Show that every convex quadrilateral $K$ in $\mathbb{R}^2$ admits an extension operator $P:W^{1,p}(K)\to W^{1,p}(\mathbb{R}^2)$ . Deduce the same property for any triangle $K$ in $\mathbb{R}^2$ .
There is a result that if we consider two open sets $A_1,A_2$ in $\mathbb{R}^2$ and define a $C^1$-bijection $H:A_2\to A_1$ by $x=H(y)$ , then for $u\in W^{1,p}(A_1)$ one has $u\circ H\in W^{1,p}(A_2)$ , provided $\displaystyle\frac{\partial H_i}{\partial y_j}\in(L^\infty(A_2))^4$ for $i,j=1,2$ and $H=(H_1,H_2)$ .
I suspect that this correspondence between these two arbitrary open sets the assertion above can be proved for any convex quadrilateral , but I don't get any closer .
Any help is appreciated . Regards .