What does $|Du| \leq 1 $ a.e. mean for $u\in L^2(\mathbb{R}^n)$.
In the paper that I am reading, the authors used the notation $|Du| \leq 1 $ a.e. I know when $u\in W^{1,p}(\mathbb{R}^n)$ then this is the same as $\|Du\|_\infty \leq 1$, but what about when $Du$ is not necessarily a function?
For example, the distributional derivative of the jump function at $0$ in $\mathbb{R}$ is $\delta_0$, I thought we can say $\delta_0 \leq 1 $ a.e. but $\|\delta_0\|_\infty$ is not well defined since $\delta_0$ is not a function.
Here is the link to the full paper. And this notation is even used in the abstract.
Thank you very much!
It seems that the author mean the following: $u$ does beong to $L^2(\mathbb{R}^N)$ and its distributional derivative can be identified with a function, which is bounded a.e. by another function.
To support my guess, note that the variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is defined by
$$W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):\ |\nabla u|\in L^{p(x)}(\Omega)\}.$$
As you can see, they do not talk explicitly about weak derivatives or distributional derivatives, so it is safe to assume that the above definition means that the distributions $\partial u/\partial x_i$ can be identified with functions in $L^{p(x)}(\Omega)$.