Parameter dependence of solutions of the Cauchy-Riemann equation on spaces of weighted smooth functions

Abstract

We study the inhomogeneous Cauchy-Riemann equation on spaces $\mathcal{E}\mathcal{V}(\Omega ,E)$ of weighted ${\mathcal{C}}^{\infty }$-smooth $E$-valued functions on an open set $\Omega \subset {\mathbb{R}}^2$ whose growth on strips along the real axis is determined by a family of continuous weights $\mathcal{V}$ where $E$ is a locally convex Hausdorff space over $\mathbb{C}$. We derive sufficient conditions on the weights $\mathcal{V}$ such that the kernel $\ker \overline{\partial }$ of the Cauchy-Riemann operator $\overline{\partial }$ in $\mathcal{E}\mathcal{V}(\Omega ):=\mathcal{E}\mathcal{V}(\Omega ,\mathbb{C})$ has the property $(\Omega )$ of Vogt. Then we use previous results and conditions on the surjectivity of the Cauchy-Riemann operator $\overline{\partial }:\mathcal{E}\mathcal{V}(\Omega )\to \mathcal{E}\mathcal{V}(\Omega )$ and the splitting theory of Vogt for Fréchet spaces and of Bonet and Domański for (PLS)-spaces to deduce the surjectivity of the Cauchy-Riemann operator on the space $\mathcal{E}\mathcal{V}(\Omega ,E)$ if $E:=F_b^{\prime }$ where $F$ is a Fréchet space satisfying the condition $(DN)$ or if $E$ is an ultrabornological (PLS)-space having the property $(PA)$. As a consequence, for every family of right-hand sides $(f_{\lambda }{)}_{\lambda \in U}$ in $\mathcal{E}\mathcal{V}(\Omega )$ which depends smoothly, holomorphically or distributionally on a parameter $\lambda$ there is a family $(u_{\lambda }{)}_{\lambda \in U}$ in $\mathcal{E}\mathcal{V}(\Omega )$ with the same kind of parameter dependence which solves the Cauchy-Riemann equation $\overline{\partial }{u}_{\lambda }=f_{\lambda }$ for all $\lambda \in U$.