Sharp global \(L^2\) estimates for oscillatory integral operators with quasi-homogeneous phases

Abstract

For one-dimensional oscillatory integral operators with real-analytic phases, the seminal work of Phong and Stein established that the Newton polygon of the mixed Hessian of the phase determines the sharp local \(L^2\)-decay. The local theory relies on spatial localization near the origin and therefore does not capture the global geometry of the real zero set of the Hessian. For globally defined operators, unbounded real zero curves of the Hessian can make the global decay strictly slower than the local decay or destroy \(L^2\)-boundedness altogether. We establish a global analogue of the Phong--Stein theory for quasi-homogeneous polynomial phases. We introduce a family of Newton polygons adapted to the regions near and away from the real zero curves of the Hessian, together with the corresponding box-area scales. The relative geometry of these Newton polygons and area scales characterizes global \(L^2\)-boundedness and determines the sharp \(L^2\)-estimates in the high- and low-frequency regimes. As an application, we determine the exact range of anisotropic homogeneous \(L^2\)-Sobolev estimates for the associated global Radon transform.