A New Characterization of the Mappings of Bounded Length Distortion

Hajłasz, Piotr; Malekzadeh, Soheil

doi:10.1093/imrn/rnv108

Cited by 5 publications

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“…In the proof of Theorem 1.1 the most involved part is to show that the radiality condition (R) implies the path-length condition (BLD) with the same constant (see also [HM15a]). This implication is true in the setting of general path-metric spaces and does not require f to be open.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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A Characterization of BLD-Mappings Between Metric Spaces

Luisto

2016

J Geom Anal

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We prove a characterization theorem for BLD-mappings between complete locally compact path-metric spaces. As a corollary we obtain a sharp limit theorem for BLD-mappings. J = N ≥k J ∩ N ≤k J . Since f is an open continuous map, the set N ≥k J is open. Thus the complementary set N ≤kSince f is discrete the sets {N ≤k J | k ∈ N} form a countable closed cover of the compact set J. Thus, by the Baire category theorem there exists a minimal index k J ∈ N for which the set N ≤k J J has interior points in J. Since k J is minimal, also the set N k J J has interior points in J. This means that there exists an open. Let also r J > 0 be a radius so small that the sets U (z i J , f, r J ) are disjoint normal neighbourhoods of the points z i for i = 1, . . . , k J satisfyingThus for any compact set K ⊂ W J ∩ J the pre-image f −1 (β(K)) is compact and as a locally injective map between compact sets in Hausdorff spaces the restriction of f to f −1 (β(K)) is a local homeomorphism. This local inverse yields maps. . , k J . Furthermore the images of these lifts cover all of U 0 ∩ f −1 (β(W J ∩ J)).and obtain a lifting interval W J for J together with the related points z 1 J , . . . , z k J J as in (LI2). We claim that W J ∈ I. Let c ∈ W J and fix a pre-For each of the intervals (a j , b j ) ⊂ W J there exists at least one lift α j of β| (a j ,b j ) with |α j | ∩ U (z i 0 J , f, r J ) = ∅; if c ∈ (a j , b j ) we take the lift of β| (a j ,b j ) containing z 0 , otherwise we take any one of the finitely many possibilities. Since

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Section: Proof Of Theorem 11mentioning

confidence: 99%

“…by Haj lasz and Malekzadeh, see [HM15b]. In Euclidean spaces the two definitions are equivalent (see [HM15a]) but, in general, weak BLD-mappings form a strictly larger class of mappings. For example the identity map H 1 → R 3 from the first Heisenberg group to the Euclidean 3-space is a weak BLD-mapping but not a BLD-mapping.…”

Section: Introductionmentioning

confidence: 99%