Estimates of the $p$-norms of solutions to difference equations and infinite systems of linear equations

We study the problem of estimating the $p$-norms ($1\le p\le\infty$) of solutions to inhomogeneous difference equations. The difference equations are considered as bi-infinite (infinite in both directions) systems of linear equations. We establish estimates in the case of a diagonally dominant Laurent matrix. Using this result and the idea of matrix decomposition into the product of matrices related to the decomposition of the characteristic polynomial, we propose some estimates in the case of an arbitrary nonsingular band Laurent matrix.