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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!

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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 $\sigma\leq s+m_1+m_2-1.$

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Answer to MO question/414577 Your question reduces to the simpler version you state,in view of Egorov elliptic Fourier conjugation.Also,for a star domain Pohozaev Identity directly computes the gradient estimated p*- norm of u in terms of the normal derivative at the boundary, noting that on the zero trace subspace,H^1 norm is equivalent to the grad(u) norm- NagarajIyengar

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