交换子在L1端点的估计及其应用

The endpoint estimate of commutator is alway an interesting and important problem. Since the singularity of commutator in higher than singular integral operator, in general, the commutator does not rely on a weak type (1,1) estimate. Joint pointwise decomposition theory of BMO function, and theory of grand Lebesgue space, we discuss the endpoint estimate of commutator. The aim is to reveal the L^{1)}(\Omega) boundedness of commutator and its application in elliptic equation. We focus the research on following problems, (1) we discuss the commutator is bounded from L^{1}(\Omega) to L^{1)}(\Omega); (2) In weighted case, we consider the commutator from L^{1}_{w}(\Omega) to L^{1)}_{w}(\Omega); (3) In bounded domain, we discuss the existence of elliptic equations. In all, the reaearch discuss the boundedness of commutator in L^{1}(\Omega) . It not only improve the estimate of commutator, but also the corresponding results are appied to the existence of elliptic equations.

交换子在端点的估计一直是一个重要而有趣的问题。由于交换子的奇性比积分算子稍高，因此端点的估计无法做到弱(1,1)有界。本研究结合BMO函数的逐点分解原理以及极大Lebesgue函数空间理论，对交换子的端点问题展开研究。目的是揭示交换子在L1端点处的估计及其在椭圆型方程中的应用，重点研究：(1)有界域\Omega上，讨论交换子从L^{1}(\Omega)空间到L^{1)}(\Omega)空间的有界性；(2)加权情形下，讨论交换子从L^{1}_{w}(\Omega)空间到L^{1)}_{w}(\Omega)空间的有界性；(3)在端点L^{1)}(\Omega)空间上，讨论非散度椭圆型方程解的存在性及其不等式估计。

本项目主要探求交换子在L1端点的估计及其应用。项目组成员围绕该类问题展开研究。主要内容有：交换子在加权Morrey空间上的有界性估计；带变量核的Marcinkiewicz积分交换子的加权Campanato估计；Campanato函数与\theta型积分算子的交换子估计；多线性Marcinkiewicz高阶交换子在变指数Herz-Morrey 空间上的有界性估计以及在非散度分数次拉普拉斯方程组解的端点估计。这些结果使得我们能更全面的了解交换子的性质及其相关应用。但在交换子在L1端点的估计这一关键你问题上未取得实质性的突破。