Weighted estimates for the discrete Laplacian on the cubic lattice

We consider the discrete Laplacian $\Delta$ on the cubic lattice $\mathbb Z^d$, and deduce estimates on the group $e^{it\Delta}$ and the resolvent $(\Delta-z)^{-1}$, weighted by $\ell^q(\mathbb Z^d)$-weights for suitable $q\geq 2$. We apply the obtained results to discrete Schr\"odinger operators in dimension $d\geq 3$ with potentials from $\ell^p(\mathbb Z^d)$ with suitable $p\geq 1$.