Restricting a quasiconformal homeomorphism of the disc to the boundary gives a surjective homomorphism from $QC(D^2)$ (quasiconformal homeos of $D^2$) to $QS(S^1)$ (quasisymmetric homeos of the circle). Surjetivity here follows, for example, from the existence of natural extensions of QS homeos like DouadyEarle.
The same restrictiontotheboundary map from the group of area preserving smooth diffeos (symplectomorphisms if you will) to Diff$(S^1)$ is also surjective  one way to see this is to construct a byhand extension defined in a collar neighborhood and then use Moser's theorem (here), and details of perhaps another approach have been written up here.
My question is: what are the possible "boundary values" of area preserving QC homeos of the disc? My guess is that the map from area preserving QC homeos to $QS(S^1)$ is not surjective... perhaps someone with a good understanding of symplectic homeos has an idea of what the image is.



Claim. A QS homeomorphism $f$ of the circle extends to a QC area preserving map of the disk if and only if $f$ is BL (biLipschitz). Proof.
I was assuming here that $f$ is orientationpreserving. However, by composing with a symmetry of the disk, the general case reduces to this one, provided that you allow QC maps to reverse orientation. 

