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The boundary absolute continuity of quasiconformal mappings (II).

Juha Heinonen — 1996

Revista Matemática Iberoamericana

In this paper a quite complete picture is given of the absolute continuity on the boundary of a quasiconformal map B → D, where B is the unit 3-ball and D is a Jordan domain in R with boundary 2-rectifiable in the sense of geometric measure theory. Moreover, examples are constructed, for each n ≥ 3, showing that quasiconformal maps from the unit n-ball onto Jordan domains with boundary (n - 1)-rectifiable need not have absolutely continuous boundary values.

Quasiconformal mappings onto John domains.

Juha Heinonen — 1989

Revista Matemática Iberoamericana

In this paper we study quasiconformal homeomorphisms of the unit ball B = B = {x ∈ R: |x| < 1} of R onto John domains. We recall that John domains were introduced by F. John in his study of rigidity of local quasi-isometries [J]; the term John domain was coined by O. Martio and J. Sarvas seventeen years later [MS]. From the various equivalent characterizations we shall adapt the following definition based on diameter carrots, cf. [V4], [V5], [NV].

Smooth quasiregular mappings with branching

Mario BonkJuha Heinonen — 2004

Publications Mathématiques de l'IHÉS

We give an example of a 𝒞 3 - ϵ -smooth quasiregular mapping in 3-space with nonempty branch set. Moreover, we show that the branch set of an arbitrary quasiregular mapping in-space has Hausdorff dimension quantitatively bounded away from . By using the second result, we establish a new, qualitatively sharp relation between smoothness and branching.

Quasiconformal mappings with Sobolev boundary values

Kari AstalaMario BonkJuha Heinonen — 2002

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

We consider quasiconformal mappings in the upper half space + n + 1 of n + 1 , n 2 , whose almost everywhere defined trace in n has distributional differential in L n ( n ) . We give both geometric and analytic characterizations for this possibility, resembling the situation in the classical Hardy space H 1 . More generally, we consider certain positive functions defined on + n + 1 , called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems for them....

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