双参数振荡积分算子及海森堡群上的Moser-Trudinger不等式

本项目拟运用调和分析,泛函分析的的思想和方法研究双参数振荡奇异积分算子及海森堡群上的几何不等式.首先我们将研究最简单的一类双参数振荡积分算子即带振荡项的双参数Hilbert变换的有界性.然后我们将尝试建立适当的覆盖引理和插值理论，然后利用这些工具去探讨最一般的双参数振荡奇异积分算子的有界性.最后我们将深化项目申请人之前发现的研究几何不等式的非对称重排方法，然后使用该方法建立海森堡群上具有最佳增长的Moser-Trudinger不等式及其最佳常数.

The main purpose of this proposal is to study the boundedness of bi-parameter oscillatory singular integrals and Moser-Trudinger inequality with exact growth condition on Heisenberg group with the ideas and methods of harmonic analysis and functional analysis. First, we will try to set up the boundedness of the simplest case of bi-parameter oscillatory singular integrals that's the oscillatory biparameter hilbert transform . After that, we will develop some covering lemma and Interpolation tool, and using these tools we will study the Then we will study the boundedness of bi-parameter oscillatory singular integrals. At last, we will deepen the rearrangment-free method used by the proposer to consider the geometric inequality before, and build the Moser-Trudinger inequality with exact growth condition on Heisenberg group with this deepen rearrangment-free method.