  In the Euclidean setting the Sobolev spaces $W^{\alpha,p}\cap L^\infty$ are
algebras for the pointwise product when $\alpha>0$ and $p\in(1,\infty)$. This
property has recently been extended to a variety of geometric settings. We
produce a class of fractal examples where it fails for a wide range of the
indices $\alpha,p$.
