Language：English   Publishing type：Research paper (scientific journal)  

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^ay^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<b, local zeta functions can be meromorphically extended to the half-plane Re(s)>−1/a and their poles on the half-plane are contained in the set {−k/b:k∈Nwithk<b/a}.

DOI： 10.1016/j.jfa.2019.108408

Repository Public URL： http://hdl.handle.net/2324/4795995

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1090/tran/7771

Language：English   Publishing type：Research paper (scientific journal)  

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．

Language：Japanese   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

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。

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.1090/tran/6528

Language：English   Publishing type：Research paper (international conference proceedings)  

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。

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.2969/jmsj/06520521

Language：English   Publishing type：Research paper (international conference proceedings)  

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。

Language：English   Publishing type：Research paper (scientific journal)  

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。

Language：English   Publishing type：Research paper (scientific journal)  

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.

DOI： 10.2206/kyushujm.65.349

Language：English   Publishing type：Research paper (other academic)  

On the Bergman fuction for semipositive holomorphic line bundles

Language：Japanese   Publishing type：Research paper (scientific journal)  

Language：English   Publishing type：Research paper (scientific journal)  

We give a precise decay rate of the Bergman kernel and metric at infinity on model domains, characterized in terms of certain convex polyhedron.

DOI： 10.1007/s00209-004-0676-6

Language：English   Publishing type：Research paper (scientific journal)  

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 ℂⁿ⁺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.

DOI： 10.1007/s00209-003-0554-7

Language：English   Publishing type：Research paper (scientific journal)  

For any weakly pseudoconvex tube domain in ℂ² 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.

DOI： 10.1007/PL00004843

Language：English   Publishing type：Research paper (scientific journal)  

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(x₁,⋯, x_n) such that for each (m₁,⋯, m_n) ∈ (ℕ \ {0})ⁿ the translates of the function p(x₁,⋯, x_n)exp (-∑_j=1ⁿx_j^2mj) generate L¹(ℝⁿ). Finally, we discuss the problem of finding the minimum number of monomials pα(x₁,⋯, x_n), α ∈ A, which have the property that the translates of the functions pα(x₁,⋯, x_n)exp(-∑_j=1ⁿx_j^2mj), α ∈ A, generate L¹ℝⁿ), for a given (m₁,⋯,m_n) ∈ (ℕ\{0})ⁿ.

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.3792/pjaa.75.12

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.2206/kyushujm.52.249

Publisher：Journal of Functional Analysis  

In this paper, we attempt to resolve the singularities of the zero variety of a C∞ function of two variables as much as possible by using ordinary blowings up. As a result, we formulate an algorithm to locally express the zero variety in the “almost” normal crossings form, which is close to the normal crossings form but may include flat functions. As an application, we investigate analytic continuation of local zeta functions associated with C∞ functions of two variables. As is well known, the desingularization theorem of Hironaka implies that the local zeta functions associated with real analytic functions admit the meromorphic continuation to the whole complex plane. On the other hand, it is recently observed that the local zeta function associated with a specific (non-real analytic) C∞ function has a singularity different from the pole. From this observation, the following questions are naturally raised in the C∞ case: how wide the meromorphically extendible region can be and what kinds of information essentially determine this region? This paper shows that this region can be described in terms of some kind of multiplicity of the zero variety of each C∞ function. By using our blowings up algorithm, it suffices to investigate local zeta functions in the almost normal crossings case. This case can be effectively analyzed by using real analysis methods; in particular, a van der Corput-type lemma plays a crucial role in the determination of the above region.

DOI： 10.1016/j.jfa.2023.110185

Web of Science

Scopus

Publisher：Springer Proceedings in Mathematics and Statistics  

In this paper, we investigate the asymptotic behavior of the Bergman kernel at the boundary for some pseudoconvex model domains. This behavior can be described by the geometrical information of the Newton polyhedron of the defining function of the respective domains. We deal with not only the finite type cases but also some infinite type cases.

