Multiplications and Convolutions in L. Schwartz' Spaces of Test Functions and Distributions and their Continuity

TL;DR

This paper investigates the properties of multiplication and convolution operations within Schwartz's spaces of test functions and distributions, focusing on their structure and continuity aspects.

Contribution

It explicitly characterizes the multiplier and convolutor spaces in Schwartz's distribution theory and clarifies the continuity of these operations.

Findings

Identifies the multiplier and convolutor spaces for Schwartz's spaces

Determines the continuity properties of multiplication and convolution

Provides a detailed structural analysis of these operations in distribution spaces

Abstract

We list multiplier and convolutor spaces of the spaces occurring in L. Schwartz' "Th\'eorie des distributions". Furthermore we clarify whether the multiplications and convolutions are continuous or not.