p进数域上的奇异积分算子、小波变换及其在通信领域的应用研究

The field of p-adic numbers is a completion of the field of rational number, which is equipped with a measure of non-Archimedes. And, it is the local field of characteristic zero. With the extensive application in number theory, quantum physics, information coding, dynamic system and so on, the p-adic analysis has attracted many scholars’ attention. It is an important topic of harmonic analysis to study the bounedeness of some p-adic singular integral operators, wavelet transform and its application in communication. Compared with the classical harmonic analysis, the research for the harmonic analysis in the field of p-adic numbers is still very young. And there are many problems to be resolved. Hence, based on the previous works, we tend to consider the following questions in the field of p-adic numbers: the boundedness of the singular integral operator and its commutator on some function spaces; the weighted inequalities for the singular integral operator and its commutaor; the p-adic transform of wavelet and its applications in PDEs and communication. The research for the project can further enrich and develop the theory of singular integral operator and the characterization of function space, as well as can provide the new ideas and new methods for the research of p-adic analysis and its application in the field of communication and so on.

p进数域是有理数域装备了非阿基米德度量后进行拓扑完备化得到的完备数域，是特征为零的局部域。随着p进数域在数论、量子物理、信息编码、动力系统等领域的广泛应用，p进分析受到广大学者的重视。以p进数域为底空间，研究奇异积分算子有界性、小波变换及其在通信等领域的应用是调和分析的一个重要课题。 相对于经典调和分析，p进数域上调和分析的研究还很年轻，有许多问题有待研究。因此，我们拟在以往科研工作的基础上，研究p进数域上的以下几个问题：奇异积分算子及其交换子在某些函数空间的有界性；奇异积分算子及其交换子的加权有界性； p进数域上的小波变换在偏微分方程及通信等领域的应用。这些问题的研究不仅是对调和分析的算子理论、函数空间刻画的丰富和完善，同时也为研究p进数域上的分析及其在通信等领域的应用提供新的思路和方法。

p进数域是一类特殊的局部域，它是有理数域装备了非阿基米德度量后进行拓扑完备化得到的完备数域。随着p进数域的广泛应用，p进数域上的分析问题受到广大学者的关注。本项目以p进数域为底空间，主要研究了p进域上的奇异积分算子及其交换子在某些函数空间的有界性；p进域上的奇异积分算子及其交换子的加权有界性； p进数域上的分析在偏微分方程等领域的应用。本项目的研究达到了预期目标，取得的主要成果有：p进域上的Riesz位势及其交换子在广义Morrey空间的有界性； p进域上的一类奇异积分算子及其交换子在广义Morrey空间的有界性；Littlewood-Paley面积积分μ_(Ω,S)及g_λ^*函数在广义局部 Morrey 空间的有界性；带粗糙核的分数次积分算子在消失广义加权Morrey空间的有界性。这些研究成果丰富和完善了p进域上调和分析的算子理论及函数空间刻画。