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。

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.