I'm interested in Sobolev space estimates for commutators involving a pseudodifferential operator and a Fourier multiplier. More specifically, suppose $p = p(x,\xi) \in S_{1,0}^{m_1}$ and let $q = q(\xi) \in S_{1,0}^{m_2}$, where $S_{\rho,\delta}^m$ denotes the standard Hörmander symbol class. Does anyone know of a reference for a commutator estimate of the form $$\| [\mathrm{Op}(p),\mathrm{Op}(q)](u) \|_{H^\sigma} \lesssim \|u\|_{H^s}$$ for some range of $\sigma, s$?

Or even something simpler like $$\| [\mathrm{Op}(p),\partial_j](u) \|_{H^\sigma} \lesssim \|u\|_{H^s}?$$

Any ideas would be appreciated! Thanks!