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!



Reference for commutator estimate
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 $\sigma\leq s+m\_1+m\_2-1.$


