This work deals with global solvability and global hypoellipticity of complex vector fields of the form L. The solvability and hypoellipticity depend on condition (P) and also on Diophantine properties of the coefficients.
We present conditions on the coefficients of a class of vector fields on the torus which yield a characterization of global solvability as well as global hypoellipticity, in other words, the existence and regularity of periodic solutions. Diophantine conditions and connectedness of certain sublevel sets appear in a natural way in our results.
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