Estimate for the commutator \([A(x,D), I_s]\) in \({\mathcal B}_{1,\sigma} (\mathbb{R}^n)\) spaces (Q1568173)

Let \(B_{1,\sigma}= \{T\in {\mathcal S}^\prime (\mathbb{R}^n)\), \(\widehat{T} \in L^1_{\text{loc}}(\mathbb{R}^n)\), \(\int_{\mathbb{R}^n}(1+|\xi|^2)^\frac{s}{2}|\widehat{T}(\xi) d\xi<\infty\), \(\sigma \in \mathbb{R}\}\), \(I_s:B_{1,\sigma}\mapsto B_{1,\sigma-s}\), \(I_sT= F^{-1}((1+|\xi|^2)^\frac{s}{2} \widehat{T}(\xi))\) and \(A(x,D)\), \({\mathcal A}(x,D)\) the pseudo-differential operators corresponding to \(C^\infty(\mathbb{R}^n\times \mathbb{R}^n \setminus \{0\})\) and zero-homogeneous symbols \(a(x,\xi)\) \[ (A(x,D)u)^\wedge=a(\infty,\xi)\widehat{u}(\xi)+(2\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^n} \widehat{a}^\prime(\xi-\eta,\eta)\widehat{u}(\eta)d\eta, \] \[ \mathcal A(x,D)u=\mathcal F^{-1}(a(\infty,\xi)\widehat{u}(\xi)+(2\pi)^{-\frac{n}{2}}\int_{\mathbb{R}^n} \widehat{a}^\prime(\xi-\eta,\eta)\widehat{u}(\eta)d\eta) \quad (\forall) u\in B_{1,\sigma} \] \(\widehat{a}^\prime(\lambda,\eta)\) representing the Fourier transform with respect to \(x\) of the symbol \(a^\prime(x,\eta)=a(x,\eta)-a(\infty,\eta)\). The author proves that the commutator \([A,I_s]=AI_s-I_sA\) is linear continuous from \(B_{1,\sigma}\) in \(B_{1,\sigma+1-s}\) as well as \([\mathcal A,I_s].\)