Global solvability and hypoellipticity on the torus for a class of differential operators withvariable coefficients

Global hypoellipticity and global solvability in Gevrey classes on the n-dimensional torus

“…We remark that recently several authors have studied global hypoellipticity and solvability in this setting (see, for example, [8][9][10][11][14][15][16][25][26][27]33] and the references quoted therein). We recall that generalized Diophantine conditions have been given to characterize global hypoellipticitysolvability of classes of operators with variable coefficients (see [3,[8][9][10][14][15][16]). The simplest such condition was first used by Greenfield and Wallach [12] to show that the operator P ¼ @ t À a@ x ; aAR; is globally hypoelliptic in T 2 if and only if a is not a Liouville number.…”

Global Solvability on the Torus for Certain Classes of Operators in the Form of a Sum of Squares of Vector Fields

“…These operators include certain classes of systems of vector fields and certain classes of second order operators, but do not include operators in the form of a sum of squares of vector fields (see, e.g., [1,2,3,4,5,8]). In [9] the global solvability on the torus for a particular class of operators in the form of a sum of squares of vector fields with constant coefficients was studied.…”

Partial Fourier series on compact Lie groups

“…The study of global properties for vector fields and systems of vector fields defined on closed manifolds has made a great advance in the last decades, especially in the case where the manifold is a torus or a product of a torus by another closed manifold. See, for example, the impressive list of articles [2,3,7,[14][15][16][17][18][20][21][22] and references therein, discussing problems related to the global solvability and global hypoellipticity in the smooth, analytical and ultradifferentiable senses.…”