# Convolutions with the continuous primitive integral

## TL;DR

This paper develops a convolution theory for distributions integrable via the continuous primitive integral, establishing properties, estimates, and a Fubini theorem within this framework.

## Contribution

It introduces convolution operations for distributions with the continuous primitive integral, extending classical properties and proving a Fubini theorem in this context.

## Key findings

- Convolution is well-defined for integrable distributions and functions of bounded variation or in L^1.
- Key properties like commutativity, associativity, and translation invariance are established.
- Provides estimates for the convolution norm and results on differentiation and integration of convolutions.

## Abstract

If $F$ is a continuous function on the real line and $f=F'$ is its distributional derivative then the continuous primitive integral of distribution $f$ is $\int_a^bf=F(b)-F(a)$. This integral contains the Lebesgue, Henstock--Kurzweil and wide Denjoy integrals. Under the Alexiewicz norm the space of integrable distributions is a Banach space. We define the convolution $f\ast g(x)=\intinf f(x-y)g(y) dy$ for $f$ an integrable distribution and $g$ a function of bounded variation or an $L^1$ function. Usual properties of convolutions are shown to hold: commutativity, associativity, commutation with translation. For $g$ of bounded variation, $f\ast g$ is uniformly continuous and we have the estimate $\|f\ast g\|_\infty\leq \|f\|\|g\|_\bv$ where $\|f\|=\sup_I|\int_If|$ is the Alexiewicz norm. This supremum is taken over all intervals $I\subset\R$. When $g\in L^1$ the estimate is $\|f\ast g\|\leq \|f\|\|g\|_1$. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.