Local representation of smooth functions and asymptotic analysis in harmonic analysis

2023 Fiscal Year Final Research Report

• Project/Area Number: 19K14563
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 12010:Basic analysis-related
• Research Institution: Fukuoka Institute of Technology
• Principal Investigator: Nose Toshihiro 福岡工業大学, 工学部, 助教 (90637993)
• Project Period (FY): 2019-04-01 – 2024-03-31
• Keywords: 局所ゼータ関数 / 解析接続 / 漸近解析 / 無限階微分可能関数 / 平坦関数 / トーリック・ブローアップ

Outline of Final Research Achievements

I study local zeta functions, which are holomorphic functions on the right half-plane defined by using infinitely differentiable smooth functions.
(1) We define a quantity determined from a smooth function in two dimensions to express the size of the region to which the local zeta function associated with the smooth function can be meromorphically continued. Lower estimates of these quantities are investigated for model functions represented by the sum of a monomial and flat functions. (2) The optimality of the above estimates in some sense is obtained. In particular, it is shown that in certain cases, the local zeta functions have both polar and non-polar singularities simultaneously. (3) It is resolved affirmatively whether the estimate obtained in (1) is optimal for all monomial exponents in the model functions.

Free Research Field

調和解析

Academic Significance and Societal Importance of the Research Achievements

本研究課題では無限階微分可能関数の局所表示に関する研究が主なテーマであったが、それに関連する形で無限階微分可能関数のモデル関数に対する局所ゼータ関数の解析接続可能領域や特異点での振る舞いについて詳細な結果を得た。調和解析の他の問題において、これらの結果およびその証明方法が無限階微分可能関数を取り扱う際の指針の一つとなることが期待される。