A counterexample to the ${\displaystyle L^p}$-Hodge decomposition

We construct a bounded domain ${\displaystyle \Omega \subset {ℝ}^2}$ with the cone property and a harmonic function on Ω which belongs to ${\displaystyle W_0^{1,p}(\Omega )}$ for all 1 ≤ p < 4/3. As a corollary we deduce that there is no ${\displaystyle L^p}$-Hodge decomposition in ${\displaystyle L^p(\Omega ,{ℝ}^2)}$ for all p > 4 and that the Dirichlet problem for the Laplace equation cannot be in general solved with the boundary data in ${\displaystyle W^{1,p}(\Omega )}$ for all p > 4.