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 a strictly hyperbolic linear operator. For concreteness, let $P$ be 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

$$P\phi=P(1-\Delta)^s \psi=(1-\Delta)^s P \psi+ [P,(1-\Delta)^s]\psi $$


$$\| P \psi\|_s \le \Arrowvert P\phi\Arrowvert_{-s} +\|[P,(1-\Delta)^s]\psi\|_{-s} $$

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 two 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 desire result?



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