Quasiconformal mappings with Sobolev boundary values

Astala, Kari; Bonk, Mario; Heinonen, Juha

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 3, pp. 687-731.

We consider quasiconformal mappings in the upper half space ${ℝ}_{+}^{n+1}$ of ${ℝ}^{n+1}$, $n\ge 2$, whose almost everywhere defined trace in ${ℝ}^n$ has distributional differential in $L^n\left({ℝ}^n\right)$. 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 ${ℝ}_{+}^{n+1}$, called conformal densities. These densities mimic the averaged derivatives of quasiconformal mappings, and we prove analogous trace theorems for them. The abstract approach of general conformal densities sheds new light to the mapping case as well.

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