An example related to Whitney's extension problem for $L^{2,p}({\mathbb{R}}^2)$ when $1<p<2$

In this paper, we prove the existence of a bounded linear extension operator $T:L^{2,p}(E)\to L^{2,p}({\mathbb{R}}^2)$ when $1<p<2$, where $E\subset {\mathbb{R}}^2$ is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman-Klartag on the existence of linear extension operators for radially symmetric binary trees.