A paper titled “A Dichotomy for Hörmander-type Oscillatory Integral Operators”, co-authored by Prof. Guo Shaoming, a Chair Professor at the Chern Institute of Mathematics at Nankai University, along with Wang Hong from the Courant Institute of Mathematical Sciences at New York University and Zhang Ruixiang from the University of California, Berkeley, has been published online in the prestigious mathematics journal Inventiones Mathematicae.

This paper presents a significant advancement in the study of Hörmander-type oscillatory integral operators in harmonic analysis. The research builds on the foundational work of Fields Medalist Jean Bourgain, who systematically investigated the problem of oscillatory integrals posed by another Fields Medalist, Lars Hörmander, in 1973, in his seminal 1991 paper. Prof. Guo Shaoming and his collaborators have further explored the Hörmander’s problem by proposing a new curvature condition, known as Bourgain condition, which is crucial for understanding phase functions related to the Fourier restriction problem and the Bochner-Riesz problem. This condition integrates several previously isolated issues in harmonic analysis, including local smoothing estimates for the Schrödinger equations and resolvent estimates for the Laplacian in Euclidean space. By unifying these studies under a cohesive framework, the three researchers have provided the best current estimates for Hörmander-type oscillatory integral operators and opened new avenues for future research in the field.

Prof. Guo Shaoming specializes in harmonic analysis and related fields, including analytic number theory, geometric measure theory, and partial differential equations. He has made significant academic contributions in these areas. In 2024, he joined the Chern Institute of Mathematics at Nankai University as a full-time faculty member.

Link to the paper:https://link.springer.com/article/10.1007/s00222-024-01288-8

（Edited and translated by Nankai News Team.)