中文摘要
在齐型空间上建立了与奇异积分算子相关的极大算子、极大交换子的带一般权的加权Lp估计和弱端点估计；给出了带非光滑核的奇异积分算子、相应的极大算子、交换子以及与交换子相应的极大算子的带一般权的加权估计；研究了齐型空间上Hp空间的分子特征、弱Hp空间的新的分解特征并建立某些重要算子在Hp空间、弱Hp空间上的有界性，并由此得到了与Monge-Ampere奇异积分算子在Hp空间和弱Hp空间上的性质。关于非齐型空间，引进了John-Stromberg sharp极大算子，进而给出了非齐型空间上的RBMO空间的一些新的特征；给出了非齐型H1空间的新特征并建立了相关的内插定理。建立了带非双倍测度的John-Stromberg sharp极大算子和局部极大算子之间的关系式并证明了带非双倍测度的奇异积分算子、相应交换子的加权估计；研究了非齐型空间上的双倍方体的特征并给出了非齐型局部型空间h1，bmo等的新的特征刻画并研究了Littlewood-Paley算子在这些空间上的性质，引进了某些新的非齐型空间并讨论了一些算子在这些空间上的性质.
英文摘要
In this program, we first considered the weighted estimates with general weights for the maximal operator associated singular integral operators, and the corresponding maximal commutators on spaces of homogeneous type. We introduced a new sharp operator from which we established weighted estimates for the singular integrals with non-smooth kernels, the corresponding maximal operator, the commutator and the maximal commutator. Also, we devoeloped some new characterization for Hp spaces (weak Hp spaces ,respectively) on spaces of homogeneous type, and considered the boundedness of some important operators on Hp and weak Hp spaces, from which we obtained the behavior on Hp and weak Hp space for Monge-Ampere singular integrals. In the second part of this program, we introduced the John-Stromberg sharp maximal operators with non-doubling measure, and gave some new characterizations of RBMO space；we also gave a new characterization of H1 space and established an new interpolation theorem; using John-Stromberg sharp maximal operator, we obtained the weighted estimates for Calderon-Zygmund operators with non-doubling measures and the corresponding commutators; moreover, we studied the properties of doubling cubes in the setting of non-homogeneous space, and characterized some spaces of local type, such as h1, blo; meanwhile, we introduced some new function spaces and considered the behavior of some operators on these spaces.
结题摘要
在齐型空间上建立了与奇异积分算子相关的极大算子、极大交换子的带一般权的加权Lp估计和弱端点估计；给出了带非光滑核的奇异积分算子、相应的极大算子、交换子以及与交换子相应的极大算子的带一般权的加权估计；研究了齐型空间上Hp空间的分子特征、弱Hp空间的新的分解特征并建立某些重要算子在Hp空间、弱Hp空间上的有界性，并由此得到了与Monge-Ampere奇异积分算子在Hp空间和弱Hp空间上的性质。关于非齐型空间，引进了John-Stromberg sharp极大算子，进而给出了非齐型空间上的RBMO空间的一些新的特征；给出了非齐型H1空间的新特征并建立了相关的内插定理。建立了带非双倍测度的John-Stromberg sharp极大算子和局部极大算子之间的关系式并证明了带非双倍测度的奇异积分算子、相应交换子的加权估计；研究了非齐型空间上的双倍方体的特征并给出了非齐型局部型空间h1，bmo等的新的特征刻画并研究了Littlewood-Paley算子在这些空间上的性质，引进了某些新的非齐型空间并讨论了一些算子在这些空间上的性质.