  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, $0 < p,q \leq \infty$ 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.
