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Distance boundary condition domains and Hlder continuity

WU Jie-chao,WANG Fang (1.College of Computer and Communication Engineering,Changsha University of Science and Technology,Changsha 410076,China;2.College of Mathematics and Computing Science,Changsha University of Science and Technologe,Changsha 410076,China)  
This paper utilizes the definition of distance boundary condition domains and Hlder continuitiy,obtains four necessary and sufficient conditions and one sufficient condition of quasiconformal mapping and Hlder continuity when kD(x1,x2) and jD(x1,x2) satisfies some conditions.If D is a k-BC domian in n and f is a K-quasiconformal mapping from D to D′,then f is Hlder continuous in each open ball contained in D if and only if D′=f(D) is a k-BC domain;if D is a bounded uniform domain in n and f is a K-quasiconformal mapping from D to D′,then f is Hlder continuous in D if and only if D′=f(D) is a k-BC domain;if D is a bounded uniform domain and f is a K-quasiconformal mapping,then f is Hlder continuous in D if and only if D′=f(D) is a j-BC domain;a bounded uniform domain D is invariant for quasiconformal mapping if and only if f is Holder continuous in D;if D is a uniform domain and j-BC domain,then D is a John domain in n.
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