  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 $L^p$-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 \cite[Theorem
7.7.5]{Hormander} 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.
