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

Question

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!!

Answer 1

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 $1<p<\infty$. 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.