Examples of $d-$sets with irregular projection of Hausdorff measures

Given positive integers $\mathrm{\ell }<n$ and a real $d\in (\mathrm{\ell },n)$, we construct sets $K\subset {\mathbb{R}}^n$ with positive and finite Hausdorff $d-$measure such that the Radon-Nikodym derivative associated to all projections on $\mathrm{\ell }-$dimensional planes is not an $L^p$ function, for all $p>1$.