Uniform continuity of quasiconformal mappings onto generalized John domains

Abstract

Abstract. We study uniform continuity of quasiconformal mappings onto δ-Gromov-hyperbolic ϕ-John domains. The general ϕ-John case is also investigated.

Generalized quasidisks and conformality II

“…This is not a big surprise from the technical point of view since our aim is to estimate the scalewise distortion of our conformal welding G. It is fairly easy to observe that this is closely related to the modulus of continuity of G −1 . On the other hand, combining the duality results in [11] with the global Hölder continuity estimates of conformal mappings in [10,18], one can immediately see how the role of Ω being ϕ-LC-1 or ψ-LC-2 is related the modulus of continuity of G and G −1 . In fact, this is exactly the way we proved [12,Theorem 4.4].…”

Sharpness of uniform continuity of quasiconformal mappings onto s-John domains

“…In [4], the following uniform continuity result for general s-John domains was established. Theorem 1.1.…”

“…Notice that for all z, w ∈ Ω, |z − w| ≤ D I (z, w); hence the left-hand side of (1.4) can be replaced with |f (x ′ ) − f (y ′ )|. It was shown in [4] that the restriction s < 1 + 1 n−1 can be disposed of if Ω equipped with the quasihyperbolic metric is Gromov hyperbolic.…”

Sharp Capacity Estimates in S-John Domains

“…The recent studies [1,5,7] on mappings of finite distortion have generated new interest in the class of s-John domains. In particular, uniform continuity of quasiconformal mappings onto s-John domains was studied in [4,6].…”

“…When q < min{(n − 1)s + 1 − p, log 2 (2 n − 1)}, the s-John domain Ω constructed in Theorem 1.3 is in fact Gromov hyperbolic in the quasihyperbolic metric. This is very surprising, since it was proven in [4] that for all Gromov hyperbolic s-John domains Ω, an estimate of the form as in (1.4) holds when p = n, q = 1 and E ⊂ Ω is a continuum. Our example shows that one can not replace the assumption being a continuum by just being compact, and still obtain the estimate for all s-John domains.…”