An example related to Whitney’s extension problem for L 2,p (R2) when 1 < p < 2

In this paper, we prove the existence of a bounded linear extension operator T : L 2 , p ( E ) → L 2 , p ( R 2 ) $T:{L}^{2,p}\left(E\right)\to {L}^{2,p}\left({\mathbb{R}}^{2}\right)$ when 1 < p < 2, where E ⊂ R 2 $E\subset {\mathbb{R}}^{2}$ is a certain discrete set with fractal structure. Our proof makes use of a theorem of Fefferman–Klartag (“Linear extension operators for Sobolev spaces on radially symmetric binary trees,” Adv. Nonlinear Stud., vol. 23, no. 1, p. 20220075, 2023) on the existence of linear extension operators for radially symmetric binary trees.