Study of harmonic analysis and applications to partial differential equations

调和分析及其在偏微分方程中的应用研究

• 批准号: 11440037
• 负责人: ARAI Hitoshi
• 金额: $ 6.4万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2001
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440037/
• 关键词: Harmonic functions; Negatively curved manifolds; Schrodinger equations; Nonlinear Schrodinger equations; Hardy spaces; Fourier multiplier; F.B.I transform; Bloch function; 非線形シュレーディンガー方程式; F.B.I.変換; 負曲率多様体上の調和解析; 離散ハーディー空間; 非線形波動方程式; 非線形クライン・

Arai studied harmonic analysis on negatively curved manifolds. Let M be a complete, simply connected Riemannian manifold whose sectional curvatures K_M satisfy -∞ < -k^2_2 【less than or equal】 K_M 【less than or equal】 -k^2_1 < 0, where k_1 and k_2 are positive constants. Arai obtained several results on elliptic harmonic functions on M. In particular he established fundamental part of harmonic analysis on M by proving theorems related to Hardy spaces, BMO, VMO, Carleson measures, Green's potential. As applications, he also studied Bloch function theory on manifolds and the regularity problem of degenerate harmonic measures. Ozawa studied by using real variable method nonlinear Schrodinger equations, nonlinear wave equations and nonlinear Krein-Gordon equations. Yajima obtained some results on the fundamental solutions of Schrodinger equations. Kanjin proved Paley's inequality for Jacobi expansions and studied the Hausdorff operator acting on real Hardy spaces.

新井研究调和分析负弯曲流形。设M是一个完备的单连通黎曼流形，其截面曲率K_M满足-∞ < -k^2_2 [小于或等于] K_M [小于或等于] -k^2_1 < 0，其中k_1和k_2为正常数。Arai得到了M上椭圆调和函数的几个结果。特别是他建立的基本部分调和分析M证明定理有关的哈代空间，BMO，VMO，Carleson措施，绿色的潜力。作为应用，他还研究了布洛赫函数理论的流形和正则性问题的退化调和措施。Ozawa用真实的变量方法研究了非线性薛定谔方程、非线性波动方程和非线性Krein-Gordon方程。Yajima得到了薛定谔方程基本解的一些结果。Kanjin证明了Jacobi展开的Paley不等式，并研究了Hausdorff算子作用于真实的哈代空间.

项目成果

K.Yajima: "On the behavior at intinity of the fundamental solution of the dime dependent Schrodinear equctions"Revies in Math. Phys.. 13. 891-920 (2001)

K.Yajima：“论一毛钱相关薛定尼尔方程的基本解的无穷大行为”数学评论。

K.Yajima: "On the bahavior at infinity of the fundamental solution of the time dependent Schrodinger equations"Revies in Math. Physics. 13. 891-920 (2001)

K.Yajima：“关于与时间相关的薛定谔方程的基本解的无穷大行为”数学评论。

新井仁之: "Hardy spaces,Carleson measures and a gradient estimate for harmonic functions on negatively curved manifolds"Adv.Stud. in Pure Math. (印刷中). 1-49

Hitoshi Arai：“负弯曲流形上的调和函数的哈迪空间、卡尔森测量和梯度估计”《纯数学》高级研究（正在出版）。

Galtbayer: "L^P-boundedness of wave operators for one dimensional Schrodinger operators"J.Math.Sci.Univ.Tokyo. 7. 221-240 (2000)

Galtbayer：“一维薛定谔算子的波算子的 L^P 有界性”J.Math.Sci.Univ.Tokyo。

H.Arai: "Hardy spaces, Carleson measures and a gradient estimate for harmenic functions on negatively curved manifolds"Advanced Studies in Pure Mathematics. 31. 1-49 (2001)

H.Arai：“哈代空间、卡尔森测量和负曲流形上谐波函数的梯度估计”纯数学高级研究。