# Metric on the phase space

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?

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.
• When you are given a positive-definite quadratic form $Q$ on a symplectic vector space, you may define $Q^\sigma$ the symplectic inverse (you can check the definition in the Hörmander reference given above). With $Q=g_X$, Hörmander's conditions on the metric involve $g_X$ and $g_X^\sigma$. – Bazin Jun 7 at 8:10