複雑集合の内と外のポテンシャル解析

• Project/Area Number: 13874023
• Research Category: Grant-in-Aid for Exploratory Research
• Allocation Type: Single-year Grants
• Research Field: Basic analysis
• Research Institution: Shimane University
• Principal Investigator: 相川 弘明 島根大学, 総合理工学部, 教授 (20137889)
• Co-Investigator(Kenkyū-buntansha): 熊谷 隆 京都大学, 数理解析研究所, 助教授 (90234509) ; 杉江 実郎 島根大学, 総合理工学部, 教授 (40196720) ; 山崎 稀嗣 島根大学, 総合理工学部, 教授 (70032935)
• Project Period (FY): 2001 – 2003
• Project Status: Completed (Fiscal Year 2003)
• Budget Amount: ¥2,000,000 (Direct Cost: ¥2,000,000); Fiscal Year 2003: ¥700,000 (Direct Cost: ¥700,000); Fiscal Year 2002: ¥600,000 (Direct Cost: ¥600,000); Fiscal Year 2001: ¥700,000 (Direct Cost: ¥700,000)
• Keywords: 容量密度条件 / 内部一様領域 / 調和関数 / Fatou定理 / Martin境界 / Carleson評価 / Besov空間 / フラクタル / Denjoy領域 / John領域 / 一様領域 / 境界Harnack原理 / Dirichlet解 / 非線形方程式 / Lienard系 / 確率過程 / ベゾフ空間

Research Abstract

・境界の容量密度条件の下で,一様領域,内部一様領域,John領域が境界Harnack原理や調和測度の評価で特徴付けられることを示した.また,非可積分な核に対する境界挙動を調べ,Fatou型の定理とLittlewood型の定理を導いた.3G不等式を内部一様領域に対して示すとともに,3次元以上の次元ではMartin境界が位相境界と一致するにもかかわらず,Cranston-McConnellの不等式が成立せず,その結果3G不等式が成立しない領域の例を構成した.p-調和関数に関するCarleson評価を導いた.
・ベソフ空間の理論の応用として、フラクタルを典型例として持つd-setと呼ばれるクラスの上に構成した飛耀型対称確率過程が、Triebel氏を初めとした関数空間の専門家が研究している作用素とどのような関係にあるかを調べ、論文にまとめた。

