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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.

26A39 Denjoy and Perron integrals, other special integrals
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