Sharp global L2 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 L2-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 L2-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 L2-boundedness and determines the sharp L2-estimates in the high- and low-frequency regimes. As an application, we determine the exact range of anisotropic homogeneous L2-Sobolev estimates for the associated global Radon transform.
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BibTeX
@misc{kang2026sharp,
title = {{Sharp global \(L^2\) estimates for oscillatory integral operators with quasi-homogeneous phases}},
author = {Minbeom Kang and Joonil Kim},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36969},
eprint = {2610.36969},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36969},
}
RIS
TY - UNPB
TI - Sharp global L^2 estimates for oscillatory integral operators with quasi-homogeneous phases
AU - Minbeom Kang
AU - Joonil Kim
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36969
UR - https://arxiv.org/abs/2610.36969
AB - 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.
ER -