## Convolution operators for the one-sided Laplace transformation.(English)Zbl 0584.46032

Let $${\mathcal L}_{0\gamma}$$ be the space of all distributions $$f\in {\mathcal D}'({\mathbb{R}})$$ such that supp $$f\subset [\alpha,\infty)$$ for some $$\alpha\in {\mathbb{R}}$$ and $$e^{-\alpha x}f\in L^ 2({\mathbb{R}})$$. Then for $$p\in {\mathbb{N}}$$ denote by $${\mathcal L}_{p\gamma}$$ the space of derivatives of order p of the elements from $${\mathcal L}_{0\gamma}$$ endowed with an inductive limit topology. The image $$H_{p\gamma}$$ of $${\mathcal L}_{p\gamma}$$ under the Laplace transformation is studied and the convolution operators for $${\mathcal L}_{\gamma}=\cup_{p\in {\mathbb{N}}}{\mathcal L}_{p\gamma}$$ are identified with distributions with some property related to their Laplace transforms. This space of operators can be written as $$C^{\gamma}=\cap_{p\in {\mathbb{N}}}\cup_{q\leq p}C^{\gamma}_{pq}$$ where $$C^{\gamma}_{pq}$$ is the set of distributions f for which the convolution $$g\to f^*g$$ is continuous mapping from $${\mathcal L}_{p\gamma}$$ to $${\mathcal L}_{q\gamma}$$. The image $$M^{\gamma}_{pq}$$ of $$C^{\gamma}_{pq}$$ under the Laplace transformation is the space of multipliers of $$H_{p\gamma}$$ to $$H_{q\gamma}$$.

### MSC:

 46F12 Integral transforms in distribution spaces 44A30 Multiple integral transforms 44A10 Laplace transform 46F05 Topological linear spaces of test functions, distributions and ultradistributions
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