Continuity in the Alexiewicz norm

Abstract

If $f$ is a Henstock--Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\|f\|=\sup_I|\int_I f|$ where the supremum is taken over all intervals $I\subset\R$. Define the translation $τ_x$ by $τ_xf(y)=f(y-x)$. Then $\|τ_xf-f\|$ tends to 0 as $x$ tends to 0, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\|τ_xf-f\|$ can tend to 0 arbitrarily slowly. In general, $\|τ_xf-f\|\geq {\rm osc}f |x|$ as $x\to 0$, where ${\rm osc}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\|τ_xF-F\|\leq \|f\||x|$. An example shows that the function $y\mapsto τ_xF(y)-F(y)$ need not be in $L^1$. However, if $f\in L^1$ then $\|τ_xF-F\|_1\leq \|f\|_1|x|$. For a positive weight function $w$ on the real line, necessary and sufficient conditions on $w$ are given so that $\|(τ_xf-f)w\|\to 0$ as $x\to 0$ whenever $fw$ is Henstock--Kurzweil integrable. Applications are made to the Poisson integral on the disc and half-plane. All of the results also hold with the distributional Denjoy integral, which arises from the completion of the space of Henstock--Kurzweil integrable functions as a subspace of Schwartz distributions.