Research on Fourier multiplier by operating functions on function spaces

函数空间上函数运算的傅里叶乘子研究

基本信息

批准号：
06804010
负责人：
HATORI Osamu
金额：
$ 0.77万
依托单位：
Niigata University (1995-1996)Tokyo Medical University (1994)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1996
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-06804010/
关键词：
commutative Banach algebra multiplier operating function function space locally compact abelian group Wiener-Pitt phenomenum L^P-multiplier 極大イデアル空間 locallycompact abelian group regular Banach algebra decomposable operator natural sp

项目摘要

In this research I study Fourier multiplier on locally compact abelian groups G.The maximal ideal space of p-q multplier on a comact abelian group is identified. In particular, I proved that the dual group of G is dense in the maximal ideal space, henceforce naturality of sspectra of p-q multiplier is proved. The operating functions of p-2 multplier is also identified. Let C_0M_p (G) denote the algebra of L^p-multiplier whose Fourier transforms vanish at infinity. I proved that the Apostol algebra coincides with the greatest regular closed subalgebra RegC_0M_p (G) and they are maximal, in a sense, in C_0M_p (G). The proof depends on the general results concerning abstract algebras of continuous functions which are modeled after Fourier multipliers. I also proved that if the maximal ideal space of the algebra in thin, they the greatest regular closed subalgebra coincides with the set of functions with natural spectra. Laursen and Neumann proved that if p=1 or G is compact, then RegC_0M_p (G) is the closed ideal C_<00>M_p (G) which consists of multplier whose Gelfand transforms vanish of the dual group of G.I proved that if p*1, then RegC_0M_p (R^n) is not an ideal of C_0M_p (R^n) and C_<00>M_p (R^n)= {0}. Let G be a non-discrete locally comapct abelian group. I prove that there exists a bounded regular Borel measure outside of the radical of L^1 (G) with a natural spectrum. In particular if G is not compact, then the Fourier-Stieltjes transform of the measure can be vanish at infinity on the dual group, which answers the question posed by Eschimier, Laursen and Neumann. I also study BSE-algebras.

在这项研究中，我研究了局部紧阿贝尔群G上的傅立叶乘子。确定了紧阿贝尔群上p-q乘子的最大理想空间。特别是，我证明了G的对偶群在最大理想空间中是稠密的，从而证明了p-q乘子谱的自然性。 p-2乘法器的操作功能也被确定。令C_0M_p (G) 表示L^p-乘子的代数，其傅里叶变换在无穷远处消失。我证明了 Apostol 代数与最大正则闭子代数 RegC_0M_p (G) 一致，并且从某种意义上说，它们在 C_0M_p (G) 中是最大的。证明取决于有关连续函数抽象代数的一般结果，这些连续函数是根据傅里叶乘数建模的。我还证明了如果代数的最大理想空间很薄，则它们的最大正则闭子代数与具有自然谱的函数集相一致。 Laursen 和 Neumann 证明，如果 p=1 或 G 是紧的，则 RegC_0M_p (G) 是闭合理想 C_<00>M_p (G)，其由 G 的对偶群的 Gelfand 变换消失的乘法器组成。 1，则RegC_0M_p (R^n)不是C_0M_p (R^n)和C_<00>M_p的理想值(R^n)={0}。令 G 为非离散局部 comapct 阿贝尔群。我证明了在具有自然谱的 L^1 (G) 根之外存在有界正则 Borel 测度。特别是如果 G 不是紧的，那么测度的 Fourier-Stieltjes 变换可以在对偶群上无穷远处消失，这回答了 Eschimier、Laursen 和 Neumann 提出的问题。我还学习 BSE 代数。