Local solvability of some classes of linear differential operators with multiple characteristics (Q2756130)

The author considers operators of form \( P(x,\xi)= (p(x,\xi)) ^{m} +p _{m-1}(x,\xi) + \cdots + p _{0} (x,\xi)\), where ord\(_{\xi } p _{m-j}= m-j\) and \(p(x,\xi)\) is a first order real-valued symbol of principal type. Denote by \( p' _{m-1}\) the sub-principal symbol of \(P\) and assume that \( \operatorname {Re} p' _{m-1}( \rho ^{0})\neq 0\) if \(p(\rho ^{0})=0\). Assuming \(m\) odd, the author gives conditions on the imaginary part of \( p '_{m-1}\) under which the operator will not be locally solvable. An explicit example is the operator \(D _{1} ^{m}+(a+ib x _{1} ^{k})D _{2} ^{m-1} +(c +id x _{1} ^{\ell })D _{2} ^{m-2} D _{1}\), \( m \geq 3\), odd, \(a,b,c,d\) real constants, \(a\neq 0, b \neq 0, d\neq 0\) and \(k\) an even integer, \(k \geq 2 \ell +2\).