On convolvability conditions for distributions
type: Item
instance of: scholarly article
title: On convolvability conditions for distributions
zbMATH Open document ID: 1213.46034
published in: Monatshefte für Mathematik
publication date: 2010-08-13T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46A11
zbMATH DE Number: 5770245
zbMATH Keywords: convolution
zbMATH Keywords: distributions
zbMATH Keywords: tensor products
author: Norbert Ortner
reviewed by: María Carmen Gómez-Collado
MaRDI profile type: Publication
full work available at URL: s00605-008-0087-6
OpenAlex ID: W2061748688
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DOI: 10.1007/S00605-008-0087-6
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description: scientific article; zbMATH DE number 5770245