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Henstock-Kurzweil integral; Alexiewicz norm; distributional Denjoy integral; Poisson integral
If $f$ is a Henstock-Kurzweil integrable function on the real line, the Alexiewicz norm of $f$ is $\Vert f\Vert =\sup _I|\int _I f|$ 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 $\Vert \tau _xf-f\Vert $ tends to $0$ as $x$ tends to $0$, i.e., $f$ is continuous in the Alexiewicz norm. For particular functions, $\Vert \tau _xf-f\Vert $ can tend to 0 arbitrarily slowly. In general, $\Vert \tau _xf-f\Vert \ge \mathop {\text{osc}}f|x|$ as $x\rightarrow 0$, where $ \mathop {\text{osc}}f$ is the oscillation of $f$. It is shown that if $F$ is a primitive of $f$ then $\Vert \tau _xF-F\Vert \le \Vert f\Vert |x|$. 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 $\Vert \tau _xF-F\Vert _1\le \Vert f\Vert _1|x|$. For a positive weight function $w$ on the real line, necessary and sufficient conditions on $w$ are given so that $\Vert (\tau _xf-f)w\Vert \rightarrow 0$ as $x\rightarrow 0$ 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.
[1] P.-Y. Lee: Lanzhou lectures on Henstock integration. Singapore, World Scientific, 1989. MR 1050957 | Zbl 0699.26004
[2] P. Mohanty, E. Talvila: A product convergence theorem for Henstock-Kurzweil integrals. Real Anal. Exchange 29 (2003–2004), 199–204. MR 2061303
[3] H. Reiter, J. Stegeman: Classical harmonic analysis and locally compact groups. Oxford, Oxford University Press, 2000. MR 1802924
[4] D. W. Stroock: A concise introduction to the theory of integration. Boston, Birkhäuser, 1999. MR 1658777 | Zbl 0912.28001
[5] C. Swartz: Introduction to gauge integrals. Singapore, World Scientific, 2001. MR 1845270 | Zbl 0982.26006
[6] E. Talvila: The distributional Denjoy integral. Preprint. MR 2402863 | Zbl 1154.26011
[7] E. Talvila: Estimates of Henstock-Kurzweil Poisson integrals. Canad. Math. Bull. 48 (2005), 133–146. DOI 10.4153/CMB-2005-012-8 | MR 2118770 | Zbl 1073.26004
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