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 \(H_{q\gamma}\).


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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