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We prove that quasiconformal maps onto domains which satisfy a quasihyperbolic boundary condition are globally H?lder continuous in the internal metric. The primary improvement here over existing results along these lines is that no assumptions are made on the source domain. We reduce the problem to the verification of a capacity estimate in domains satisfing a quasihyperbolic boundary condition, which we establish using a combination of a chaining argument involving the Poincaré inequality on Whitney cubes together with Frostman's theorem. We also discuss related results where the quasihyperbolic boundary condition is slightly weakened; in this case the H?lder continuity of quasiconformal maps is replaced by uniform continuity with a modulus of continuity which we calculate explicitly. Received: June 16, 2000  相似文献
2.
This paper is devoted to the study of fractional(q,p)-Sobolev-Poincaré inequalities in irregular domains.In particular,the author establishes(essentially) sharp fractional(q,p)-Sobolev-Poincaré inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions.When the order of the fractional derivative tends to 1,our results tend to the results for the usual derivatives.Furthermore,the author verifies that those domains which support the fractional(q,p)-Sobolev-Poincaré inequalities together with a separation property are s-diam John domains for certain s,depending only on the associated data.An inaccurate statement in [Buckley,S.and Koskela,P.,Sobolev-Poincaré implies John,Math.Res.Lett.,2(5),1995,577-593] is also pointed out.  相似文献
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