Multianisotropic Gevrey Regularity and Iterates of Operators with Constant Coefficients

Abstract

We consider linear partial differential operators with constant coefficients $P$ and show that the inclusion of the Gevrey classes $G^d_P$ defined by the iterates of $P$ in some multianisotropic Gevrey classes implies a growth condition on the symbol of $P$. Under the hypothesis of hypoellipticity, the converse implication is also true. These results are also related to the regular weight of hypoellipticity, that gives a precise description of the growth of the symbol of $P$ with respect to its derivatives.