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 $\mathcal{EV}(\Omega,E)$ 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 $\operatorname{ker}\overline{\partial}$ of the Cauchy-Riemann operator $\overline{\partial}$ in $\mathcal{EV}(\Omega):=\mathcal{EV}(\Omega,\mathbb{C})$ has the property $(\Omega)$ of Vogt. Then we use previous results and conditions on the surjectivity of the Cauchy-Riemann operator $\overline{\partial}\colon\mathcal{EV}(\Omega)\to\mathcal{EV}(\Omega)$ and the splitting theory of Vogt for Fr\'{e}chet spaces and of Bonet and Doma\'nski for (PLS)-spaces to deduce the surjectivity of the Cauchy-Riemann operator on the space $\mathcal{EV}(\Omega,E)$ if $E:=F_{b}'$ where $F$ is a Fr\'{e}chet 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 $\mathcal{EV}(\Omega)$ which depends smoothly, holomorphically or distributionally on a parameter $\lambda$ there is a family $(u_{\lambda})_{\lambda\in U}$ in $\mathcal{EV}(\Omega)$ with the same kind of parameter dependence which solves the Cauchy-Riemann equation $\overline{\partial}u_{\lambda}=f_{\lambda}$ for all $\lambda\in U$.