Recently, "A dichotomy for Hörmander-type oscillatory integral operators" by Shaoming Guo, chair professor of Chern Institute of Mathematics, Nankai University, was published online in the top international mathematics journal Inventiones mathematicae. The paper is co-authored by Hong Wang from NYU Courant Institute of Mathematical Sciences and Ruixiang Zhang from UC Berkeley.

This paper studies the Hörmander-type oscillatory integral operators in harmonic analysis. In 1991, Fields Medalist Jean Bourgain studied systematically, in his ground breaking work, the problem of oscillatory integrals first posed in 1973 by another Fields Medalist Lars Hörmander. Shaoming Guo and his collaborators further studied Hörmander's problem and proposed the Bourgain condition for Hörmander-type oscillatory integral operators, which holds for many problems in harmonic analysis and related fields, including the Fourier restriction problem, the Bochner-Riesz problem, the problem of the local smoothing estimate for the Schrdinger equations, and the problem of the resolvent estimate for the Laplacian in Euclidean space. These problems were all studied separately before, until Shaoming Guo and his collaborators put them in the same framework and gave the best estimate so far, which provided a new direction for research on Hörmander-type oscillatory integral operators.

Shaoming Guo has long been engaged in research on harmonic analysis and related fields such as analytic number theory, geometric measure theory and partial differential equation. In these fields he has made remarkable achievements. He was selected for national high-level talent program in 2023 and joined Chern Institute of Mathematics, Nankai University in 2024.

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

(Resource: NKU News; translated by CIM)