Global Hypoellipticity for a Class of Complex-Valued Evolution Equations on Compact Lie Groups

We present necessary and sufficient conditions to have global hypoellipticity for a class of complex-valued coefficient first-order evolution equations defined on T¹ × G T 1 × G , where G is a compact Lie group. First, we show that the global hypoellipticity of the constant coefficient operator related to this operator is a necessary condition, but not a sufficient condition. Under certain hypothesis, we show that the global hypoellipticity of this class of operator is completely characterized by Nirenberg–Treves’ condition (P) ( P ) .