Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ of $h$, is a diffeomorphism of $D$ and homeomorphism of $\bar{D}$.


My question is : suppose $h:S^1\to S^1$ is a k-quasisymmetric homeomorphism, then is the complex harmonic extension $H=E(h)\hspace{2mm} K$-quasiconformal with $K$ depending only on $k$ ?


Answers/resources/papers related to this would be greatly appreciated !



Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$
For the unit disk see: <http://poincare.matf.bg.ac.rs/~pavlovic/QC_DIF.PDF>
For the halp-plane: <http://poincare.matf.bg.ac.rs/~pavlovic/QC_DIF.PDF>
For a generalization to Jordan domains: <http://arxiv.org/abs/0910.4950> (see also my paper in MATH z)


