# Domains in $\mathbb{R}^n$ for which Hajlasz-Sobolev spaces and Sobolev Spaces are the same

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!!

Your argument is not correct. If a property $$P$$ fails for $$Y$$ and $$X\subset Y$$, it does not follow that it fails for $$X$$. For example $$X=\{0\}\subset\mathbb{R}=Y$$ but there are many properties true for $$X$$ and not true for $$Y$$.

You always have $$M^{1,p}(\Omega)\subset W^{1,p}(\Omega)$$ for all $$1\leq p\leq\infty$$. However, the inclusion is usually strict.

The spaces $$W^{1,p}(\Omega)$$ and $$M^{1,p}(\Omega)$$ are equal if for example $$\Omega$$ is a bounded extension domain. There is in fact the following characterization:

Theorem. Let $$1. Then a bounded domain $$\Omega\subset\mathbb{R}^n$$ is a $$W^{1,p}$$-extension domain if and only if $$M^{1,p}(\Omega)=W^{1,p}(\Omega)$$ and there is $$C>0$$ such that $$\begin{equation} |B(x,r)\cap\Omega|\geq Cr^n \quad \text{for all x\in\Omega and r\leq\operatorname{diam}(\Omega).} \end{equation}$$

Example. Take a disc with a radius removed. The above condition for the measure is satisfied, but the domain is not an extension domain so $$M^{1,p}$$ cannot be equal to $$W^{1,p}$$.

For a good source for basic properties, see

P. Hajłasz, Sobolev spaces on metric-measure spaces. (Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002)), 173--218, Contemp. Math. , 338, Amer. Math. Soc., Providence, RI, 2003.

The above theorem is from

P. Hajłasz, P. Koskela, H. Tuominen, Sobolev embeddings, extensions and measure density condition J. Funct. Anal. 254 (2008), 1217--1234.