The Metivier inequality and ultradifferentiable hypoellipticity
Paulo D. Cordaro, Stefan Fürdös
TL;DR
The paper extends Métivier's inequality-based characterization of analytic and Gevrey hypoellipticity to ultradifferentiable Denjoy-Carleman classes defined by admissible weight sequences. It introduces a weighted Sobolev framework with the norm $|||u|||_{U,\mathbf{M},k}$ and proves that a differential operator $P$ with coefficients in $\mathcal{E}^{\{\mathbf{M}\}}(U)$ is $\{\mathbf{M}\}$-hypoelliptic at a point $x_0$ iff a local a priori estimate holds: $\|u\|_{H^k(V)}\le C L^k (|||Pu|||_{U,\mathbf{M},k}+M_k\|u\|_{L^2(U)})$ for all $k$ and suitable $V\Subset U\subset U_0$. The results hold under a Right-Inverse condition $PR=\mathrm{Id}$ and extend to hyperfunctions, with implications for Hörmander’s sum-of-squares operators and for monotonicity with respect to weight sequences. Collectively, the work unifies analytic, Gevrey, and Denjoy-Carleman hypoellipticity theory, providing new tools for regularity analysis in ultradifferentiable classes and broadening the scope of Métivier-type characterizations.
Abstract
In 1980 M{é}tivier 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.