DOI： 10.1007/978-981-99-9506-6_10

Web of Science

Scopus

Language：English   Publisher：Faculty of Mathematics, Kyushu University  

<p>The purpose of this article is to describe the singularities of one-dimensional oscillatory integrals, whose phases have a certain singularity, in the form of an asymptotic expansion. In the case of the Laplace integral, an analogous result is also given.</p>

DOI： 10.2206/kyushujm.77.319

Scopus

CiNii Research

CiNii Research

Language：English   Publishing type：Research paper (scientific journal)  

Language：Japanese   Publishing type：Research paper (bulletin of university, research institution)  

古典的なハルトークスの拡張定理をめぐって、多変数関数論の入門的な解説を行っている。

Language：English   Publishing type：Research paper (scientific journal)  

DOI： 10.3792/pjaa.72.51

Event date： 2022.7

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：神奈川県葉山町   Country：Japan  

Event date： 2022.5

Language：English   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Austria  

Event date： 2022.5

Language：English   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Mexico  

可微分関数に関するある種の「特異点解消定理」を示し、その応用として局所ゼータ関数の解析接続に関する問題について考察した。実際に、解析性を持つ関数の場合に関する局所ゼータ関数は全平面に有理型関数として解析接続されることが知られており、さらにその極の分布や位数に関しても、かなり詳細に調べられているが、解析性を仮定しない場合に関しては、一般的な成果が得られていなかっただけでなく、局所ゼータ関数が極以外の特異性を持つという例まで見つかっている。私は、可微分関数の場合に、極以外の特異性がどこに現れるかという問題に関して、関数のある種の不変量を導入し、その不変量を用いて、ある種の解答を与えた。その際に、先に述べた特異点解消定理が必要となる。

Event date： 2022.3

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：埼玉大学   Country：Japan  

Event date： 2022.3

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：熊本大学   Country：Japan  

Event date： 2021.3

Language：English   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2020.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2020.11

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：オンライン   Country：Japan  

Event date： 2020.3

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：日本大学, 東京   Country：Japan  

Event date： 2020.3

Language：English   Presentation type：Oral presentation (general)  

Venue：イーグレ姫路、兵庫県   Country：Japan  

Event date： 2020.2

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：鹿児島大学   Country：Japan  

Event date： 2019.11

Language：English   Presentation type：Oral presentation (general)  

Venue：京大数理解析研究所   Country：Japan  

Event date： 2019.9

Language：Japanese   Presentation type：Oral presentation (invited, special)  

Venue：金沢大学   Country：Japan  

この講演では，$\C^n$内のなめらかな実超曲面に関して「ニュートン多面体」という概念を導入し，多変数関数論のいくつかの問題に応用する．ニュートン多面体は，特異点論などの分野において，有用な道具として重要な役割を果たしている．さらに，近年，実解析の分野においても，この概念を用いることにより非常に多くの成果が得られている．それに倣って多変数関数論においても，有用な概念となることを期待して，具体的に，D'Angeloの特異型の定量的な決定とベルグマン核の境界挙動に関する問題に関して，ニュートン多面体を用いて解析を行う．そのおかげで，現在までに得られているこれらの問題に関する多くの成果が，統一的に理解され，さらに新しい成果ももたらされる．この2つの研究対象は，特異点論の分野で盛んに研究されてきた，\L ojasiewicz指数の決定と振動積分の漸近挙動に関する問題とそれぞれ類似するものであり，ニュートン多面体を用いた解析が自然なアプローチであることがわかる．

Event date： 2019.8

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：静岡県伊豆の国市   Country：Japan  

Event date： 2019.1

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：沖縄県市町村自治会館   Country：Japan  

Event date： 2018.12

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：東京大学   Country：Japan  

多変数関数論で重要な不変量であるD'Angeloの定義したタイプについて，ニュートン多面体を用いて、詳細な研究を行っている。

Event date： 2018.11

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：広島大学   Country：Japan  

Event date： 2018.10

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：京大数理解析研究所   Country：Japan  

Event date： 2018.1

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：金沢市   Country：Japan  

特異点論的な概念であるニュートン多面体の幾何学的な研究を、様々な解析の分野に応用した。特に、多変数関数論において、重要な、接触位数に関する研究に関して、非常に興味深い結果を得たことを報告した。

Event date： 2017.10

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：京大数理解析研   Country：Japan  

滑らかな関数に関する局所ゼータ関数の特異点として、極以外のものが存在するような例を見つけた。

Event date： 2016.10

Language：English   Presentation type：Oral presentation (general)  

Venue：京大数理解析研究所   Country：Japan  

局所ゼータ関数の解析接続に関して、現在までの研究および最新の研究の成果について、発表した。

Event date： 2015.6

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：京大数理解析研究所   Country：Japan  

振動積分と局所ゼータ関数の漸近解析について、最新の結果を発表した。

Event date： 2014.2

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：鹿児島大学理学部   Country：Japan  

Newton　polyhedra and oscillatory integrals

Event date： 2012.11

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：金沢大学   Country：Japan  

Newton polyhedra and asymptotic analysis of the Bergman kernel

Event date： 2012.7

Language：English   Presentation type：Oral presentation (general)  

