# Principal type, partial differential operator of

with constant coefficients

An operator $A( D)$ whose principal part $P( D)$( cf. Principal part of a differential operator) satisfies the condition

$$\tag{* } \sum _ {j = 1 } ^ { n } \left | \frac{\partial P ( x) }{\partial x _ {j} } \ \right | ^ {2} \neq 0$$

for any vector $\mathbf x = ( x _ {1} \dots x _ {n} ) \in \mathbf R ^ {n}$. Another formulation is: Any real hyperplane that is characteristic with respect to $P( D)$ must be a simple characteristic. Condition (*) is necessary and sufficient for the domination of $A( D)$ by any operator of lower order. Operators with identical principal parts $P( D)$ are equally strong if and only if condition (*) is satisfied. If the coefficients are variable, the condition to the effect that $A( D)$ is of principal type is usually formulated using special inequalities estimating the derivatives of functions with compact support by the values of the operator. If condition (*) holds pointwise, a supplementary condition regarding the order of the commutator $[ P( x, D), \overline{P}\; ( x, D)]$ is sufficient to ensure that $A( D)$ is in fact an operator of principal type.

Let $A( D)$, $B( D)$ be constant-coefficient linear partial differential operators on $\mathbf R ^ {n}$ with principal parts $P( D)$, $Q( D)$, respectively. Put $\widetilde{A} ( \xi ) = \sum _ {| \alpha | \geq 0 } | A ^ {( \alpha ) } ( \xi ) | ^ {2}$( the sum is finite, since $A( \xi )$ is a polynomial), and similarly for $B ( \xi )$. Then $A( D)$ is said to be stronger than $B( D)$, written $B \prec A$, if
$$\frac{\widetilde{B} ( \xi ) }{\widetilde{A} ( \xi ) } < C ,\ \ \xi \in \mathbf R ^ {n} ,$$
for some constant $C$. $A( D)$ and $B( D)$ are said to be equally strong if $B \prec A \prec B$.