若干积分算子的研究

本项目拟重点就三类问题展开研究：.第一、积分的微分问题。我们将主要围绕Halo猜测、Kakeya型微分问题与区间基微分问题开展研究。.第二、奇异积分的有界性问题。我们将主要围绕具粗糙球面函数齐次核CZ算子的端点有界性问题、多线性奇异积分算子的有界性问题等展开研究。.第三、Fourier积分算子的有界性问题。我们将主要围绕乘子算子特别是幺模乘子的有界性问题、超奇性振荡核积分算子特别是沿曲线情形的相关问题展开研究。.我们将尽力能为本项目的三大核心问题（一般微分基的Halo猜测、H1球面粗糙齐次核CZ算子的端点有界性问题、Lp乘子的特征刻划）的解决做些贡献，但至少我们能在与它们有关联的问题的研究上有所突破。

The project is intended to focus on three types of issues:.1. The differentiation problem of integrals. We will focus on the Halo conjecture, the Kakeya-type differentiation problem and the interval-based differentiation issues. .2. The boundedness problem of singular integrals. We will mainly focus on the end-point boundedness of CZ operators of homogeneous kernels with rough spherical functions, boundedness of multi-linear singular integral operators, etc. .3. The boundedness problem of Fourier oscillatory integral operators. We will mainly focus on the boundedness problem of multiplier operators, especially the unimodular multipliers, and the super-singular oscillating integral operators, especially the related problems along curves. .We hope to make a breakthrough in the research on some of the three core issues of this project: the Halo conjecture of the general differential bases, the endpoint boundedness of CZ operators of homogeneous kernels with H1 spherical functions, the characterization of the Lp multipliers.