Venue：東京大学   Country：Japan  

On oscillatory integrals with smooth phases

Event date： 2011.12

Presentation type：Oral presentation (invited, special)  

Venue：広島   Country：Japan  

Newton polyhedra and oscillatory integrals

Event date： 2011.11

Presentation type：Oral presentation (general)  

Venue：京大数理解析研究所   Country：Japan  

Newton polyhedra and oscillatory integrals

Event date： 2011.11

Presentation type：Oral presentation (general)  

Venue：奈良   Country：Japan  

Asymptotic analysis of oscillatory integrals via the Newton polyhedra of
the phase and the amplitude

Event date： 2011.9

Presentation type：Oral presentation (general)  

Venue：京都大学数理解析研究所   Country：Japan  

Newton polyhedra and asymptotic analysis of oscillatory integrals

Presentation type：Oral presentation (general)  

Venue：慶州   Country：Korea, Republic of  

Presentation type：Oral presentation (general)  

Venue：京都大学数理解析研究所   Country：Japan  

Presentation type：Oral presentation (general)  

Venue：京大数理解析研究所   Country：Japan  

Presentation type：Oral presentation (general)  

Venue：名古屋大学   Country：Japan  

Presentation type：Oral presentation (general)  

Venue：葉山   Country：Japan  

Presentation type：Oral presentation (general)  

Venue：京大数理解析研究所   Country：Japan  

Presentation type：Oral presentation (general)  

Venue：京都大学数理解析研究所   Country：Japan  

Presentation type：Oral presentation (general)  

Venue：東京大学   Country：Japan  

Event date： 2018.9

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：岡山大学   Country：Japan  

Event date： 2018.9

Language：Japanese  

Venue：岡山大学   Country：Japan  

Event date： 2018.8

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：福岡県北九州市   Country：Japan  

Event date： 2017.9

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：福岡県柳川市   Country：Japan  

Event date： 2014.8

Language：English  

Venue：Gyeongju   Country：Korea, Republic of  

We investigate meromorphic continuation of local zeta functions and properties of their poles. In the real analytic case, local zeta functions can be meromorphically continued to the whole complex plane and, moreover, properties of the poles have been precisely investigated. However, in the only smooth case, the situation of meromorphic continuation is very different. Actually, there exists an example in which a local zeta function has a singularity different from poles. We give a sufficient condition for that the first finitely many poles samely appear as in the real analytic case and exactly investigate properties of the first pole.

Event date： 2014.5

Language：Japanese   Presentation type：Oral presentation (general)  

Venue：東京大学大学院数理学研究科   Country：Japan  

Resolution of singularities via Newton polyhedra and its application to analysis

Event date： 2013.1

Language：Japanese  

Venue：広島大学   Country：Japan  

Newton polyhedra and oscillatory integrals

Other Link： Newton polyhedra and oscillatory integrals

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：3

Proceedings of International Conference Number of peer-reviewed papers：2

Type：Competition, symposium, etc. 

Type：Competition, symposium, etc. 

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：3

Proceedings of International Conference Number of peer-reviewed papers：2

Type：Competition, symposium, etc. 

Number of participants：25

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：4

Proceedings of International Conference Number of peer-reviewed papers：2

Proceedings of domestic conference Number of peer-reviewed papers：0

Type：Competition, symposium, etc. 

Number of participants：35

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：4

Proceedings of International Conference Number of peer-reviewed papers：1

Proceedings of domestic conference Number of peer-reviewed papers：1

Type：Competition, symposium, etc. 

Number of participants：30

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：6

Proceedings of International Conference Number of peer-reviewed papers：2

Type：Competition, symposium, etc. 

Number of participants：30

Type：Peer review 

Number of peer-reviewed articles in foreign language journals：5

Proceedings of International Conference Number of peer-reviewed papers：1

Type：Competition, symposium, etc. 

Number of participants：50

Type：Competition, symposium, etc. 

Number of participants：60

Type：Competition, symposium, etc. 

Number of participants：30

Type：Competition, symposium, etc. 

Number of participants：80

Type：Competition, symposium, etc. 

Number of participants：50

Type：Competition, symposium, etc. 

Number of participants：20

Type：Competition, symposium, etc. 

Number of participants：30

Type：Competition, symposium, etc. 

Number of participants：500