Kernel identities and vectorial regularisation
The aim of this article is to present a new method to prove some known and new kernel identities. More precisely, let \(\dot{{\mathcal B}}\) denote the space of smooth functions with derivatives vanishing at infinity and let \(\dot{{\mathcal B}}'\) be the closure of compactly supported distributions in the class of bounded distributions. In Section, 2 the authors obtain the identities \[ \dot{{\mathcal B}}_{xy} = \dot{{\mathcal B}}_{x}\widehat{\otimes}_\varepsilon \dot{{\mathcal B}}_{y},\;\;\dot{{\mathcal B}}'_{xy} = \dot{{\mathcal B}}'_{x}\widehat{\otimes}_\varepsilon \dot{{\mathcal B}}'_{y}, \] (Propositions 3 and 4), while in Section 3, the authors prove \[ {\mathcal D}_{L_1,xy} = {\mathcal D}_{L_1,x}\widehat{\otimes}_\pi {\mathcal D}_{L_1,y}, \;\;{\mathcal D}'_{L_1,xy} = {\mathcal D}'_{L_1,x}\widehat{\otimes}_\pi {\mathcal D}'_{L_1,y}, \] (Propositions 9 and 10). All the identities are obtained after applying appropriate vectorial regularization properties obtained in Propositions 1 and 7. For instance, the identity \({\mathcal D}_{L_1,xy} = {\mathcal D}_{L_1,x}\widehat{\otimes}_\pi {\mathcal D}_{L_1,y}\) follows from \[ K(x,z)\in {\mathcal D}_{L_1,x}\widehat{\otimes}_\pi E_z \Leftrightarrow K(x-y,z)\in \left({\mathcal E}_y\widehat{\otimes}L_x^1\right)\widehat{\otimes}_\pi E_z. \]
scientific article; zbMATH DE number 6894843
scientific article