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Continuity in the Alexiewicz norm. (English) Zbl 1112.26011
Summary: 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 $$I\subset \mathbb R$$. 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, $$\| \tau _xf-f\| \geq \text{osc}\, f| x|$$ as $$x\to 0$$, where $$\text{osc}\, f$$ 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 $$\| (\tau _xf-f)w\| \to 0$$ 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.

MSC:
 26A39 Denjoy and Perron integrals, other special integrals
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