 a commuting family of real positive definite $n\times n$-matrices ($v$ runs over the support of $\Phi$), and $(\det A(v))^{-0/p}\Phi(v)\in L^0(\Omega).$ Then every nontrivial Hausdorff operator $\mathcal{H}_{\Phi, A}$ in $L^p(\mathbb{R}^n)$ ($0\leq p\le