# Reference for commutator estimate

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!

Given your assumptions on $$p$$ and $$q$$, $$[Op(p),Op(q)]\in S^{m_1+m_2-1}_{1,0}.$$ Given any $$A\in OPS^m_{\rho,\delta}$$ with $$0\leq\delta<\rho\leq 1$$, one has that $$A:H^s\rightarrow H^{s-m}$$ boundedly (follows from a combination of Calderon-Valliancourt and the Fourier multiplier $$\Lambda_s$$ with symbol $$\langle \xi\rangle^s$$). In particular, given $$u\in H^s,$$ we have that $$\|[Op(p),Op(q)]u\|_{H^\sigma}\lesssim \|u\|_{H^s}$$ for any $$\sigma\leq s+m_1+m_2-1.$$