I'm just beginning the study of pseudo-differential operators with focus on symbols in $S_{1,0}^m$ and I'm having a hard time grasping why we require a symbol $p(x,\xi)$ to satisfy
$$|D_{x}^{\beta}D_{\xi}^{\alpha}p(x,\xi)|\leq C_{\alpha\beta}(1+|\xi|^2)^{\frac{m-|\alpha|}{2}}$$
I'm sure there's a logical reason behind this and I hope someone can help me in the right path to understanding why. Thanks in advance.
Initially, one may pseudodifferential operators for "nice" symbols, like Schwartz or compactly-supported symbols. These symbols have various limitations, and one often extends to wider ranges of classes. One such example is functions whose derivative is bounded by an order function (of which $\langle\xi\rangle^m$ is an example). The problem with a class like this is that they're not all invariant under a change of coordinates (i.e. preserve the symbol class). In fact, such a class is so restrictive that it leaves out differential operators. The issue that arises is that, when you pull back, you may pick up extra powers of $\xi.$
Kohn-Nirenberg symbols make it so that taking derivatives in $\xi$ improves the decay in $\xi$. These symbols are invariant under changing coordinates, which is an important quality when one wants to define pseudodifferential operators on manifolds.