We derive sufficient conditions for the surjectivity of the Cauchy–Riemann operator $\overline{\mathrm{\partial }}$ between weighted spaces of smooth Fréchet-valued functions. This is done by establishing an analog of Hörmander's theorem on the solvability of the inhomogeneous Cauchy–Riemann equation in a space of smooth $\mathbb{C}$-valued functions whose topology is given by a whole family of weights. Our proof relies on a weakened variant of weak reducibility of the corresponding subspace of holomorphic functions in combination with the Mittag–Leffler procedure. Using tensor products, we deduce the corresponding result on the solvability of the inhomogeneous Cauchy–Riemann equation for Fréchet-valued functions.

我们推导出了柯西-黎曼算子的满射性的充分条件$\overline{\mathrm{\partial }}$在平滑 Fréchet 值函数的加权空间之间。这是通过建立关于非齐次 Cauchy-Riemann 方程在光滑空间中可解性的 Hörmander 定理的模拟来完成的$\mathbb{C}$值函数，其拓扑由整个权重族给出。我们的证明依赖于全纯函数的相应子空间的弱可约性的弱化变体与 Mittag-Leffler 过程相结合。使用张量积，我们推导出 Fréchet 值函数的非齐次 Cauchy-Riemann 方程的可解性的相应结果。