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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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