非交换奇异积分算子交换子及在非交换几何中的应用

奇异积分算子理论是经典调和分析中非常重要的研究内容，对于调和分析本身，及数学物理、偏微分方程等其他数学分支领域影响深远。奇异积分算子的交换子是奇异积分算子理论中重要的研究对象，其有界性、紧性、奇异值序列的可和性等刻画涉及Sobolev、Besov、BMO等非常基础的函数空间。在非交换几何领域，奇异积分算子交换子与Connes所定义的量子化微分吻合，因此在非交换分析领域对奇异积分算子交换子的刻画尤为重要，将应用到非交换几何领域中对量子微分的研究。同时，如同经典情形，非交换拟微分算子理论将作为非交换分析和非交换几何的桥梁，帮助我们更好地将非交换分析中的方法和结果应用到量子微分和积分的计算中。.申请人希望通过非交换奇异积分算子交换子的研究，在非交换流形上刻画量子可微性，并结合拟微分算子理论，给出这些量子微分Dixmier迹的精确计算公式。

Singular integral operator theory is a very important topic in the classical harmonic analysis, which is fundamental for harmonic analysis itself, as well as for many branches of mathematics such as mathematical physics and PDE. Commutators of singular integral operators are main objects in singular integral operator theory; their characterizations of boundedness, compactness or boundedness as Schatten operators involve many basic function spaces such as Sobolev, Besov, BMO spaces. In Alain Connes’ Noncommutative Geometry theory, commutators of singular integral operators are nothing but quantized differentials. Therefore, the characterizations of these commutators using noncommutative harmonic analysis play a crucial role in the study of quantized differentials in noncommutative geometry. Meanwhile, as in the classical setting, noncommutative pseudo-differential operator theory connects noncommutative analysis and noncommutative geometry, which may help us apply better the results and techniques in noncommutative analysis to quantum calculus..We will focus on the study of commutators of singular integral operators in the noncommutative analysis setting, trying to give precise characterizations of quantum differentiability; we will also develop noncommutative pseudo-differential operator theory, so as to calculate precisely the Dixmier traces of these quantum differentials.