非齐次距离空间上几类积分算子的有界性及其应用

The theory of function spaces and the boundedness of operators play a key role in harmonic analysis. Since Hytonen introduced the non-homogeneous metric measurable spaces, some classical results on Calderon-Zygmund theory have been generalized to this kind of space. Unfortunately, there are still other ones on Calderon-Zygmund theory remaid to be established. This projector will make a further investigation. In this proposal, we firstly focus on the boundedness of Calderon-Zygmund type operators and square type Littlewood-Paley operators and their commutators on non-homogeneous metric measure spaces, such as Lebesgue spaces, Morrey spaces, Hardy spaces etc. Secondly, we concern with the boundedness of multilinear operators and their commutators on some non-homogeneous product spaces and establish their multiple weighted estimates. Finally, we investigate the existence of solutions for integral equations and integral systems associated with potential operators and the properties of solutions,such as symmetry, monotonicity, uniqueness, asympotical behavior and so on.

函数空间和算子的有界性理论一直是调和分析研究的核心问题。自Hytonnen引进非齐次可测距离空间以来，已有一些经典的C-Z理论被推广到非齐次可测距离空间中，还有许多问题没有解决，本项目将做进一步的研究。首先主要致力于研究Calderon-Zygmund奇异积分算子、平方型Littlewood-Paley算子及其交换子在各类非齐次可测距离空间(如：Lebesgue空间、Morrey空间、Hardy型空间等)上的有界性。其次研究多线性算子及其交换子在各类乘积型非齐次距离空间上的有界性和多权估计。最后研究与位势型算子相关的积分方程与方程组解的存在性以及解的对称性、单调性、唯一性、渐进性等性质。

本项首先研究了广义分次积分算子及其交换子在非齐次距离可测Morrey空间上的端点估计,建立了多线性平方函数和Lipschitz函数生成的交换子在一些乘积空间上的端点估计。其次研究了一类磁流体方程组局部强解在Morrey-Campanato型空间上的正则性判别准则，得到了一类抛物-抛物型Keller-Segel方程组在Fourier-Besov-Morrey型空间上的小初值整体适定性。同时我们还研究了一类极小化问题解的可达性，同时得到到达函数的的衰减性。最后研究了与Schrodinger算子相关的Riesz位势及其交换子的有界性。