Mathematical Sciences: Oscillatory Integrals, Singular Integrals, and Their Applications

摘要：潘 潘的调查将集中在各种奇异积分和振荡积分运营商自然产生的许多问题在调和分析和相关领域，包括复杂的分析和偏微分方程理论。随着大量数学问题的应用，这些积分因其重要性而得到越来越多的认可。虽然在这一领域的研究已经深入和富有成果，但仍有更多的东西有待了解。这个项目将涵盖当前感兴趣的许多主题，完成后将朝着充分理解这些运算符和数学分析中的相关问题迈出一大步。 被调查的对象形成的物理现象，如热传导和波动的数学建模的重要组成部分。它们也出现在涉及电路和信号处理的工程问题中。因此，了解这些物体不仅具有理论价值，而且对其他科学和工程领域具有指导意义。

Abstract: Pan Pan's investigation will be focused on various singular integral and oscillatory integral operators which arise naturally in many problems in harmonic analysis and related fields including complex analysis and the theory of partial differential equations. With applications to a large number of problems in mathematics, these integrals have received an increasing amount of recognition for their importance. While research in this area has been intensive and fruitful, much more remains to be understood. This project will cover many topics of current interest, which when finished will make a big stride toward the full understanding of these operators and related problems in mathematical analysis. The objects under investigation form an important part of the mathematical modeling of physical phenomena such as heat conducting and wave motion. They also appear in engineering problems involving electric circuits and signal processing. Thus, gaining understanding of such objects will not only have theoretical values but also provide guidance in other areas of science and engineering.