Surjectivity criteria for convolution operators in \(A^{-\infty}\) (Q960986)

Let \(\Omega\) be a convex bounded domain in \(\mathbb C^n\) and \(d(z) = \inf_{\zeta\in \partial\Omega} |z-\zeta|\), \(z\in\Omega\). Let \({\mathcal O}(\Omega)\) be the space of holomorphic functions on \(\Omega\), and \(A^{-\infty}(\Omega)\) be its subspace of those holomorphic functions, which have polynomial growth near the boundary \(\partial\Omega\). The following criterion of surjectivity of convolution operators is proved. Theorem. Let \(\mu\) be an analytic functional on \(\mathbb C^n\), carried by a compact convex set \(K\), and \(\mu* A^{-\infty}(\Omega+K)\subseteq A^{-\infty}(\Omega)\) for any convex bounded domain \(\Omega\in\mathbb C^n\). The convolution operator \(\mu*: A^{-\infty}(\Omega+K)\to A^{-\infty}(\Omega)\) is surjective for every \(\Omega\) if and only if the regularized radial indicator \(h^*_\mu(\zeta)\) of the Fourier-Borel transformation \(\widehat\mu\) of \(\mu\) coincides with \(H_K\) and \(\widehat\mu\) as a function of exponential type satisfies the following condition: \[ \exists s,\;N>0\;\forall \zeta \in\mathbb C^n,\;|\zeta| > N,\;\exists \zeta'\in\mathbb C,\;|\zeta-\zeta'|< \log(1+|\zeta|): \log|\widehat\mu|\geq |\zeta|h^*_\mu \bigg(\frac\zeta{|\zeta|}\bigg)- s\log |\zeta|. \]