Symbol estimates using metric on the phase space

Question

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:

1. 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 $g$?

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

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

Thank you in advance.

Answer 1

You are given a metric on the phase space, i.e. at each point $(x,\xi)$ in $\mathbb R^{2n}$ a positive definite quadratic form $g_{x,\xi}$ on $\mathbb R^{2n}$. You will use that metric (and the affine connection on $\mathbb R^{2n}$) to express the conditions on the symbols.

In the first place, the metric $g$ must be slowly varying, temperate and should satisfy the uncertainty principle according to the references to Lerner's book. Next, you introduce a weight function $m$ defined on the phase space and valued in $(0,+\infty)$, with slow variation and temperate requirements as in the same reference.

Then you define the space of symbols $S(m,g)$ as the set of smooth functions $a$ on the phase space such that $$\forall k\in \mathbb N, \exists C_k,\forall X\in \mathbb R^{2n},\forall T\in \mathbb R^{2n}, \quad \vert a^{(k)}(X) T^k\vert\le C_k m(X)g_X(T)^{k/2}. \tag 1$$ Since the tensor products $T_1\otimes\dots\otimes T_k$ are sums of powers $S^{\otimes k}$ where $S$ is a linear combination of the vectors $T_j$, it turns out that $(1)$ is equivalent to \begin{multline}\forall k\in \mathbb N, \exists C_k,\forall X\in \mathbb R^{2n},\forall \{T_j\}_{1\le j\le k}\in \mathbb R^{2n}, \quad \\ \vert a^{(k)}(X) (T_1,\dots, T_k)\vert\le C_k m(X)\prod_{j=1}^{j=k}g_X(T_j)^{1/2}. \tag{2}\end{multline} When you have chosen some symplectic coordinates such that, for a given point $(x,\xi)$ in the phase space, you have
$$ g_{x,\xi}(t_1,\dots,t_n, \tau_1,\dots, \tau_n)=\frac{\vert t_1\vert^2}{r_1^2} +\dots+\frac{\vert t_n\vert^2}{r_n^2} + \frac{\vert \tau_1\vert^2}{\rho_1^2} +\dots+\frac{\vert \tau_n\vert^2}{\rho_n^2}, $$ the estimates $(1), (2)$ are equivalent to $$ \bigl\vert\bigl(\partial_{t_1}^{b_1}\dots\partial_{t_n}^{b_n} \partial_{\tau_1}^{\beta_1}\dots\partial_{\tau_n}^{\beta_n} a\bigr)(t,\tau)\bigr\vert \le C_{b_1,\dots, b_n,\beta_1,\dots,\beta_n} m(t,\tau)\Bigl(\prod_{j=1}^{j=n} r_j^{-b_j}\Bigr)\Bigl(\prod_{j=1}^{j=n} \rho_j^{-\beta_j}\Bigr). $$ Note that in this framework, you have $$ g^\sigma_{x,\xi}= {\vert t_1\vert^2}{\rho_1^2} +\dots+{\vert t_n\vert^2}{\rho_n^2} + {\vert \tau_1\vert^2}{r_1^2} +\dots+{\vert \tau_n\vert^2}{r_n^2}, $$ so that the "uncertainty principle" condition $g_{x,\xi}\le g^\sigma_{x,\xi}$ reads $$ \min_{1\le j\le n}\rho_j r_j\ge 1. $$