Hölder continuity of conformal mappings and non-quasiconformal Jordan curves

Becker, Jochen; Pommerenke, Ch.

doi:10.1007/bf02565858

Cited by 55 publications

(38 citation statements)

References 3 publications

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“…3. The examples in § 2 show that the exponent À1 a=n in Theorem 5.2 is sharp for the entire class of densities r. However, for a ®xed density r, a better estimate holds at least when m r B n < 1: in this case the claim of Theorem 5.2 holds with À1 a=n replaced by À1 a=n «, where « «n; B.…”

Section: Asymptotic Behaviour Of Rmentioning

confidence: 88%

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Conformal Metrics on the Unit Ball in Euclidean Space

Bonk

Koskela

Rohde

1998

Proceedings of the London Mathematical Society

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We study densities ρ on the unit ball in euclidean space which satisfy a Harnack type inequality and a volume growth condition for the measure associated with ρ. For these densities a geometric theory can be developed which captures many features of the theory of quasiconformal mappings. For example, we prove generalizations of the Gehring‐Hayman theorem, the radial limit theorem and find analogues of compression and expansion phenomena on the boundary. 1991 Mathematics Subject Classification: 30C65.

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Section: Asymptotic Behaviour Of Rmentioning

confidence: 88%

“…In the case of conformal mappings, this is related to a result of Gerasch [13], as extended in [23], and to geometric consequences of Ho Èlder continuity as in [3,20,29]. …”

Section: Hausdorff Dimension Of the Boundarymentioning

confidence: 92%

Conformal Metrics on the Unit Ball in Euclidean Space

Bonk

Koskela

Rohde

1998

Proceedings of the London Mathematical Society

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“…The term Hölder domain is derived from the planar case R 2 , where a simply connected domain Ω is a Hölder domain if and only if a Riemann mapping from the unit disk onto Ω is Hölder continuous; see [1] and [24]. Hölder domains can also be characterised by H-chain sets.…”

Section: Length Metric Hölder Domains and H-chain Setsmentioning

confidence: 99%

Global integrability of p-superharmonic functions on metric spaces

Maasalo

2008

J Anal Math

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“…Overall, sharp constants are not essential, though it will be necessary to have specified constants. For now, one should focus on the fact that supp(µ K ) and K have comparable diameters, that hcap(K) is bounded from above by a multiple of diam(K) 2 , and that the map f does not move points by more than distance 3 diam(K).…”

Section: The Quasi-hyperbolic Metricmentioning

confidence: 99%

Loewner chains and Hölder geometry

Kinneberg

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. The Loewner equation provides a correspondence between continuous real-valued functions λ t and certain increasing families of half-plane hulls K t . In this paper we study the deterministic relationship between specific analytic properties of λ t and geometric properties of K t . Our motivation comes, however, from the stochastic Loewner equation (SLE κ ), where the associated function λ t is a scaled Brownian motion and the corresponding domains H\K t are Hölder domains. We prove that if the increasing family K t is generated by a simple curve and the final domain H\K T is a Hölder domain, then the corresponding driving function has a modulus of continuity similar to that of Brownian motion. Informally, this is a converse to the fact that SLE κ curves are simple and their complementary domains are Hölder, when κ < 4. We also study a similar question outside of the simple curve setting, which informally corresponds to the SLE regime κ > 4. In the process, we establish general geometric criteria that guarantee that K t has a Lip(1/2) driving function.

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Hölder continuity of conformal mappings and non-quasiconformal Jordan curves

Cited by 55 publications

References 3 publications

Conformal Metrics on the Unit Ball in Euclidean Space

Conformal Metrics on the Unit Ball in Euclidean Space

Global integrability of p-superharmonic functions on metric spaces

Loewner chains and Hölder geometry

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