\(\omega\)-hypoelliptic differential operators of constant strength (Q1883358)

The authors study the hypoellipticity of differential operators with constant strength and coefficients in \(E_\omega(\Omega)\), where \(\omega\) is a weight function. They show that any such operator which is homogeneous \(\omega\)-hypoelliptic is also \(\sigma\)-hypoelliptic for weight function \(\sigma= O(\omega)\). They also present a sufficient condition to ensure that a differential operator admits a parametrix. As a consequence they obtain some conditions on the weights \((\omega, G)\) to conclude that, for any operator \(P(x,D)\) with constant strength, the \(\sigma\)-hypoellipticity of the frozen operator \(P(x_0,D)\) implies the \(\omega\)-hypoellipticity of \(P(x,D)\). They include an example of an \(\omega\)-hypoelliptic operator which has not constant strength.