 Solution

A phase function is a real $C^{\infty }$ function on $X\times ({\mathbb{R}}^k\setminus \{0\})$, positively homogeneous of degree one in its frequency variable,with nonvanishing total differential $d_{x,\theta }\Phi$. Vanishing of its frequency gradient alone is allowed; those critical directions are relevant to singularities.

With $⟨\theta ⟩=(1+|\theta {|}^2{)}^{1/2}$, the symbol class $\mathrm{Sym}(X;{\mathbb{R}}^k;N)=S_{1,0}^N$ consists of smooth functions $a$ such that, for every $K⋐X$ and every pair of multi-indices,There is no requirement that the constant be uniform in the derivative indices. We use $D=(1/i)\partial$; the factors of $i$ do not affect these estimates.

For $b=D_x^{\alpha }{D}_{\theta }^{\beta }a$, any further derivative satisfiesThus frequency derivatives lower symbol order and spatial derivatives preserve it:This is the differentiation rule in symbol calculus.