I am reading an article of D. Tataru and I found the following situation,
Suppose we have a differential operator $P(x,D)$ of order $m$ in $\mathbb{R}^n$. Let $\Sigma$ be an oriented hypersurface in $\mathbb{R}^n$, which we represente as a nondegenerate level set of a smooth function, $\Sigma=\{\phi=0\}$ and $\Sigma^+=\{\phi>0\}$, $\Sigma^-=\{\phi<0\}$.
If $P$ has a real symbol and is not elliptic, then we can consider high frequency solutions which are localized near null bicharacterizistic rays of $P$, given by \begin{equation*} \dot{x}=p_{\xi}\qquad \dot{\xi}=-p_x \end{equation*} on $\{p(x,\xi)\}=0$ where $p(x,\xi)$ is the principal symbol of $P$. Then it is natural to ask that all such rays which intersect a small neighborhood of $\Sigma$ in $\Sigma^-$ must also intersect $\Sigma^+$. A strong form of this condition can be expressed in the form \begin{equation} \{p,\phi\}=p=0\implies\{p,\{p,\phi\}\}>0\qquad\text{ on }T_{\Sigma}^{*}\mathbb{R}^{n}, \end{equation} where $\{\cdot,\cdot\}$ is the Poisson bracket of two symbols, $\{p,q\}=p_{\xi}q_{x}-p_{x}q_{\xi}$.
I don't understand why this condition can be expressed in the form \begin{equation} \{p,\phi\}=p=0\implies\{p,\{p,\phi\}\}>0\qquad\text{ on }T_{\Sigma}^{*}\mathbb{R}^{n}, \end{equation}
Cause, $\{p,\phi\}=p_{\xi}\phi_{x}-p_{x}\phi_{\xi}=0$ implies that $p_{\xi}\phi_{x}=p_{x}\phi_{\xi}$, Where does that condition come from and why you need the tangent space for the definition??
Morover, if we have $\{p,\phi\}=p=0$ then $$\{p,\{p,\phi\}\}=\{p,0\}=0$$
I really don't understand this condition and what is the meaning of that
Thanks!! Any idea or hint will be appreciated!!