Consider the symbols introduced by Beals and Fefferman in which the symbol estimates are of the form: \begin{equation}\tag{1} \label{eq1} |\partial_x^\beta\partial_\xi^\alpha a(x,\xi)| \leq C_{\alpha \beta} M(x,\xi) \Phi(x,\xi)^{-|\alpha|} \varphi(x,\xi)^{-|\beta|}, \quad (x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n, \end{equation} where the positive functions $M,\Phi, \varphi$ satisfy certain conditions.

Now from the book- Metric on the Phase Space and Non-Selfadjoint Pseudodifferential Operators by N. Lerner (Chapter 2 Section 2.2), it is said that by considering the following Riemannian structure on $\mathbb{R}^{2n}$ $$ g_{x,\xi} = \frac{|dx|^2}{\varphi(x,\xi)^2} + \frac{|d\xi|^2}{\Phi(x,\xi)^2}: \; \text{ for }(t,\tau)\in \mathbb{R}^n \times \mathbb{R}^n, g_{x,\xi}(t,\tau) = \frac{|t|^2}{\varphi(x,\xi)^2} + \frac{|\tau|^2}{\Phi(x,\xi)^2}, $$ one can write the symbol estimate in (\ref{eq1}) for $|\alpha|+|\beta|=1,$ in terms of the metric as below: \begin{equation}\label{eq2}\tag{2} |\nabla a(X)\cdot T|\leq C g_{X}(T)^{1/2}, \end{equation} for $X=(x,\xi), T=(t,\tau).$

I tried to see the estimate (\ref{eq2}), in the following two ways:

For the sake of demonstration let's take $n=1.$

(I) Considering $\nabla$ w.r.t Euclidean metric: I have $$ \begin{aligned} |\nabla a(X)\cdot T| &= |(\partial_xa,\partial_\xi a) \cdot (t,\tau)|\\ &= |t\partial_xa +\tau \partial_\xi a|\\ &\leq C \Big( |t| \frac{M}{\varphi} + |\tau| \frac{M}{\Phi}\Big)\\ &= CM \Big\{\frac{|t|}{\varphi} + \frac{|\tau|}{\Phi}\Big\}\\ &\leq CM(g_X(T)^{1/2} + g_X(T)^{1/2})\\ &= 2CMg_X(T)^{1/2} \end{aligned} $$

(II) Considering $\nabla$ w.r.t metric $g$, in which case $$ \nabla a = (\varphi^2\partial_x a,\Phi^2\partial_\xi a). $$ Then $$ \begin{aligned} |\nabla a(X)\cdot T| &= |(\varphi^2\partial_x a,\Phi^2\partial_\xi a) \cdot (t,\tau)|\\ &= |t\varphi^2\partial_xa +\tau \Phi^2\partial_\xi a|\\ &\leq C \Big( \varphi^2|t| \frac{M}{\varphi} + \Phi^2 |\tau| \frac{M}{\Phi}\Big)\\ &= CM \{\varphi|t| + \Phi|\tau|\}, \end{aligned} $$ after this I am unable to go further to establish (\ref{eq2}).

**My questions**:

When the author (N .Lerner) says to consider the metric $g$ what does he mean? Should we consider gradient $\nabla$ w.r.t Euclidean metric or w.r.t $g$?

If $\nabla$ has to be considered w.r.t $g$, how do I proceed further in my calculations?

If $\nabla$ has to be considered w.r.t Euclidean metric, then what exactly is the role of $g$?

Thank you in advance.