1996 Fiscal Year Final Research Report Summary
Research on Fourier multiplier by operating functions on function spaces
| Project/Area Number |
06804010
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
解析学
|
|---|
| Research Institution | Niigata University (1995-1996)
Tokyo Medical University (1994) |
|---|
Principal Investigator |
HATORI Osamu Niigata University Griduate School of Science and Technology Associate Professor, 自然科学研究科, 助教授 (70156363)
|
|---|
| Project Period (FY) |
1994 – 1996
|
| Keywords | commutative Banach algebra / multiplier / operating function / function space / locally compact abelian group / Wiener-Pitt phenomenum |
|---|
| Research Abstract |
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.
|
