## Distance boundary condition domains and Hlder 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 Hlder continuitiy,obtains four necessary and sufficient conditions and one sufficient condition of quasiconformal mapping and Hlder 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 Hlder 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 Hlder 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 Hlder 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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