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Sobolev Algebra Counterexamples

In the Euclidean setting the Sobolev spaces $W^{α,p}\cap L^\infty$ are algebras for the pointwise product when $α>0$ 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 $α,p$.