Ben-Artzi, M.; Saut, J.-C. Uniform decay estimates for a class of oscillatory integrals and applications. (English) Zbl 1016.35006 Differ. Integral Equ. 12, No. 2, 137-145 (1999). 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. Reviewer: Jan Franců (Brno) Cited in 27 Documents MSC: 35B45 A priori estimates in context of PDEs 35Q53 KdV equations (Korteweg-de Vries equations) 35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients Keywords:long-time decay estimate; short-time decay estimate; linearized Kadomtsev-Petviashvili equations; Strichartz type estimates × Cite Format Result Cite Review PDF