Sobolev Algebra Counterexamples

Abstract

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 $\alpha ,p$.