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arXiv · 1810.05069

Surjectivity of the $\overline{\partial}$-operator between weighted spaces of smooth vector-valued functions

Karsten Kruse
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Abstract

We derive sufficient conditions for the surjectivity of the Cauchy-Riemann operator $\overline{\partial}$ between spaces of weighted smooth Fr\'echet-valued functions. This is done by establishing an analog of H\"ormander's theorem on the solvability of the inhomogeneous Cauchy-Riemann equation in a space of smooth $\mathbb{C}$-valued functions whose topologyis 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 Fr\'echet-valued functions.

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Karsten Kruse. 2018-10-11. Surjectivity of the $\overline{\partial}$-operator between weighted spaces of smooth vector-valued functions. https://doi.org/10.1080/17476933.2021.1945587

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