A dichotomy for Hormander-type oscillatory integral operators

• 成果类型: Article
• 署名作者: Guo, Shaoming; Wang, Hong; Zhang, Ruixiang
• 署名单位: Nankai University; University of Wisconsin System; University of Wisconsin Madison; New York University; University of California System; University of California Los Angeles; University of California System; University of California Berkeley
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01288-8
• 发表日期: 2024
• 页码: 503-584

摘要

In this paper, we first generalize the work of Bourgain (Geom. Funct. Anal. 1(4):321-374, 1991) and state a curvature condition for H & ouml;rmander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for H & ouml;rmander-type oscillatory integral operators satisfying Bourgain's condition, they satisfy the same L-p bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain's condition fails, then the L-infinity -> L-q boundedness always fails for some q=q(n)>2n/n-1, extending Bourgain's three-dimensional result (Geom. Funct. Anal. 1(4):321-374, 1991). On the other hand, if Bourgain's condition holds, then we prove L-p bounds for Hormander-type oscillatory integral operators for a range of p that extends the currently best-known range for the Fourier restriction conjecture in high dimensions, given by Hickman and Zahl (A note on Fourier restriction and nested polynomial wolff axioms, 2020, arXiv:2010.02251). This gives new progress on the Fourier restriction problem, the Bochner-Riesz problem on R-n, the Bochner-Riesz problem on spheres S-n, etc.