An alternate definition of Sobolev space $W^{1,p}(\Omega)$ when $1<p\leq\infty$ and consequences

Question

Suppose that we define the Sobolev space $W^{1,p}(\Omega)$ with $1<p\leq \infty$, where $\Omega\subset\mathbb{R}^d$ ($d\geq 1$) is an open set (not necessarily bounded), in the following manner.

Definition. We say $u\in W^{1,p}(\Omega)$ if there is a constant $C$ such that for every open set $U$ with $\overline{U}\subset\Omega$ we have $$ \|\tau_hu-u\|_{L^p(U)}\leq C|h|,\;\;\forall h\in\mathbb{R}^n\,\, \text{ s.t } \,0<|h|< d(U,\partial \Omega).\label{1}\tag{$*$}$$

($\tau_h$ is the translation map). We know that the usually defined weakly differentiable functions satisfy \eqref{1}. On the other hand, any function $u$ satisfying \eqref{1} can be shown to satisfy $$\exists C\in\mathbb{R}, \,\text{ s.t }\left|\int_{\Omega}u\varphi'\right|\leq C\|\varphi\|_{L^{p'}(\Omega)}, \;\forall \varphi\in C^1_c(\Omega).\label{2}\tag{$**$}$$ By virtue of the Riesz representation theorem on $L^p$ and Hahn-Banach/extension by continuity, \eqref{2} implies that $u$ is weakly differentiable when $1<p\leq\infty$. Therefore, the definitions are equivalent.

However, \eqref{1} is easier for me to grasp (especially when $p=\infty$) than your usual technical definition (that requires $u$ to satisfy integration by parts with functions in $C^1_c$), as \eqref{1} seems to insinuate that "$u$ is differentiable on average" when $p<\infty$ and that "$u$ is Lipschitz a.e" (so differentiable a.e) when $p=\infty$. So here are my questions:

1. What more can be said about the intuition behind \eqref{1}, and what is a nice example of a function $u\not\in C^1(\Omega)$, for which we can prove \eqref{1} directly?

2. How can we use \eqref{1}, while possibly avoiding Riesz or even defining the weak derivative, to show that $$W^{1,p}(\Omega)\subset C(\overline{\Omega}) \text{ for all } \begin{cases}1<p\leq \infty & \text{ if } \Omega\subset \mathbb{R},\\ d<p\leq\infty & \text{ if }\Omega\subset\Bbb R^d,\;d\geq 2,\;\Omega \text{ regular enough.} \end{cases}$$
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   Possible approach to question 2 when $\Omega=\Bbb R$.
   
   I think one can use convolutions to go from the integral inequality \eqref{1} to a point-wise inequality. My intuition comes from the following inequality: if $u\in W^{1,p}(\Bbb R)$ (as defined by \eqref{1}) with $1<p<\infty$ and $\{\rho_n\}$ is the standard sequence of mollifiers then $$\|\rho_n\star (\tau_h u-u)\|_{L^\infty(\Bbb R)}\leq \underbrace{\|\tau_h u-u\|_{L^p(\Bbb R)}}_{\leq C|h|}\cdot\underbrace{\|\rho_n\|_{L^{p'}(\Bbb R)}}_{\leq Kn^{1/p}}.$$ Maybe this inequality will help if we also assume that $u\in L^\infty(\Bbb R)$, since this assumption implies $$\|\rho_n\star (\tau_h u-u)\|_{\infty}\to \|\tau_h u-u\|_\infty.$$

3. Do you think this approach might lead somewhere?

Answer 1

Preliminary remark. The equivalence of the usual definition of Sobolev spaces and the property \eqref{1} can for instance be found in Proposition 9.3 in this book by Brezis. It is even true in the vector-valued case under appropriate assumptions - see Theorem 2.2 in this article by Arendt and Kreuter, or Proposition 2.5.7 of this book by Hytönen, van Neerven, Veraar and Weis.

I would not say that it yields an alternative approach to the theory of Sobolev spaces, but rather that it is a useful tool the complements the classical definition via weak derivatives. I called it a tool because it is useful in the theory of Sobolev spaces on various occasions - also if you still you the classical definition.

Applications. Here are several applications of the characterization \eqref{1} of $W^{1,p} := W^{1,p}(\Omega)$. Let us assume, for the sake of simplicity, that $\Omega$ is bounded. (For more general result, please see the references provided below.) Fix $p \in (1,\infty]$

1. On bounded domains, the property \eqref{1} is obviously satisfied by every Lipschitz continuous function. So it follows that all Lipschitz continuous functions on $\Omega$ are in $W^{1,p}$ (as also pointed out in a comment by Hannes).

2. One can also deduce from \eqref{1} that for every Sobolev function $u \in W^{1,p}$ and every Lipschitz continuous $f: \mathbb{R} \to \mathbb{R}$, the composition $f \circ u$ is again in $W^{1,p}$.

More details about 1. and 2. can, for instance, be found in Corollary 2.7 in Arendt and Kreuter - which even deals with the vector-valued case.

3. In Remark 7 on page 269 in Brezis it is noted that \eqref{1} implies that every function in $W^{1,\infty}$ has a continuous representative.

4. Property \eqref{1} is used in the proof of the Rellich–Kondrachov theorem in Brezis (Theorem 9.16).

5. In Theorem 2.5.17 of Hytönen et. al., properpty \eqref{1} is used (in the vector-valued case) is used to prove an interpolation property of Sobolev spaces.

Summary. The property \eqref{1} is very useful - but I'd consideration it a complement of the classical definition rather than an alternative.