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''with constant coefficients''

An operator  $  A( D) $
whose principal part  $  P( D) $(
cf. [[Principal part of a differential operator|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|characteristic]] with respect to  $  P( D) $
must be a simple characteristic. Condition (*) is necessary and sufficient for the [[Domination|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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>