Basic examples of functions in Hörmander class The Hörmander class $S_{\rho,\delta}^m$ (with $\rho,\delta\in[0,1]$) consists of smooth functions $p(x,\xi)$ with
$$|D_x^\beta D_\xi^\alpha p(x,\xi)|\leq C_{\alpha\beta}(1+|\xi|^2)^{(m-\rho|\alpha|+\delta|\beta|)/2}.$$
I'm a newcomer to pseudodifferential calculus so my question is: what are the basic examples to have in mind when thinking about this class? Relatedly, why is this class a natural thing to consider?
The only example apparent to me is that if
$$p(x,\xi)=\sum_{|\alpha|\leq k} a_\alpha(x)\xi^\alpha$$
with $a_\alpha\in C_c^\infty$, then $p\in S_{0,\delta}^m$ for any $\delta\geq 0.$
 A: I don't know if the following example is nontrivial enough for you, but I'll let you decide that. Let $\psi$ be a Schwartz function such that $\widehat{\psi}$ has support in the annulus $\{2^{-1/2}\leq|\xi|\leq 2^{1/2}\}$. Consider the symbol $a$ given by
$$a(x,\xi)=\sum_{j=1}^{\infty}a_{j}(x)\widehat{\psi}(2^{-j}\xi), \tag{1}$$
where the $a_{j}$ are $C^{\infty}$ functions with uniformly bounded derivatives. Set $\psi_{j}$ so that $\widehat{\psi_{j}}=\psi(2^{-j}\cdot)$. You can check that $\widehat{\psi_{j}}$ is supported in the annulus $\{2^{j-1/2}\leq|\xi|\leq 2^{j+1/2}\}$, and therefore for each $\xi$, at most one term in (1) is nonzero. It is evident that $a\in C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})$. And when $2^{j-1/2}\leq|\xi|\leq 2^{j+1/2}$, $j\geq 1$, we have that
$$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\leq C_{\alpha}2^{-j|\beta|}\|\partial^{\beta}\widehat{\psi}\|_{\infty}\leq C_{\alpha\beta}(1+|\xi|^{2})^{-|\beta|/2},\quad\forall \xi\in\mathbb{R}^{n}$$
Lastly, we note that $a(x,\xi)$ vanishes for $|\xi|<2^{-1/2}$.
One source of motivation for considering the Hörmander class of symbols is because they generalize space multipliers and Fourier multipliers, which give rise to $L^{2}$-bounded operators. In the latter case, we have the Hörmander-Mikhlin multiplier theorem which gives conditions for a sufficiently smooth function on $\mathbb{R}^{n}\setminus\{0\}$ function $m(\xi)$ to yield an $L^{p}$-bounded operator for $1<p<\infty$. More precisely, it gives conditions for the associated operator to be a Calderón-Zygmund operator. 
On this last point, certain subclasses of symbols give rise to Calderón-Zygmund operators. The standard symbol class $S_{1,0}^{0}$ is such an example; this is precisely the Calderón-Vaillancourt theorem. I'm sorry I can't say more; I'm still incredibly new to the subject as well.
