The following is Exercise 3 of Chapter 3 in Stein & Shakarchi Book 4:
Show that a bounded function $f$ on $\mathbb{R}^d$ satisfies a Lipschitz condition $$|f(x)-f(y)| \leq C|x-y| \qquad\text{ for all } x,y\in\mathbb{R}^d,$$ if and only if $f\in L^\infty$ and all the first order partial derivatives $\partial f/\partial x^j$ ($1\leq j\leq d$), belong to $L^\infty$ in the sense of distributions.
The right side of the $\Leftrightarrow$ doesn't make any sense to me. Of course we can take derivatives of $f$ in the sense of distributions, but what would it mean for a distribution to be $L^\infty$? The book doesn't make mention of it, and it looks like it could easily be a typo. My google search got me nothing, and I can't find errata for this book. Distributions are linear functions, so they can't strictly speaking have an $L^\infty$ norm in the usual sense. My best guess is that it's a bounded linear functional on $D(\Omega)$.
Is this a common notation? Do any of you know what it might mean?
What is probably meant is that the derivative of $f$ with respect to $x^j$ in sense of distributions can be represented by an $\mathbb L^\infty$ function. This means that there exists a function $g\in \mathbb L^\infty$ such that for any $\varphi\in\mathcal DV(\Omega)$, $$-\int_\Omega f(x)\partial _j \varphi(x)\mathrm dx=\int_{\Omega}g(x)\varphi(x)\mathrm dx.$$