac{1}{p}} \end{equation*} holds for every sequence of nonnegative numbers $\{a_n{\}}_{n\in \mathbb{Z}}$ where $U$ is a kernel satisfying certain regularity condition, $0<p,q\le \mathrm{\infty }$ and $\{v_n{\}}_{n\in \mathbb{Z}}$ and $\{w_n{\}}_{n\in \mathbb{Z}}$ are fixed weight sequences. We do the same for the inequality \begin{equation*} \Bigg( \sum{n\in$\mathbb{Z}$} w_n \Big[ \sup_{-\infty<i\le n} U(i,n) \sum_{j=-\infty}^{i} a_j \Big]^q \Bigg)^{\frac{1}{q}} \le C \Bigg( \sum_{n\in$\mathbb{Z}$} a_n^p v_n \Bigg)^{\frac{1}{p}}. \end{equation*} We characterize these inequalities by conditions of both discrete and continuous nature.