Characterization of the matrix whose norm is determined by its action on decreasing sequences:The exceptional cases

Let $A=(a_{j,k})_{j,k ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $|A|_{ell_p,ell_q}$ are determined by their actions on non-negative decreasing sequences, where one of $p$ and $q$ is 1 or $infty$. The conditions forcing on $A$ are sufficient and they are also necessary for non-negative finite matrices.