BV and Sobolev homeomorphisms between metric measure spaces and the plane
Keyword(s):
Abstract We show that, given a homeomorphism f : G → Ω {f:G\rightarrow\Omega} where G is an open subset of ℝ 2 {\mathbb{R}^{2}} and Ω is an open subset of a 2-Ahlfors regular metric measure space supporting a weak ( 1 , 1 ) {(1,1)} -Poincaré inequality, it holds f ∈ BV loc ( G , Ω ) {f\in{\operatorname{BV_{\mathrm{loc}}}}(G,\Omega)} if and only if f - 1 ∈ BV loc ( Ω , G ) {f^{-1}\in{\operatorname{BV_{\mathrm{loc}}}}(\Omega,G)} . Further, if f satisfies the Luzin N and N - 1 {{}^{-1}} conditions, then f ∈ W loc 1 , 1 ( G , Ω ) {f\in\operatorname{W_{\mathrm{loc}}^{1,1}}(G,\Omega)} if and only if f - 1 ∈ W loc 1 , 1 ( Ω , G ) {f^{-1}\in\operatorname{W_{\mathrm{loc}}^{1,1}}(\Omega,G)} .
2012 ◽
Vol 2012
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pp. 1-15
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2012 ◽
Vol 273
(3-4)
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pp. 613-632
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2012 ◽
Vol 61
(1)
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pp. 63-85
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2018 ◽
Vol 2018
(737)
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pp. 161-187
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2011 ◽
Vol 9
(3)
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pp. 245-282
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