函数空间与积分算子理论中若干问题的研究

主要研究目标：通过对频率空间单位分解产生的Besov型与Triebel型函数空间及其上的积分算子的研究，建立一套更为广泛的新型函数空间理论以及其上的积分算子理论，并通过这套理论探讨Lusin猜测型问题与PDE适定性问题研究的新路子，以求得到新突破。..主要研究内容：(1).通过对频率空间的单位分解结合Lp与lq，引进Besov型与Triebel型函数空间，发现与构造出有意义的新型函数空间，进而研究这些函数空间的性质及相互关系。(2).拟研究一些重要的积分算子在新型函数空间上的有界性，这些算子原本在Lp上的有界性对指标p有限制，进而探索研究高维Lusin猜测的新路子。(3).拟在新型函数空间及其上积分算子研究的基础上，探索若干非线性PDE适定性问题研究的新路子。(4)、将针对Liflyand的系列公开问题开展深入研究，期望能就其中若干问题的研究取得突破性进展。

Main Research Goal：By studying function spaces of Besov-type and Triebel-type associated with some unit decomposition of frequency space and integral operators on these spaces, we are planning to set up a new kind theory of function spaces and integral operators.Furthermore, we are trying to find some new approaches to problems of Lusin's conjecture-type and well-posedness of some important PDEs...Main Research Topics：(1). By combining unit decomposition of frequency space with Lp and lq spaces, we try to construct and find some new interesting function spaces,and further study their properties. (2). We plan to study boundedness of some important integral operators on these new type spaces, which are originally not bounded on some Lebesgue type spaces. And furthermore, we plan to find some new approach to study problems of Lusin's conjecture type. (3). On the base of the study of boundedness of operators on these new type spaces, we try to find some new approach .to well-posedness of nonlinear PDEs. (4). Try to solve some of Liflyand's open problems on Hausdorff operators.