p-adic域上的拟微分算子

Pseudo-differential operators over p-adic fields

本文研究p-adic域Qp上的一类拟微分算子{Tα:α∈R},其中算子Tα在Qp的检验函数空间S(Qp)以及相应的分布空间S(Qp)中作为一个运算是封闭的.本文构造了算子Tα的卷积核,指出算子与其卷积核在解决p-adic域上的相关问题中所起的重要作用.最后研究算子Tα的谱性质,构造了算子Tα的全体固有值与固有函数,并证明全体固有函数所成的集合组成L2(Qp)空间中的一组完整正交基.

This paper investigates a class of pseudo-differential operators Tα over p-adic fields. The test function class S（Qp） and its distribution class S＇（Qp） are invariant under these operators. The convolution kernel of the operator Ta is constructed which plays a key role in the study related to lots of operator problems. Spectral properties of Tα are studied, and an orthonormal basis of eigen-functions of Tα is constructed.