Hipoelipticidade global para sublaplacianos, perturbações de ordem inferior, resolubilidade e hipoelipticidade global para uma classe de campos vetoriais

Abstract

We start this work by recalling a class of globally hypoelliptic sublaplacians defined on the N-dimensional torus introduced by Cordaro and Himonas in 1994 and studied by Himonas and Petronilho in 2000. We consider a new class of sublaplacians that generalizes this one and prove that it is globally {omega}-hypoelliptic if and only if the coefficients satisfy a diophantine condition involving a new concept of simultaneous approximability with exponent {omega}. We also recall the Petronilho's conjecture (2006) for the smooth hypoellipticity and present a new class of sublaplacians for which the Petronilho's conjecture holds true in the ultradifferentiable functions setup.

Motivated by the works of Petronilho and Zani (2008) and Chini and Cordaro (2017), we considered a vector field in the 2-dimensional torus with constant and real coefficients and we analyze the hypoellipticity and the solvability of this vector field when it is perturbed by a negative-order pseudodifferential operator. We find a non-negative order \sigma_0 such that for all operators with order less than to \sigma_0, the perturbed operator preserves both the global hypoellipticity and the global solvability of the initial vector field. This study was done in both the smooth case and the Gevrey case.

Finally, we generalize, for a particular class, the work of Petronilho and Zani for the N-dimensional torus. Our study was motivated by the study of the Gevrey solvability of perturbations by Gevrey functions of the vector field system given in the first level of the complex studied by Bergamasco and Petronilho (1999).