Joe NMN Kamimoto | Last modified date：2023.11.28 |

Graduate School

Undergraduate School

Homepage

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Reseacher Profiling Tool Kyushu University Pure

##### https://www2.math.kyushu-u.ac.jp/~joe/

Academic Degree

PHD Math. Sci.

Country of degree conferring institution (Overseas)

No

Field of Specialization

Complex Analysis, Real Analysis, Partial differential equations.

Total Priod of education and research career in the foreign country

00years00months

Outline Activities

I am studying complex analysis of several complex variables.

It is important to understand the properties of holomorphic

functions on many kinds of domains in complex space.

In particular, the boundary behavior of these functions

can be represented in terms of the geometry of the

boundary of the respective domains.

I am interested in the class of pseudoconvex domains of finite

type. My viewpoint is from the singularity theory.

My students are also studying these thema from my viewpoints.

It is important to understand the properties of holomorphic

functions on many kinds of domains in complex space.

In particular, the boundary behavior of these functions

can be represented in terms of the geometry of the

boundary of the respective domains.

I am interested in the class of pseudoconvex domains of finite

type. My viewpoint is from the singularity theory.

My students are also studying these thema from my viewpoints.

Research

**Research Interests**

- Complex analysis, Harmonic Analysis, Partial differential equations

keyword : holomorphic functions, asymptotic expansion, partial differential equation, complex geometry, Singularity theory

2000.10complex analysis.

**Academic Activities**

**Papers**

1. | Joe Kamimoto, Hiromichi Mizuno, Asymptotic expansion of oscillatory integrals with singular phases, Kyushu Journal of Mathematics, 2023.10. |

2. | Joe Kamimoto, On Holomorphic Curves Tangent to Real Hypersurfaces of Infinite Type, The Journal of Geometric Analysis, https://doi.org/10.1007/s12220-020-00567-z, 2021.08, [URL], The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D’Angelo infinite type in Cn. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra, which is an important concept in singularity theory. More precisely, equivalence conditions are given in the case of some model hypersurfaces.. |

3. | Joe Kamimoto, Newton polyhedra and order of contact on real hypersurfaces, J. Math. Soc. Japan, 2021.01, The purpose of this paper is to investigate order of contact on real hypersurfaces in $\C^n$ by using Newton polyhedra which are important notion in the study of singularity theory. To be more precise, an equivalence condition for the equality of regular type and singular type is given by using the Newton polyhedron of a defining function for the respective hypersurface. Furthermore, a sufficient condition for this condition, which is more useful, is also given. This sufficient condition is satisfied by many earlier known cases (convex domains, pseudoconvex Reinhardt domains and pseudoconvex domains whose regular types are 4, etc.). Under the above conditions, the values of the types can be directly seen in a simple geometrical information from the Newton polyhedron.. |

4. | Joe Kamimoto, Toshihiro Nose, Meromorphy of local zeta functions in smooth model cases, Journal of Functional Analysis, 10.1016/j.jfa.2019.108408, 278, 6, 2020.04, 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)x^{a}y^{b}+ 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 |

5. | Joe Kamimoto, Toshihiro Nose, Nonpolar singularities of local zeta functions in some smooth case, Transactions of the American Mathematical Society, 10.1090/tran/7771, 372, 1, 661-676, 2019.01, It is known that local zeta functions associated with real analytic functions can be analytically continued as meromorphic functions to the whole complex plane. In this paper, the case of specific (nonreal analytic) smooth functions is precisely investigated. Indeed, asymptotic limits of the respective local zeta functions at some singularities in one direction are explicitly computed. Surprisingly, it follows from these behaviors that these local zeta functions have singularities different from poles.. |

6. | 神本 丈, 野瀬敏洋, Asymptotic limit of oscillatory integrals with certain smooth phases, RIMS K\^oky\^uroku Bessatsu, 2017.09, 平坦な関数項を含む相関数について、振動積分の漸近挙動を正確に計算している．. |

