Global analytic and Gevrey hypoellipticity of sublaplacians under Diophantine conditions (Q2718972)

The author addresses the problem of analytic and Gevrey hypoellipticity for operators with semidefinite principal part. The result proved in this paper is that an operator of the form \( - \Delta_{t} - (\sum_{j=1}^{n} a_{j}(t) \partial_{x_{j}})^{2} \), defined on the torus \( T^{m+n}_{(t,x)} \), is Gevrey (analytic) hypoelliptic if, after a possible renaming of the variables \( x \), the coefficients of the \( x \)-derivatives satisfy a certain diophantine condition involving the Gevrey index.