The Metivier inequality and ultradifferentiable hypoellipticity

Abstract

In 1980, Métivier characterized the analytic (and Gevrey) hypoellipticity of $L^2$-solvable partial linear differential operators by a priori estimates. In this note, we extend this characterization to ultradifferentiable hypoellipticity with respect to Denjoy–Carleman classes given by suitable weight sequences. We also discuss the case when the solutions can be taken as hyperfunctions and present some applications.