I'm reading Heinonen's book on metric measure spaces. He writes that for general domains $\Omega \subset \mathbb{R}^n$, $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$ where the former are Hajlasz-Sobolev spaces (defined in 5.4) and the latter are Sobolev spaces.

Later in 5.17, he remarks on a way to see that $M^{1,p}(\Omega)$ and $W^{1,p}(\Omega)$ are not equivalent by constructing a domain for which the Poincare inequality fails. I don't understand this comment though: if the Poincare inequality fails for $u \in W^{1,p}(\Omega)$, it will also fail for $M^{1,p}(\Omega)$ because $M^{1,p}(\Omega) \subset W^{1,p}(\Omega)$.

What is Heinonen trying to say when he's using the Poincare inequality to show non-equality of $M^{1,p}(\Omega)$ and $W^{1,p}(\Omega)$? Thanks!!