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 $\tau_x$ by $\tau_xf(y)=f(y-x)$. Then $\|\tau_xf-f\|$ tends to $0$ as $x$ tends to $0$, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\|\tau_xf-f\|$ can tend to 0 arbitrarily slowly. In general, $\|\tau_xf-f\|\geq\osc f|x|$ as $x\to0$, where $ \osc f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\|\tau_xF-F\|\leq\|f\||x|$. An example shows that the function $y\mapsto\tau_xF(y)-F(y)$ need not be in $L^1$. However, if $f\in L^1$ then $\|\tau_xF-F\|_1\leq\|f\|_1|x|$. For a positive weight function $w$ on the real line, necessary and sufficient conditions on $w$ are given so that $\|(\tau_xf-f)w\|\to0$ as $x\to0$ 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.

Keywords: Henstock-Kurzweil integral, Alexiewicz norm, distributional Denjoy integral, Poisson integral

Classification (MSC2000): 26A39, 46Bxx

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