粗糙核奇异积分算子的若干问题研究

The study of singular integral operators with non-smooth kernels is one of the most important project in harmonic analysis. And it plays an important role in partial differential equations. Recently, the sharp dependence of the singular integral operator norm on weighted spaces on the weight constant has been attracting attentions of many research workers. Until recently, the sharp Ap weighted estimates for Calderon-Zygmund singular integral operator have been absolutely solved by T. Hytonen. This Program will consider the sharp Ap weighted estimates for singular integral operators with non-smooth kernels, including Littlewood-Paley operators associateed to operators、Riesz transforms associateed to operators、fractional integral operator associateed to operators、Spectral multiplier associateed to operators and singular integral operators with non-smooth kernels of Duong-McIntosh and so on.

粗糙核奇异积分算子的研究是调和分析的一个重要课题之一，其在微分方程中有着重要的应用。另一方面，近年来有关奇异积分算子的加权模和权常数的最佳依赖关系问题引起了学者们的广泛注意。直到最近，T. Hytonen等才彻底解决了Calderon-Zygmund 奇异积分算子的最佳Ap 权估计问题。本项目将在此基础上研究粗糙核奇异积分算子的最佳Ap 权估计。所研究的算子包括：与算子相联系的Littlewood-Paley函数、Riesz变换、分数次积分算子、谱乘子以及满足Duong-McIntosh 核条件的非光滑核奇异积分算子等。

近年来有关奇异积分算子的加权模和权常数的最佳依赖关系问题引起了学者们的广泛注意。直到最近，T. Hytonen 等才彻底解决了Calderon-Zygmund 奇异积分算子的最佳Ap 权估计问题。本项目主要研究了与微分算子相联系的Littlewood-Paley 函数的最佳Ap权估计，Hormander 型谱乘子的加权估计，加权局部Hardy空间的等价刻画等问题。