Uniform decay estimates for a class of oscillatory integrals and applications
The paper deals with estimates for one dimensional oscillatory integrals of the form \[ I_\alpha (x,t)=\int _0^\infty \xi ^\alpha \text{e}^{it(p(\xi)-\xi x)}d\xi , \quad t>0, \quad x\in {\mathbb R}, \] where \(p(\xi)\) is a real polynomial of degree \(m\geq 3\). In the paper long-time and short-time uniform estimates of \(I_\alpha (x,t)\) for the case \(\alpha \in (0,m/2-1)\) are derived. These decay estimates are applied to a linearized system of Kadomtsev-Petviashvili equations \[ u_t+p(\partial _x)u+v_y=0, \qquad u_y=v_x , \] where \(p\) is a real odd polynomial of degree \(m\geq 3\). Strichartz type estimates with smoothing for linearized KP equations close the paper.
linearized Kadomtsev-Petviashvili equations
scientific article; zbMATH DE number 1858740