On Pavlovic’s Theorem in Space

Astala, Kari; Manojlović, Vesna

doi:10.1007/s11118-015-9475-4

Cited by 20 publications

(19 citation statements)

References 21 publications

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“…This is a natural geometric quantity which, for n = 2 and f conformal, agrees with |f ′ (z)|. Both a f and Theorem 1.1 have found various applications, for instance in connection with the global distortion properties of quasiconformal mappings [3], diameter bounds for images of curves [29], in the studies of conformal metrics [9], and more recently related to harmonic quasiconformal mappings [4]. We address counterparts of some of these results as well as their generalizations.…”

Section: Introductionmentioning

confidence: 87%

A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

Adamowicz

Fässler

Warhurst

2019

Annali di Matematica

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We prove a Koebe distortion theorem for the average derivative of a quasiconformal mapping between domains in the sub-Riemannian Heisenberg group H 1 . Several auxiliary properties of quasiconformal mappings between subdomains of H 1 are proven, including BMO-estimates for the logarithm of the Jacobian. Applications of the Koebe theorem include diameter bounds for images of curves, comparison of integrals of the average derivative and the operator norm of the horizontal differential, as well as the study of quasiconformal densities and metrics in domains in H 1 . The theorems are discussed for the sub-Riemannian and the Korányi distances. This extends results due to Astala-Gehring, Astala-Koskela, Koskela and Bonk-Koskela-Rohde.

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Section: Introductionmentioning

confidence: 87%

A Koebe distortion theorem for quasiconformal mappings in the Heisenberg group

Adamowicz

Fässler

Warhurst

2019

Annali di Matematica

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“…Then (7) and our assumption on g verify that ∆h ∈ L q/2 (D). Since h vanishes continuously on the boundary ∂D, we may apply Lemma 1(ii) to obtain that ∇h ∈ L 2q/(4−q) (D) which yields the claim according to (5).…”

Section: Introductionmentioning

confidence: 92%

Quasiconformal maps with controlled Laplacian

Kalaj

Saksman

2019

JAMA

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We establish that every K-quasiconformal mapping w of the unit disk D onto a C 2 -Jordan domain Ω is Lipschitz provided that ∆w ∈ L p (D) for some p > 2. We also prove that if in this situation K → 1 with ∆w L p (D) → 0, and Ω → D in C 1,α -sense with α > 1/2, then the bound for the Lipschitz constant tends to 1. In addition, we provide a quasiconformal analogue of the Smirnov theorem on absolute continuity over the boundary.

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“…The same conclusion was obtained in [2] by Božin and Mateljević for merely C 1,α domains. Further results in the two-dimensional case can be found in [13] and for several dimensions in [1] and [14]. For a different setting for the class of quasiconformal harmonic mappings, we refer to the papers [16,18].…”

Section: 13])mentioning

confidence: 99%