Global $${\mathcal {M}}-$$Hypoellipticity, Global $${\mathcal {M}}-$$Solvability and Perturbations by Lower Order Ultradifferential Pseudodifferential Operators

Abstract

We introduce a new class of ultradifferentiable pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity, in ultradifferentiable classes, with a finite loss of derivatives of a system of pseudodifferential operators, is stable under perturbations by lower order pseudodifferential operators whose order depends on the loss of derivatives. The key point in our study is our definition of loss of derivatives. We also give an easy proof of the fact that if a system of pseudodifferential operators is globally $${\mathcal {M}}$$ -hypoelliptic then its transpose is globally solvable in $$D'_{\mathcal {M}}\left( {\mathbb {T}}^N\right) $$ . Finally we present an application of our results.