Global hypoellipticity and simultaneous approximability in ultradifferentiable classes

We start this work by recalling a class of globally hypoelliptic sublaplacians defined on the N-dimensional torus introduced by Himonas and Petronilho and we consider a new class of sublaplacians that generalizes this one and prove that it is globally {ω}-hypoelliptic if and only if the coefficients satisfy a diophantine condition involving a new concept of simultaneous approximability with exponent {ω}. Furthermore we prove that this new class is globally {ω}-hypoelliptic if and only if certain perturbations of its vector fields, by adding more derivatives with respect to other variables, are globally {ω}-hypoelliptic. We also recall the Petronilho's conjecture for the smooth hypoellipticity and present a new class of sublaplacians for which the Petronilho's conjecture holds true in the ultradifferentiable functions setup.