In 1998, P. Schultz established the sharp dispersive estimates for the discrete wave equations on ℤ^d for d = 2, 3. We extended the results to the cases d = 4 and 5. The key step in our proof is uniform estimates for the degenerate oscillatory integrals, which is achieved by using an algorithm introduced by V. N. Karpushkin in 1983. I will also discuss some results in higher dimensions.

This talk is based on the joint work with Jiawei Cheng and Bobo Hua.