Analytic and Gevrey hypoellipticity for a class of pseudodifferential operators in one variable (Q2434789)

Consider equations of the form \[ P(x,D_{x})u(x)=f(x), \] where \[ P(x,D_{x})\sim \sum_{j\geq 0}a_{j}(x,D_{x}) \] denotes a pseudodifferential operator with analytic symbol. The hypoellipticity of such operators in one variable when a positivity assumption is made on a suitable principal part is studied. It is proved that the found Gevrey regularity is optimal.