Cross-post from math.sx.

Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the operators $\eta(-i\nabla)$, which is a well-defined bounded and selfadjoint operator by the functional calculus, and $M_f$, which is the multiplication operator given by $f$, acting on $L^2(\mathbb R^n)$.

If $f$ is Lipschitz-continuous, I can prove the commutator is bounded by using properties of the Fourier transform. However, I would like to use weaker assumptions (for example locally Hölder continuous) on $f$, as my assumption on $\eta$ is rather strong. References, proofs and counterexamples are all warmly welcome.