Let $\Omega$ be a bounded domain with lipschitz boundary. Given $f\in W^{1,1}(\Omega)$ . Let $g\in C^{\infty}(\Omega)$ such that $fg\in W^{1,\infty}(\Omega)$, does there exist a constant $C$ such that $$ |fg|_{1,\infty}\leq C\left(|f|_{1,1}|g|_{0,\infty}+|f|_{0,1}|g|_{1,\infty}\right)\quad? $$
For an example, let $\Omega\subset\mathbb{R}^3$ and $f=\frac{1}{|x|},x\in\Omega$, does there exist a constant $C$ such that for any $g$ satisfy $$g=(a\cdot x+b)^2,\quad\forall a\in \mathbb{R}^3,b\in\mathbb{R}$$ the above inequality holds ?
This cannot be true. In particular, for $g \equiv 1$ you ask whether $$|f|_{1,\infty} \le C ( |f|_{1,1} + |f|_{0,1} )$$ can hold for all $f \in W^{1,\infty}(\Omega)$, but this is blatantly false.