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Controlled diffeomorphic extension of homeomorphisms. (English) Zbl 1405.31001

If \(\Omega \subset \mathbb{C}\) is a bounded convex domain and \(\varphi\) is a homeomorphism from the unit circle \(\mathbb{S}\) onto \(\partial \Omega\), then the complex-valued Poisson extension \(h\) of \(\varphi\) is a homeomorphism from \(\overline{\mathbb{D}}\) onto \(\overline{\Omega}\) and a diffeomorphism in \(\Omega\) but its derivatives are not necessarily uniformly bounded. If the boundary \(\partial \Omega\) is a \(C^1\)-regular Jordan curve, then K. Astala et al. [Proc. Lond. Math. Soc. (3) 91, No. 3, 655–702 (2005; Zbl 1089.30013)] proved that the square integrability of the derivatives of \(h\) is equivalent to the condition \[ \int_{\partial \Omega}\int_{\partial \Omega}|\log |\varphi^{-1}(\xi)-\varphi^{-1}(\eta)| | |d\xi| |d\eta| <+\infty\, . \] In the paper under review, the authors generalize the above-mentioned result and show that if \(\Omega\) is an internal chord-arc domain then the homeomorphism \(\varphi\) has finite dyadic energy if and only if \(\varphi\) has a diffeomorphic extension \(h\) which has finite energy.

MSC:

31A05 Harmonic, subharmonic, superharmonic functions in two dimensions

Citations:

Zbl 1089.30013
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References:

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