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<p>We present necessary and sufficient conditions to have global hypoellipticity for a class of complex-valued coefficient first-order evolution equations defined on <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {T}^1 \times G\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> <mn>1</mn> </msup> <mo>×</mo> <mi>G</mi> </mrow> </math></EquationSource> </InlineEquation>, where <i>G</i> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((\mathcal {P})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="script">P</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

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

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  • 发表日期：

    2026年4月20日

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  • Wagner Augusto Almeida de Moraes

摘要

We present necessary and sufficient conditions to have global hypoellipticity for a class of complex-valued coefficient first-order evolution equations defined on \(\mathbb {T}^1 \times G\) 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 \((\mathcal {P})\) ( P ) .