# 微分方程式セミナー

2019/7/5(Fri)

15:30--17:00

Joonil Kim

Yonsei University

Uniform Estimates of Oscillatory Integrals and Linear Dispersive PDEs

Given real valued polynomials $P$ on $\mathbb{R}^n$ and various unbounded domains $D \subset \mathbb{R}^n$, we consider the oscillatory integrals $$I(P, D, \lambda) = \int_D e^{i\lambda P(y)} dy.$$ We establish a criterion on $(P, D)$ to determine the convergence of these integrals, and find the oscillation indices when they converge. These indices are described in terms of a generalized notion of Newton polyhedra associated with $(P, D)$. Next we discuss about the estimates of the oscillatory integrals associated with $P_\alpha(y)$ uniformly in $\alpha\in \mathbb{R}^m$. When $(P, D)$ for $D=\mathbb{R}^n$ satisfies the criterion of the vector polynomial version $(y_1, \cdots, y_n, P(y))$, we obtain the Strichartz estimates for the following general linear propagators: $e^{itP(\partial)}(f)(x) \text{ where } \partial=\left(\frac{\partial_{x_1}}{I}, \cdots, \frac{\partial_{x_n}}{i} \right).$