9月14日，TOP小编从南开大学了解到，该校陈省身数学所教授郭少明与合作者在全球四大顶尖数学期刊《Inventiones mathematicae》上发表题为“A dichotomy for Hörmander-type oscillatory integral operators”的研究论文。
值得注意的是，南开上一篇数学四大刊论文是中国科学院压实张伟平2017年在《Annals of Mathematics》发表的独著，论文题为“Positive Scalar curvature on foliations”。
论文摘要
In this paper, we first generalize the work of Bourgain (Geom. Funct. Anal. 1(4):321–374, 1991) and state a curvature condition for Hörmander-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 Hörmander-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 \(L^{\infty } \to L^{q}\) boundedness always fails for some \(q= q(n) > \frac{2n}{n-1}\), 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 Hörmander-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 \(\mathbb{R}^{n}\), the Bochner-Riesz problem on spheres \(S^{n}\), etc.
孙崧简介
郭少明，于2024年5月入职南开大学，在陈省身数学研究所担任教授。回国前任威斯康星大学麦迪逊分校副教授。
2010年本科毕业于于北京邮电大学，2012年和2015年在德国博恩大学获得硕士、博士学位。曾在美国印第安纳大学布鲁明顿分校、美国加州伯克利MSRI从事博士后研究工作。
编辑、审核：大可
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