Let $$\varOmega $$ Ω be an open subset of $$\mathbb {R}^{2}$$ R 2 and E a complete complex locally convex Hausdorff space. The purpose of this paper is to find conditions on certain weighted Fréchet spaces $$\mathcal {EV}(\varOmega )$$ EV ( Ω ) of smooth functions and on the space E to ensure that the vector-valued Cauchy–Riemann operator $${\overline{\partial }}:\mathcal {EV}(\varOmega ,E)\rightarrow \mathcal {EV}(\varOmega ,E)$$ ∂ ¯ : EV ( Ω , E ) → EV ( Ω , E ) is surjective. This is done via splitting theory and positive results can be interpreted as parameter dependence of solutions of the Cauchy–Riemann operator.

中文翻译：

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柯西-黎曼方程解在光滑函数加权空间上的参数相关性

令 $$\varOmega $$Ω 是 $$\mathbb {R}^{2}$$ R 2 的开子集，E 是一个完全复局部凸 Hausdorff 空间。本文的目的是在光滑函数的某些加权 Fréchet 空间 $$\mathcal {EV}(\varOmega )$$ EV ( Ω ) 和空间 E 上找到条件，以确保向量值 Cauchy-Riemann 算子$${\overline{\partial }}:\mathcal {EV}(\varOmega ,E)\rightarrow \mathcal {EV}(\varOmega ,E)$$∂¯ : EV ( Ω , E ) → EV ( Ω , E ) 是满射。这是通过分裂理论完成的，正面的结果可以解释为柯西-黎曼算子解的参数依赖性。