Let $A$ be an bounded operator on $L^2(\mathbb{R})$ with norm bounded by $1$, $$ \|A\|_{op}\leq 1. $$ Suppose that the commutator $[x,A] = xA-Ax$, against the multiplication operator $x$, also satisfies $$\|[x,A]\|_{op}\leq 1. $$
The commutator $[x,A]$ somehow measures how nonlocal the operator $A$ is. Indeed, if $[x,A]=0$ then one would have that $A$ must be multiplication against some function.
I am interested in whether $[x,A]$ controls commutators of the form $[V,A]$, where $V(x)$ is a Lipschitz function. More precisely, I would like to know whether it holds that $$ \|[V,A]\|_{op} \leq \|V\|_{Lip} $$ given the assumptions above.
Any help would be greatly appreciated!