# Principal type, partial differential operator of

*with constant coefficients*

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

(*) |

for any vector . Another formulation is: Any real hyperplane that is characteristic with respect to must be a simple characteristic. Condition (*) is necessary and sufficient for the domination of by any operator of lower order. Operators with identical principal parts are equally strong if and only if condition (*) is satisfied. If the coefficients are variable, the condition to the effect that 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 is sufficient to ensure that is in fact an operator of principal type.

#### Comments

Let , be constant-coefficient linear partial differential operators on with principal parts , , respectively. Put (the sum is finite, since is a polynomial), and similarly for . Then is said to be stronger than , written , if

for some constant . and are said to be equally strong if .

#### References

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

**How to Cite This Entry:**

Principal type, partial differential operator of.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Principal_type,_partial_differential_operator_of&oldid=15899