In this paper, we present necessary and sufficient conditions to have global analytic hypoellipticity for a class of first-order operators defined on ${\mathbb{T}}^1\times {\mathbb{S}}^3$. In the case of real-valued coefficients, we prove that an operator in this class is conjugated to a constant-coefficient operator satisfying a Diophantine condition, and that such conjugation preserves the global analytic hypoellipticity. In the case where the imaginary part of the coefficients is non-zero, we show that the operator is globally analytic hypoelliptic if the Nirenberg-Treves condition ($\mathcal{P}$) holds, in addition to an analytic Diophantine condition.

中文翻译：

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一类演化算子的​​全局解析亚椭圆度 ${\mathbb{吨}}^1\times {\mathbb{秒}}^3$

在本文中，我们提出了对定义在上的一类一阶算子具有全局解析亚椭圆度的充分必要条件 ${\mathbb{吨}}^1\times {\mathbb{秒}}^3$. 在实值系数的情况下，我们证明此类中的算子与满足丢番图条件的常数系数算子共轭，并且这种共轭保留了全局解析亚椭圆度。在系数的虚部非零的情况下，如果 Nirenberg-Treves 条件 ($\mathcal{磷}$) 成立，除了分析丢番图条件。