Isroil Ikromov - On uniform estimates for oscillatory integrals

We consider oscillatory integrals with a smooth phase depending on parameters. The estimates for oscillatory integrals by a function depending only on a large parameter are used in applications. Such kind of bounds are called uniform (with respect to the other parameters) estimates for oscillatory integrals. There are well-known uniform estimates proved by Van der Corput, Vinogradov, Duistermaat, Karpushkin, Phong, Stein, Gressman and so on. We discuss issues related to the discrete Klein-Gordon equation. Also, we plan to demonstrate a proof of uniform estimates for oscillatory integrals. This is based partly on the joint work with Jean-Claude Cuenin "Sharp time decay estimates for the discrete Klein-Gordon equation".