I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$
where $P$ is the wave operator $\square_{g}$.
Now is the energy inequality true for $-s$?
I have attempted the following:
Let $\phi\in H^{-s}$. Then we can define $\psi=(1-\Delta)^{-s} \phi\in H^s$
So we have
$$\|\phi\|_{-s}=\| \psi\|_s \le C \int \| P\psi\|_s $$
Now if we estimate $\| P\psi\|_s $ in terms of $\|\phi\|_{-s}$ and $\| P\phi\|_{-s}$ The proof will be over.
Notice that
$$P\phi=P(1-\Delta)^s \psi=(1-\Delta)^s P \psi+ [P,(1-\Delta)^s]\psi $$
Hence,
$$\| P \psi\|_s \le \Arrowvert P\phi\Arrowvert_{-s} +\|[P,(1-\Delta)^s]\psi\|_{-s} $$
Can someone point me out if there are some estimates for the commutator?
In the book "The Cauchy problem in General Relativity " by Ringstrom it is stated that the following proposition:
Let $m$ and $l$ be non-negative integers, $\alpha\le l+m$, $u\in S$ and $f\in C^{\infty}$. Then $$||f\partial^{\alpha}u||_{-m}\le C ||u||_{l}$$
gives the following bound for the commutator
\begin{equation} C(||\psi||_{s}+||\psi_{t}||_{s-1}) \end{equation} Although I am not clear how he gets it. Also he expresses the problem as a first order PDE. Is this necessary?
I also think that the result can be shown using the theory of pseudo-differential operators.
The idea will be to show that
$$[P,(1-\Delta)^s]$$ is a bounded linear operator from $H^{s}$ to $H^{-s}$.
We know that $(1-\Delta)^s\in OPS^{2s}$ and that $P\in OPS^{2}$.
Is there any theorem that might show the desired result?
