Metric on the phase space

Question

I am studying PDEs whose symbols satisfy
\begin{equation} |\partial^\alpha_\xi\partial^\beta_xp(x,\xi)| \lesssim M(x,\xi)\Psi(x,\xi)^{-|\alpha|}\Phi(x,\xi)^{-|\beta|} \end{equation} for all multi-indices $\alpha,\beta$ and $(x,\xi) \in \mathbb{R}^{2n}$.
It is said that we must look at the metric \begin{equation} g_{x,\xi}=\frac{|dx|^2}{\Phi(x,\xi)^2}+\frac{|d\xi|^2}{\Psi(x,\xi)^2} \end{equation} for the phase space to study such PDEs.
My question- How do I relate the symbol to the metric? What is reason behind such a choice of metric?

Answer 1

You can in fact reformulate the conditions on the symbol $p$ by saying that $$ \vert p^{(k)}(X) T^k\vert\le C_k M(X)g_X(T)^{k/2}, $$ with $X=(x,\xi)$. Here $p^{(k)}(X) $ stands for the $k$-th derivative (a $k$-multilinear form) and $T$ is a vector in $\mathbb R^{2n}$. Using Lars Hörmander's notations in Chapter 18 of the third volume of ALPDO (Springer-Grundlehren 274), you have $$ p\in \mathcal S(M, g). $$ The asset of this presentation is that you can for instance give rather simple sufficient conditions on the metric $g$ to ensure that symbols in $\mathcal S(1, g)$ give rise to bounded operators on $L^2(\mathbb R^n)$. Also, you can prove that you have an algebra of operators and results of type $$ P_1P_2\quad \text{has a symbol in $\mathcal S(M_1M_2, g)$}, $$ whenever $P_j$ has a symbol in $\mathcal S(M_j, g)$, under some rather simple conditions on the weights $M_j$. You may have as well some asymptotic calculus and define a "Planck" function $ h(X) $ which in your case is $(\Phi(X)\Psi(X))^{-1} $ (which is required to be $\le 1$, some type of Heisenberg Uncertainty principle) and show that, for $P_j$ as above with with symbol $p_j$, $$ P_1P_2-\text{Op}(p_1p_2)\quad \text{has a symbol in $\mathcal S(M_1M_2 h, g)$}. $$ The first occurrence of this type of conditions was given by Richard Beals and Charles Fefferman in their solution of the Nirenberg-Treves conjecture for local solvability of principal type differential operators: they had to resort to a non-homogeneous micro-localization scheme involving the metric in your question in order to obtain some factorization result which fails to hold with standard homogeneous microlocalization arguments.