Endpoint estimates for one-dimensional oscillatory integral operator

Abstract

The one-dimensional oscillatory integral operator associated to a real analytic phase $S$ is given by $$T_{\lambda }f(x)={\int }_{-\infty }^{\infty }{e}^{i\lambda S(x,y)}\chi (x,y)f(y)dy.$$ In this paper, we obtain a complete characterization for the mapping properties of $T_{\lambda }$ on $L^p(\mathbb{R})$ spaces, namely we prove that $\Vert T_{\lambda }{\Vert }_p\lesssim |\lambda {|}^{-\alpha }\Vert f{\Vert }_p$ for some $\alpha >0$ if and only if the point $(\frac{1}{\alpha p},\frac{1}{\alpha p^{\prime }})$ lies in the reduced Newton polygon of $S$, and this estimate is sharp if and only if it lies on the reduced Newton diagram.

30 pages. All comments are welcome! arXiv admin note: text overlap with arXiv:1511.05233

References in corpus (2)

• Sharp estimates for trilinear oscillatory integrals and an algorithm of two-dimensional resolution of singularities
• Maximal Decay Inequalities for Trilinear oscillatory integrals of convolution type