We characterize global Denjoy–Carleman hypoellipticity for the family of systems acting on the torus \({\mathbb {T}}_{t}^{N} \times {\mathbb {T}}_{x}\) given by \(L_{j} = \frac{\partial }{\partial t_{j}} + \left( a_{j}(t_{j}) + ib_{j}(t_{j}) \right) \frac{\partial }{\partial x} + \lambda _j\) , where \(a_j, b_j\) are real-valued \(2\pi\)-periodic elements of the classes and \(j = 1, 2,\ldots , N\).

中文翻译：

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一类复杂矢量场和摄动系统的全局Denjoy–Carleman次椭圆性

我们描述了全球的Denjoy-的Carleman亚椭圆性的家庭的作用在圆环系统\（{\ mathbb {T】} _【T} ^ {N} \次{\ mathbb【T}} _ {X} \）由下式给出\（L_ {j} = \ frac {\ partial} {\ partial t_ {j}} + \ left（a_ {j}（t_ {j}）+ ib_ {j}（t_ {j}）\ right）\ frac {\ partial} {\ partial x} + \ lambda _j \），其中\（a_j，b_j \）是类的实值\（2 \ pi \）周期元素，并且\（j = 1，2 ，\ ldots，N \）。