We consider the discrete Laplacian Δ on the cubic lattice Z^d, and deduce estimates on the group e^itΔ and the resolvent (Δ−z)⁻¹,weightedbyℓ^q (Z^d)-weights for suitable q≥2. We apply the obtained results to discrete Schrödinger operators in dimension d≥3 with potentials from ℓ^p(Z^d) with suitable p≥1.