On convolvability conditions for distributions
The author investigates several conditions for the convolvability and \(\mathcal{S}'\)-convolvability of distributions and he shows their equivalence by characterizing the partial summability of distribution kernels by multiplicative properties. More generally, partial summability to the power \(p\) and the partial vanishing at infinity of kernels are characterized by multiplicative properties. For instance, for \(1\leq p\leq \infty\) and for \(S,T \in \mathcal{D}'(\mathbb{R}^n)\), the author gives characterizations of the following conditions: (CPS):\ \(S(x-y)T(y) \in \mathcal{D}'(\mathbb{R}^n_x) \hat{\otimes}\mathcal{D}'_{L^p}(\mathbb{R}^{n}_y)\) (\(\varphi\) - C): \(\forall \varphi \in \mathcal{D}(\mathbb{R}^n): \varphi(x)S(x-y)T(y) \in \mathcal{D}'_{L^p}(\mathbb{R}^{2n}_{xy})\) (\(\varphi,\psi\) - C): \( \forall \varphi,\psi \in \mathcal{D}(\mathbb{R}^n): (\varphi* \check{S})(\psi *T) \in L^{p}(\mathbb{R}^n)\) (CR): \ \(S(x-y)T(y-z) \in \mathcal{D}'(\mathbb{R}^{2n}_{xz}) \hat{\otimes} L^p(\mathbb{R}^{n}_y)\) In Section 6 of the present paper, the author shows that the analogous convolvability conditions for distributions of finite order, in particular for measures, are not equivalent.
scientific article; zbMATH DE number 5770245
scientific article