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Characterization of convolution operators on spaces of \(C^{\infty}\)- functions admitting a continuous linear right inverse. (English) Zbl 0607.42011
Let \({\mathcal E}_{\omega}({\mathbb{R}})\) denote the Fréchet space of all ultradifferentiable functions on \({\mathbb{R}}\) in the sense of Beurling- Björck. For \(\mu\) \(\in {\mathcal E}_{\omega}({\mathbb{R}})'\) the convolution operator \(T_{\mu}: {\mathcal E}_{\omega}({\mathbb{R}})\to {\mathcal E}_{\omega}({\mathbb{R}})\) is defined by \(T_{\mu}(\phi):=\mu *\phi\). The surjective convolution operators \(T_{\mu}\) on \({\mathcal E}_{\omega}({\mathbb{R}})\) admitting a continuous linear right inverse are characterized by several equivalent conditions like: existence of fundamental solutions of \(T_{\mu}\) with support in positive and negative half lines, unique extendability of ”local” zero-solutions of \(T_{\mu}\), distribution of the zeros of the Fourier-Laplace transform \({\hat \mu}\) of \(\mu\) and global convergence of the local Fourier expansion of ”local” zero-solutions of \(T_{\mu}\). For \({\mathcal E}_{\omega}({\mathbb{R}})=C^{\infty}({\mathbb{R}})\) this implies that \(T_{\mu}\) admits a continuous linear right inverse iff \(T_{\mu}\) is hyperbolic in the sense of Ehrenpreis. These characterizations also hold for surjective convolution operators on the spaces \({\mathcal E}^{(M_ p)}({\mathbb{R}})\). Moreover, analogous characterizations hold for surjective convolution operators on the spaces \({\mathcal E}_{\{\omega \}}({\mathbb{R}})\) and \({\mathcal E}^{\{M_ p\}}({\mathbb{R}})\) of ultradifferentiable functions of Roumieu type.

42A85 Convolution, factorization for one variable harmonic analysis
46F05 Topological linear spaces of test functions, distributions and ultradistributions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
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