On a Class of Hypoelliptic Operators

Let the variables in be broken up into two groups , where and . We consider differential operators with polynomial symbols of the form where . We assume that the symbol is quasihomogeneous: and that is elliptic for . We have found a necessary and sufficient condition for operators of this class to be hypoelliptic: namely, that the equation , , have no nontrivial solutions in . Thus for example, the operator is hypoelliptic for any integers and , and the operator is hypoelliptic if and only if is not an eigenvalue of the operator in . These results are partially extended to operators with variable coefficients and to pseudodifferential operators.Bibliography: 22 references.