### 抄録

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.

元の言語 | 英語 |
---|---|

ページ（範囲） | 5301-5361 |

ページ数 | 61 |

ジャーナル | Transactions of the American Mathematical Society |

巻 | 368 |

発行部数 | 8 |

DOI | |

出版物ステータス | 出版済み - 1 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Applied Mathematics

### これを引用

*Transactions of the American Mathematical Society*,

*368*(8), 5301-5361. https://doi.org/10.1090/tran/6528

**Newton polyhedra and weighted oscillatory integrals with smooth phases.** / Kamimoto, Joe; Nose, Toshihiro.

研究成果: ジャーナルへの寄稿 › 記事

*Transactions of the American Mathematical Society*, 巻. 368, 番号 8, pp. 5301-5361. https://doi.org/10.1090/tran/6528

}

TY - JOUR

T1 - Newton polyhedra and weighted oscillatory integrals with smooth phases

AU - Kamimoto, Joe

AU - Nose, Toshihiro

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84957922766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957922766&partnerID=8YFLogxK

U2 - 10.1090/tran/6528

DO - 10.1090/tran/6528

M3 - Article

VL - 368

SP - 5301

EP - 5361

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 8

ER -