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

Abstract We present necessary and sufficient conditions to have global hypoellipticity for a class of complex-valued coefficient first-order evolution equations defined on $$\mathbb {T}^1 \times G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> , 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 $$(\mathcal {P})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> .