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.

Comments

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 $.

References

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