Principal part of a differential operator

The homogeneous differential operator formed from the given operator by discarding all the terms not containing derivatives of maximal order. The principal part of the differential operator

$$L=\sum_{|\alpha|\leq m}a_\alpha D^\alpha$$

is $\sum_{|\alpha|=m}a_\alpha D^\alpha$. The principal part of a differential operator is sometimes defined by the introduction of supplementary weights assigned to the differentiations with respect to the various arguments. For instance, the principal part of the differential operator $D_1-D_2^2+\alpha D_2$ is sometimes defined as $D_1-D_2^2$ (if $D_1$ is given weight 2 and $D_2$ weight 1).

Comments

The principal part is also called the principal symbol (cf. also Symbol of an operator).

The zero sets of the principal symbol are called the characteristics of $L$ (cf. also Characteristic).

Further, a constant-coefficient differential operator $L$ in $\mathbf R^n$ is said to be of real principal type if the principal symbol $l$ is real and if $l'(\xi)\neq0$ for $\xi\in\mathbf R^n\setminus 0$ (cf. also Principal type, partial differential operator of).

For a differential operator $L$ of order $m$ with $C^\infty$ coefficients in a $C^\infty$-manifold $X$ the principal symbol can be regarded as an invariantly-defined function on the cotangent bundle of $X$.

References

[a1]	L.V. Hörmander, "The analysis of linear partial differential operators" , 1 , Springer (1983)