On the division problem for a tempered distribution that depends holomorphically on a parameter (Q5963444)

For a halfspace \(\Pi_\gamma=\{\lambda\in\mathbb C:\mathrm{Re}(\lambda)>\gamma\}\), the author considers tempered distributions on \(\mathbb R^n\) depending holomorphically on the parameter \(\lambda\) and an operator of the form \[ P(\sigma,\lambda)u(\lambda)=\sum_{k=0}^m P_k(\sigma)\lambda^k u(\lambda) \] (that is, a test function \(\phi\in \mathcal S\) is mapped to \(\sum_k \lambda^k u(\lambda)(P_k( \phi)\)), where \(P_j\in \mathbb C[\sigma_1,\dots,\sigma_n]\) are polynomials with only finitely many common real zeros. Under the assumption that, for all \(\sigma\in\mathbb R^n\), the zeros of the polynomial \(P(\sigma,\cdot)\) do not belong to the closure of \(\Pi_\gamma\), it is shown that the equation \[ P(\sigma,\lambda) u(\lambda) =f(\lambda) \] has a solution for all holomorphic \(f\) with values in some (generalized) Sobolev space.