2003 Fiscal Year Annual Research Report

複雑集合の内と外のポテンシャル解析

Research Project

Project/Area Number	13874023
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)
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
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
(13 results)

All Other

All Publications (13 results)

[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)

2003 Fiscal Year Annual Research Report

複雑集合の内と外のポテンシャル解析

Principal Investigator

相川 弘明 島根大学, 総合理工学部, 教授 (20137889)

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)

相川弘明島根大学, 総合理工学部, 教授 (20137889)