Convolutions with the continuous primitive integral

Abstract

If $F$ is a continuous function on the real line and $f=F^{\prime }$ is its distributional derivative then the continuous primitive integral of distribution $f$ is ${\int }_a^{b}f=F(b)-F(a)$. 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 $\Vert f\Vert ={\sup }_I|{\int }_{I}f|$ is the Alexiewicz norm. This supremum is taken over all intervals $I\subset \mathbb{R}$. When $g\in L^1$ the estimate is $\Vert f\ast g\Vert \le \Vert f\Vert \Vert g{\Vert }_1$. There are results on differentiation and integration of convolutions. A type of Fubini theorem is proved for the continuous primitive integral.