Bounded Approximation Property for Sobolev spaces on simply-connected planar domains

We show that Sobolev space $W_1^1(\mathrm{\Omega })$ of any planar one-connected domain $\mathrm{\Omega }$ has the Bounded Approximation property. The result holds independently from the properties of the boundary of $\mathrm{\Omega }$. The prove is based on a new decomposition of a planar domain.