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P-adic integration and Hartogs-Stawski's theorem

P-adic 积分和 Hartogs-Stawski 定理

基本信息

批准号：
10640045
负责人：
ENDOU Mikihiko
金额：
$ 1.02万
依托单位：
RIKKYO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640045/
关键词：
Hartogs Stawski p-adic analytic function analyticity p-adic integration p-adic complex number analytic domain p-進解析関数解析性 p-進積分 p-進複素数体解析領域 p進積分非アルキメデス的Hartogsの定理 Stawskiの定理線形外測度 analytic element

项目摘要

The purpose of this reseach is to give a correct proof of Stawski's Theorem (non-archimedean version of Hartogs' Theorem) . The first step of the resarch is to put in order Stawski's outer linear measure theory. This was done by a good advice of Alain Escassut.A correct proof of Hartogs-Stawski's theorem was given when the underling field K is a complete, but not locally compact subfield of the p-adic complex field CィイD2pィエD2. If the value group |KィイD1xィエD1| is discrete, the following revised version of Stawski's theorem holds.Theorem 1 If a function f (x) = f(xィイD21ィエD2, xィイD22ィエD2, ..., xィイD2nィエD2) is analytic for each variable on the domain|xィイD21ィエD2|≦ RィイD21ィエD2,|xィイD22ィエD2|≦ RィイD22ィエD2,..., |xィイD2nィエD2|≦ RィイD2nィエD2then the function f(x) is an analytic function in the whole variables on the domain|xィイD21ィエD2|< RィイD21ィエD2,|xィイD22ィエD2|≦ ィイD2qィエD2RィイD22ィエD2,..., |xィイD2nィエD2|≦ ィイD2qィエD2RィイD2nィエD2where q = |π|< 1 (πis a prime element of K) .By symmetry we slightly extended the domain of analyticity of the functions. We also see that the Theorem holds if the field K is a general non-archimedean field which is complete, but not locally compact.

本文的目的是给出Hartogs定理的非阿基米德形式的Stawski定理的正确证明。研究的第一步是对Stawski的外线性测度理论进行整理。当从属域K是p-adic复域C D2p D2的完备但非局部紧子域时，给出了Hartogs-Stawski定理的一个正确证明.如果值组|K D1x D1|定理1如果函数f（x）= f（x ∈ D21 ∈ D2，x ∈ D22 ∈ D2，...，..）对于定义域上的每个变量，|x 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000| B2R B21 B2B B2，|x 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000| R |x x x x D2 n x D2|则函数f（x）是定义域上的全变量解析函数|x 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000| <RイD21 D2，|x 100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000| ≦ィイD2qィエD2RィイD22ィエD2,..., |x x x x D2 n x D2|其中q =| π| <1（π是K的素元）。通过对称性，我们稍微扩展了函数的解析性域。如果域K是一般的非阿基米德完备域，但不是局部紧域，则该定理成立。