Let n≥2,β∈(0,n)andΩ■R^(n) be a bounded domain.Support thatφ:[0,∞)→[0,∞)is a Young function which is doubling and satisfies sup x>0∫^(1)_(0)φ(tx)/φ(x)dt/t^(β+1)<∞.If Ω is a John domain,then we show...Let n≥2,β∈(0,n)andΩ■R^(n) be a bounded domain.Support thatφ:[0,∞)→[0,∞)is a Young function which is doubling and satisfies sup x>0∫^(1)_(0)φ(tx)/φ(x)dt/t^(β+1)<∞.If Ω is a John domain,then we show that it supports a(φ^(n/(n-β)),φ)_(β)-Poincaréinequality.Conversely,assume thatΩis simply connected domain when n=2 or a bounded domain which is quasiconformally equivalent to some uniform domain when n≥3.IfΩsupports a((φ^(n/(n-β)),φ)β-Poincaréinequality,then we show that it is a John domain.展开更多
Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,the...Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.展开更多
Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,the...Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.展开更多
This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e inequalities in...This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e 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-Poincar′e 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-Poincar′e implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.展开更多
基金Supported by National Natural Science Foundation of China(Grant No.11871088)。
文摘Let n≥2,β∈(0,n)andΩ■R^(n) be a bounded domain.Support thatφ:[0,∞)→[0,∞)is a Young function which is doubling and satisfies sup x>0∫^(1)_(0)φ(tx)/φ(x)dt/t^(β+1)<∞.If Ω is a John domain,then we show that it supports a(φ^(n/(n-β)),φ)_(β)-Poincaréinequality.Conversely,assume thatΩis simply connected domain when n=2 or a bounded domain which is quasiconformally equivalent to some uniform domain when n≥3.IfΩsupports a((φ^(n/(n-β)),φ)β-Poincaréinequality,then we show that it is a John domain.
基金Sponsored by the Foundation of Pre-973 Program of China under grant2006CB708304the National NSFC under grant 10771195the NSF of Zhejiang Province under grant Y607128
文摘Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.
基金Sponsored by the Foundation of Pre-973 Program of China under grant2006CB708304+2 种基金 the National NSFC under grant 10771195 the NSF of Zhejiang Province under grant Y607128
文摘Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.
文摘This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e 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-Poincar′e 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-Poincar′e implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.
