A note on the convolution and the product \({\mathcal D}'\) and \({\mathcal S}'\) (Q1175435)

In the paper the following questions are investigated: I. Let \(f,g\in{\mathcal S}'\) and assume that \(f*g\) \((f\cdot g\), resp.) exists in \({\mathcal D}'\). Does \(f*g\) (\(f\cdot g\), resp.) belong to \({\mathcal S}'\)? II. Let \(f,g\in{\mathcal S}'\) and suppose that \(f*g\) \((f\cdot g\), resp.) belongs to \({\mathcal S}'\). Does \(f*g\) \((f\cdot g\), resp.) exist in \({\mathcal S}'\)? The authors give counter-examples for question I with product and question II with product and convolution. Question I in the case of convolution was answered by Kaminski and by Dierolf and Voigt. A counter- example for question II with convolution in \(R^ 2\) was given by Wagner but it can not be transferred to the case \(R^ 1\).