Uniform estimates for oscillatory integrals with parameter-dependent phases

We consider the oscillatory integrals with parameter-dependent phases. We decompose the integrals into a leading term and a remainder term. Instead of the pointwise estimate, we use some Lᵖ-estimate for the remainder term and get various uniform estimates when the phase functions satisfy certain conditions. This enables us to reduce the requirement of the smoothness on the phase functions, and hence improve the results in 7. 7. 5Hormander and also obtain a refined version of the well-known Van der Corput Lemma. Some applications on the uniform expansion of the Bessel functions and dispersive estimates are also given.