Convolutions with the continuous primitive integral

If $F$ is a continuous function on the real line and $f=F^{\mathrm{\prime }}$ 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 $f\ast g(x)=\text{intinf}f(x-y)g(y)dy$ 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 $\mathrm{\parallel }f\ast g{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }f\mathrm{\parallel }\mathrm{\parallel }g{\mathrm{\parallel }}_{\text{bv}}$ where $\mathrm{\parallel }f\mathrm{\parallel }={\sup }_I\mathrm{\mid }{\int }_{I}f\mathrm{\mid }$ is the Alexiewicz norm. This supremum is taken over all intervals $I\subset \mathbb{R}$. 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.