7. | Joe Kamimoto, Toshihiro Nose, Newton polyhedra and weighted oscillatory integrals with smooth phases, Transactions of the American Mathematical Society, 10.1090/tran/6528, 368, 8, 5301-5361, 2016.01, In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his result. We are especially interested in the cases that the phase is smooth and that the amplitude has a zero at a critical point of the phase. In order to exactly treat the latter case, a weight function is introduced in the amplitude. Our results show that the optimal rates of decay for weighted oscillatory integrals whose phases and weights are contained in a certain class of smooth functions, including the real analytic class, can be expressed by the Newton distance and multiplicity defined in terms of geometrical relationship of the Newton polyhedra of the phase and the weight. We also compute explicit formulae of the coefficient of the leading term of the asymptotic expansion in the weighted case. Our method is based on the resolution of singularities constructed by using the theory of toric varieties, which naturally extends the resolution of Varchenko. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation. The investigation of this paper improves on the earlier joint work with K. Cho.. |

8. | Joe Kamimoto, Toshihiro Nose, Toric resolution of singularities in a certain class of C^{\infty} functions and asymptotic analysis of oscillatory integrals, J. Math. Soc. Univ. Tokyo , 23, 425-485, 2016.05, 実解析的という条件をはずした場合の滑らかな関数に関しては、その扱いが非常に困難になることはよく知られている。このような場合について、特異点解消という代数幾何の分野では、困難な結果を得た。これを応用して、単に滑らかな場合について、振動積分や局所ゼータ関数についての詳細な結果をえた。. |

9. | 神本 丈, 野瀬敏洋, On meromorphic continuation of local zeta functions,, Proceedings of KSCV10. F. Bracci et al. (eds.), Complex Analysis and Geometry, Springer , Proceedings in Mathematics and Statistics. , 144, 187-195, 2015.08, 局所ゼータ関数の解析接続に関して、最新の結果を報告している。. |

10. | Joe Kamimoto and Toshihiro Nose, On oscillatory integrals with C^{\infty} phases, Suriken Kokyuroku, Bessatsu, B40, 31-40, 2013.05, 相関数がなめらかな振動積分について、バルチェンコの結果を一般化した。. |

11. | Koji Cho, Joe Kamimoto, Toshihiro Nose, Asymptotic analysis of oscillatory integrals via the Newton polyhedra of the phase and the amplitude, Journal of the Mathematical Society of Japan, 10.2969/jmsj/06520521, 65, 2, 521-562, 2013.08, The asymptotic behavior at infinity of oscillatory integrals is in detail investigated by using the Newton polyhedra of the phase and the amplitude. We are especially interested in the case that the amplitude has a zero at a critical point of the phase. The properties of poles of local zeta functions, which are closely related to the behavior of oscillatory integrals, are also studied under the associated situation.. |

12. | Joe Kamimoto, Toshihiro Nose, Asymptotic analysis of weighted oscillatory integrals via Newton polyhedra, Proceedings of the 19th ICFIDCAA Hiroshima 2011, 3-12, 2013.06, 重み付き振動積分の漸近挙動をニュートン多面体の情報を用いて解析している。. |

13. | Koji Cho, Joe Kamimoto, Toshihiro Nose, Asymptotics of the Bergman function for semipositive holomorphic line bundles, Kyushu Journal of Mathematics, 10.2206/kyushujm.65.349, 65, 2, 349-382, 2011.11, In this paper, an asymptotic expansion of the Bergman function at a degenerate point is given for high powers of semipositive holomorphic line bundles on compact K̈ahler manifolds, whose Hermitian metrics have some kind of quasihomogeneous properties. In the sense of pointwise asymptotics, this expansion is a generalization of the expansion of Tian- Zelditch-Catlin-Lu in the positive line bundle case.. |

14. | 趙 康治、神本 丈、野瀬敏洋, On the Bergman fuction for semipositive holomorphic line bundles, 数理解析研究所講究録, １６１３、ｐｐ１－５, 2008.09. |

