Elliptic differential operator

A constant-coefficient operator of order $N$ is elliptic when its homogeneous principal symbol $P_N(\xi )$ is nonzero for every real $\xi \ne 0$.

A parametrix for $P$ is an operator $A$ such that $PA-I$ and, when relevant, $AP-I$ are regularizing operators.

A smooth function $a(x,\xi )$ belongs to $S_{1,0}^m$ when, on every compact $K\subset X$,

Frequency differentiation lowers symbol order, position differentiation preserves it, products add orders, and finite sums have order at most the maximum order.