Universal radius of injectivity for locally quasiconformal mappings (original) (raw)

Quasiconformal mappings which increase dimension

ANNALES-ACADEMIAE SCIENTIARUM FENNICAE …, 1999

For any compact set E ⊂ R d , d ≥ 1 , with Hausdorff dimension 0 < dim(E) < d and for any ε > 0 , there is a quasiconformal mapping (quasisymmetric if d = 1) f of R d to itself such that dim(f(E)) > d − ε .

The Degree of Regularity of a Quasiconformal Mapping

Proceedings of the American Mathematical Society, 1994

T. Iwaniec has conjectured that the derivative of a locally a-Holder continuous quasiconformal mapping of W is locally integrable to any power p <-~. We disprove this conjecture by producing examples of quasiconformal mappings of the plane that are uniformly Holder continuous with exponent 5 < a < 1 but whose derivatives are not locally integrable to the power r¿- .

On the area distortion by quasiconformal mappings

Proceedings of the American Mathematical Society, 1995

We give the sharp constants in the area distortion inequality for quasiconformal mappings in the plane. Astala [I] proved the following theorem conjectured by Gehring and Reich in [3]: Theorem A. Let f be a K-quasiconformal mapping of D = { z : lzl < 1) onto itself with f ( 0 )= 0 . Then for any measurable E c D we have where I I stands for the area. The first author [2] obtained a shorter proof which did not make use of the elaborate Thermodynamic Formalism and Holomorphic Motion Theory of the original proof of Astala. In late 1992 the second author [4] circulated a minimal proof which gives sharp bounds for the constants under the normalization f E C ( K ), i.e. f is a K-quasiconformal mapping of the plane which is conformal on C\o and f ( z )= z +o(1) near oo . In the interests of having a short sharp proof we combined our efforts.

Unique extremality of quasiconformal mappings

Journal d'Analyse Mathématique, 2009

In this paper, we study the conditions under which unique extremality of quasiconformal mappings occurs and provide a broader point of view of this phenomenon. Additional information is obtained by means of specialized constructions. In particular, we generalize the construction theorem in [BLMM], thus providing a more basic understanding of it. We also generalize the notion of unique extremality and give an analytic characterization of the generalized concept.

LOCALLY UNIFORM DOMAINS AND QUASICONFORMAL MAPPINGS

We document various properties of the classes of locally uniform and weakly linearly locally connected domains. We describe the boundary behavior for quasiconformal homeomorphisms of these domains and exhibit certain metric conditions satisfied by such maps. We characterize the quasiconformal homeomorphisms from locally uniform domains onto uniform domains. We furnish conditions which ensure that a homeomorphism maps locally uniform domains to locally uniform domains. Everywhere examples are provided which illustrate the sharpness of our results.

On lipschitz continuity of quasiconformalmappings in space

Journal d'Analyse Mathématique, 2009

We study the local growth of quasiconformal mappings in the plane. Estimates are given in terms of integral means of the pointwise angular dilatations. New sufficient conditions for a quasiconformal mapping f to be either Lipschitz or weakly Lipschitz continuous at a point are given.

Mapping problems for quasiregular mappings

2012

Abstract: We study images of the unit ball under certain special classes of quasiregular mappings. For homeomorphic, ie, quasiconformal mappings problems of this type have been studied extensively in the literature. In this paper we also consider non-homeomorphic quasiregular mappings. In particular, we study (topologically) closed quasiregular mappings originating from the work of J. V\" ais\" al\" a and M. Vuorinen in 1970's. Such mappings need not be one-to-one but they still share many properties of quasiconformal mappings.