15. | Bo Yong Chen, Joe Kamimoto, Takeo Ohsawa, Behavior of the Bergman kernel at infinity, Mathematische Zeitschrift, 10.1007/s00209-004-0676-6, 248, 4, 695-708, 2004.12, We give a precise decay rate of the Bergman kernel and metric at infinity on model domains, characterized in terms of certain convex polyhedron.. |

16. | Joe Kamimoto, Newton polyhedra and the Bergman kernel, Mathematische Zeitschrift, 10.1007/s00209-003-0554-7, 246, 3, 405-440, 2004.03, The purpose of this paper is to study singularities of the Bergman kernel at the boundary for pseudoconvex domains of finite type from the viewpoint of the theory of singularities. Under some assumptions on a domainΩin ℂ^{n+1}, the Bergman kernel B(z) of Ωtakes the form near a boundary point p: B(Z) = Φ(w, ρ)/ρ^{2+2/dF} (log(1/ρ))^{mF-1}, where (w, ρ) is some polar coordinates on a nontangential cone Λ with apex at ρ and ρ means the distance from the boundary. Here Φ admits some asymptotic expansion with respect to the variables ρ^{1/m} and log(1/ρ) as ρ → 0 on Λ The values of d_{F}- > 0, m_{F} ∈ ℤ _{+} and m ∈ ℕ are determined by geometrical properties of the Newton polyhedron of defining functions of domains and the limit of Φ as ρ → 0 on Λ is a positive constant depending only on the Newton principal part of the defining function. Analogous results are obtained in the case of the Szegö kernel.. |

17. | Joe Kamimoto, Non-analytic Bergman and Szegö kernels for weakly pseudoconvex tube domains in ℂ^{2}, Mathematische Zeitschrift, 10.1007/PL00004843, 236, 3, 585-603, 2001.01, For any weakly pseudoconvex tube domain in ℂ^{2} with real analytic boundary, there exist points on the boundary off the diagonal where the Bergman kernel and the Szegö kernel fail to be real analytic.. |

18. | Joe Kamimoto, Haseo Ki, Young One Kim, On the multiplicities of the zeros of laguerre-pólya functions, Proceedings of the American Mathematical Society, 128, 1, 189-194, 2000.12, We show that all the zeros of the Fourier transforms of the functions exp(-x^{2m}), m = 1,2,⋯, are real and simple. Then, using this result, we show that there are infinitely many polynomials p(x_{1},⋯, x_{n}) such that for each (m_{1},⋯, m_{n}) ∈ (ℕ \ {0})^{n} the translates of the function p(x_{1},⋯, x_{n})exp (-∑_{j=1}^{n}x_{j}^{2mj}) generate L^{1}(ℝ^{n}). Finally, we discuss the problem of finding the minimum number of monomials pα(x_{1},⋯, x_{n}), α ∈ A, which have the property that the translates of the functions pα(x_{1},⋯, x_{n})exp(-∑_{j=1}^{n}x_{j}^{2mj}), α ∈ A, generate L^{1}ℝ^{n}), for a given (m_{1},⋯,m_{n}) ∈ (ℕ\{0})^{n}.. |

19. | Joe Kamimoto, The Bergman kernel on weakly pseudoconvex tube domains in C^{2}, Proceedings of the Japan Academy Series A: Mathematical Sciences, 10.3792/pjaa.75.12, 75, 2, 12-15, 1999.01. |

20. | Joe Kamimoto, On an integral of hardy and littlewood, Kyushu Journal of Mathematics, 10.2206/kyushujm.52.249, 52, 1, 249-263, 1998.01. |

**Presentations**

Educational

**Educational Activities**

My usual teaching activity is the following kinds of

lectures: 1. The fundamental lectures for

first or second undergraduate scientific students,

2. the lectures of special analysis for engineering

students,

3. the lectures of complex analysis for mathematical

students.

Moreover I have seminars with about 10 students.

lectures: 1. The fundamental lectures for

first or second undergraduate scientific students,

2. the lectures of special analysis for engineering

students,

3. the lectures of complex analysis for mathematical

students.

Moreover I have seminars with about 10 students.

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