Sobolev Algebra Counterexamples

TL;DR

This paper constructs fractal examples demonstrating the failure of Sobolev algebra properties in certain geometric settings, contrasting with known Euclidean results.

Contribution

It introduces fractal counterexamples showing Sobolev space algebra failure beyond Euclidean spaces, expanding understanding of geometric influences.

Findings

Sobolev algebra property fails on fractals for various indices

Counterexamples extend to a wide range of parameters

Highlights limitations of existing Sobolev algebra results

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 $\alpha,p$.