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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.

MSC:
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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