Planar Sobolev extension domains and quasiconformal mappings

In 1979, Goldshtein et al. pointed out that a bounded simply connected planar domain is a $W^{1,\,2}$-extension domain if and only if it is a quasidisk. This result was later generalized by Jone to all dimensions, showing that uniform domains are $W^{1,\,p}$-extension domain for any $1\le p<\infty$. However, there are simply connected $W^{1,\,p}$-extension domains in the plane which are not uniform when $p\neq 2$, pointed out by e.g. Maz'ya. In this talk I will present my joint work with Koskela and Rajala on this problem, and show its connection to the Ahlfors' quasiconformal reflection theorem.