与Fourier积分算子、均匀化相关的调和分析问题之研究

经典的“Calderon-Zygmund理论”已经在调和分析、偏微分方程领域发挥了重要的作用。近年来，为了研究粗糙系数的椭圆和抛物方程，人们开始研究 “Beyond Calderon-Zygmund理论”，其中包括与微分算子相联系的函数空间理论、奇异积分理论等。至今，这个课题已经取得了丰富的成果。为了研究粗糙系数的双曲方程，本项目的第一部分拟研究和发展与双曲方程相关的调和分析理论。注意到，双曲方程解算子是Fourier积分算子的特例。我们将具体研究与Fourier积分算子相关的Hardy空间刻划及对偶问题， 粗糙系数的双曲方程正则性问题。本项目的第二部分关注非光滑区域上偏微分方程的均匀化问题，这是偏微分方程研究的重要课题之一。 我们拟发展和运用调和分析的方法来研究非光滑区域上Maxwell方程在大尺度的均匀化问题。该问题的解决将是非光滑区域的均匀化理论的一个深层次进展。

Classical “Calderon-Zygmund theory” has played an important role in harmonic analysis and PDE fields. Recently, in order to study elliptic or parabolic equation with rough coefficients, researchers start to study a kind of “Beyond Calderon-Zygmund theory”, including the theory of function spaces and singular integrals associated to differential operators, and so on. Up to now, rich results have been obtained in this topic. In order to study hypobolic equation with rough coefficients, the first part of this project will study and develop a kind of harmonic theory related to hypobolic equation. Note that the solution operators to hypobolic equation are the special cases of Fourier integral operators. Specifically, we will study some equivalent characterizations and dual problem for Hardy spaces associated to Fourier integral operators, and the regularity of the solution to hypobolic equation with rough coefficients. The second part of this project will focus on homogenization problems of PDE on nonsmooth domain, which are one of the important topics in researches of PDE. We will devolep and apply harmonic methods to study the problems of Maxwell equations at large scales in homogenization in nonsmooth domain. This will be a further progress in homogenization theory.