TY - JOUR

T1 - Meromorphy of local zeta functions in smooth model cases

AU - Kamimoto, Joe

AU - Nose, Toshihiro

N1 - Funding Information:
The authors greatly appreciate that the referee carefully read the first version of this paper and gave many valuable comments. This work was supported by JSPS KAKENHI Grant Numbers JP15K04932 , JP19K14563 , JP15H02057 . Appendix A

PY - 2020/4/1

Y1 - 2020/4/1

N2 - It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general (C∞) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as u(x,y)xayb+ flat function, where u(0,0)≠0 and a,b are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each case. Our results show new interesting phenomena in one of these cases. Actually, when a−1/a and their poles on the half-plane are contained in the set {−k/b:k∈Nwithk

AB - It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. But, in the case of general (C∞) smooth functions, the meromorphic extension problem is not obvious. Indeed, it has been recently shown that there exist specific smooth functions whose local zeta functions have singularities different from poles. In order to understand the situation of the meromorphic extension in the smooth case, we investigate a simple but essentially important case, in which the respective function is expressed as u(x,y)xayb+ flat function, where u(0,0)≠0 and a,b are nonnegative integers. After classifying flat functions into four types, we precisely investigate the meromorphic extension of local zeta functions in each case. Our results show new interesting phenomena in one of these cases. Actually, when a−1/a and their poles on the half-plane are contained in the set {−k/b:k∈Nwithk

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U2 - 10.1016/j.jfa.2019.108408

DO - 10.1016/j.jfa.2019.108408

M3 - Article

AN - SCOPUS:85076526501

VL - 278

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 6

M1 - 108408

ER -