Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Parameter dependence of solutions of the Cauchy-Riemann equation on
  spaces of weighted smooth functions   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 $\lambda\in U$.

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The categories of this paper are: alg-geom
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The categories of this paper are: comp-gas
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The categories of this paper are: plasm-ph
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The categories of this paper are: physics
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The categories of this paper are: atom-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: nlin