Research Products

• [Publications] H.Aikawa, T.Lundh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. 18. 311-357 (2003)
• [Publications] 相川弘明: "複雑領域のMartin境界と境界Harnack原理"数学. 55. 1-19 (2003)
• [Publications] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc.Amer.Math.Soc.. 131. 3873-3881 (2003)
• [Publications] H.Aikawa: "Martin boundary and boundary harnack principle for non-smooth domains"Sugaku Expositions. (to appear). (2004)
• [Publications] H.Aikawa: "Potential theoretic characterizations of nonsmooth domains"Bull.London Math.Soc.. (to appear). (2004)
• [Publications] H.Aikawa: "Fatou and Littlewood theorems for Poisson integrals with respect to non-integrable kernels"Complex Variables Theory Appl.. (to appear). (2004)
• [Publications] Z.Q.Chen, T.Kumagai: "Heat kernel estimates for stable-like processes on d-sets"Stoch.Proc.Their Appl.. 108. 27-62 (2003)
• [Publications] B.M.Hambly, T.Kumagai: "Diffusion processes on fractal fields and their large deviations"Probab.Theory Relat.Fields. 127. 305-352 (2003)
• [Publications] B.M.Hambly, T.Kumagai: "Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries"Proc.of Symposia in Pure Math. (to appear). (2004)
• [Publications] T.Kumagai: "Homogenization of Finitely Ramified Fractals"Advanced Studies in Pure Math.. 41(to appear). (2004)
• [Publications] B.M.Hambly, T.Kumagai: "Heat Kernel Estimates and Law of the Iterated Logarithm for Symmetric Random Walks on Fractal Graphs"Contemporary Mathematics. (to appear). (2004)
• [Publications] T.Kumagai: "Function spaces and stochastic processes on fractals"Progr.Probab.,Birkhauser. (to appear). (2004)
• [Publications] 相川弘明(分担執筆): "数学事典第4版の4項目(調和関数,優(劣)調和関数,ディレクレ問題,ポテンシャル論,容量)"日本数学会(to appear). (2004)
• [Publications] H.Aikawa: "Holder continuity of the Dirichlet solution for a general domain"Bull. London Math. Soc.. 34. 691-702 (2002)
• [Publications] H.Aikawa, K.Hirata, T.Lund: "Martin boundary for union of convex sets"数理解析研究所講究録「ポテンシャル論とその周辺」. 1293. 1-14 (2002)
• [Publications] 相川弘明: "複雑領域のMartin境界と境界Harnack原理"数学. 55. 1-19 (2003)
• [Publications] H.Aikawa, T.Lundh, T.Mizutani: "Martin boundary of a fractal domain"Potential Anal.. (to appear). (2003)
• [Publications] H.Aikawa: "Positive harmonic functions of finite order in a Denjoy type domain"Proc. Amer. Math. Soc.. (to appear). (2003)
• [Publications] B.M.Hambly, T.Kumagal: "Asymptotics for the spectral and walk dimension as fractals approach Euclidean space"Fractals. 10. 403-412 (2002)
• [Publications] R.F.Bass, T.Kumagai: "Laws of the iterated logarithm for the range of random walks in two and three dimensions"Ann. Probab.. 30. 1369-1396 (2002)
• [Publications] B.M.Hambly, J.Kigami, T.Kumagai: "Multifractal formalisms for the local spectral and walk dimensions"Math. Proc. Cambridge Philos. Soc.. 132. 555-571 (2002)
• [Publications] T.Kumagai: "Function spaces and stochastic processes on fractals"数理解析研究所講究録. 1293. 42-54 (2002)
• [Publications] B.M.Hambly, T.Kumagai: "Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries"Proc. of Symposia in Pure Math. (AMS). (to appear). (2003)
• [Publications] 相川弘明(編集): "数理解析研究所講究録「ポテンシャル論とその周辺」1293"京都大学数理解析研究所. 198 (2002)
• [Publications] 熊谷 隆: "確率論"共立出版. 224 (2003)
• [Publications] H.Aikawa: "Holder continuity of Dirichlet solution for a general domain"Bull. London Math. Soc.. (to appear).
• [Publications] J.Sugie, K.Kita: "Oscillation criteria for second order nonlinear differential equations of Euler type"J. Math. Anal. Appl.. 253. 414-439 (2001)
• [Publications] J.Sugie: "Lienard dynamics with an open limit orbit"NoDEA Nonlinear Differential Equations Appl.. 8. 83-97 (2001)
• [Publications] J.Sugie: "A nonoscillation theorem for second order nonlinear differential equations with decaying coefficients"Bull. London Math. Soc.. 33. 299-308 (2001)
• [Publications] J.Sugie, N.Yamada, Y.Obata: "Nonoscillation theorems for a nonlinear self-adjoint differential equation"Nonlinear Anal.. 47. 4433-4444 (2001)
• [Publications] J.Sugie, N.Yamada: "An infinite sequence of nonoscillation theorems for second order nonlinear differential equations of Euler type"Nonlinear Anal.. (to appear).
• [Publications] J.Sugie, K.Kita, N.Yamaoka: "Oscillation constant of second order nonlinear self-adjoint differential equations"Ann. Mat. Pura Appl.. (to appear).
• [Publications] J.Sugie, N.Yamaoka: "Applications of phase plane analysis of a Lienard system to positive solutions of Schrodinger equations"Proc. Amer. Math. Soc.. (to appear).
• [Publications] J.Sugie, N.Yamaoka: "Nonoscillation theorems for second order nonlinear differential differential equations of Euler type"数理解析研究所講究録. 1216. 224-235 (2001)
• [Publications] B.M.Hambly, T.Kumagai: "Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gasket"Stoch. Proc. Appl.. 92. 61-85 (2001)
• [Publications] M.T.Barlow, T.Kumagai: "Transition density asymptotics for some diffusion processes with multi-fractal structures"Electronic Journal of Probability. 6. 1-23 (2001)
• [Publications] T.Kumagai: "Construction of diffusion processes penetrating fractals -An application of the theory of Besov spaces -"数理解析研究所講究録. 1235. 91-114 (2001)
• [Publications] R.F.Bass, T.Kumagai: "Laws of the iterated logarithm for the range of random walks in two and three dimensions"Ann. Probab.. (to appear).
• [Publications] B.M.Hambly, J.Kigami, T.Kumagai: "Multifractal formalisms for the local spectral and walk dimensions"Maths. Proc. Cam. Phil. Soc.. (to appear).
• [Publications] B.M.Hambly, T.Kumagai: "Asymptotics for the spectral and walk dimension as fractals approach Euclidean space"Fractals. (to appear).
• [Publications] T.Kumagai: "Some remarks for jump processes on fractals"Proc. of the Conference held in Graz. (to appear). (2001)