In the Appendix of "Commutator Estimates and the Euler and Navier-Stokes equations", Kato proves the following inequality \begin{equation}\tag{1}\label{1} \|J^s(fg)-f(J^sg)\|_{L^p}\lesssim \|\partial f\|_{L^\infty}\|J^{s-1}g\|_{L^p}+\|J^sf\|_{L^p}\|g\|_{L^\infty}\,, \end{equation} where $1<p<\infty$, $ s\geq 0 $ and $J=(I-\Delta)^{1/2}$.
Now, let $x\in \mathbb{T}\,,$ where $\mathbb{T}=\mathbb{R}/\mathbb{Z}$. Given $A_x$ a self-adjoint operator with compact resolvent, we denote by $e^{itA_x}$ the evolution group generated by $A_x\,$.
Questions:
1-I am wondering if there exists an inequality of the same type as \eqref{1}, such that we can control the $L^2$ norm of $$ \|[\partial_x,e^{itA_x}](g)\|_{L^2}\,, \hskip1cm\text{where }\quad [\partial_x,e^{itA_x}](g)=\partial_xe^{itA_x}g\,-\,e^{itA_x}\partial_xg\,, $$ and $g$ is any function in the Sobolev space $H^s$ for $s>0$ large enough.
2- Or does there exist any estimate on $ \|e^{itA_x}g\|_{H^1} ? $