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Title: Continuity in the Alexiewicz norm (English)
Author: Talvila, Erik
Language: English
Journal: Mathematica Bohemica
ISSN: 0862-7959 (print)
ISSN: 2464-7136 (online)
Volume: 131
Issue: 2
Year: 2006
Pages: 189-196
Summary lang: English
Category: math
Summary: 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. (English)
Keyword: Henstock-Kurzweil integral
Keyword: Alexiewicz norm
Keyword: distributional Denjoy integral
Keyword: Poisson integral
MSC: 26A39
MSC: 46B99
MSC: 46Bxx
MSC: 46E30
idZBL: Zbl 1112.26011
idMR: MR2242844
DOI: 10.21136/MB.2006.134092
Date available: 2009-09-24T22:25:32Z
Last updated: 2020-07-29
Stable URL:
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Reference: [2] P. Mohanty, E. Talvila: A product convergence theorem for Henstock-Kurzweil integrals.Real Anal. Exchange 29 (2003–2004), 199–204. MR 2061303
Reference: [3] H. Reiter, J. Stegeman: Classical harmonic analysis and locally compact groups.Oxford, Oxford University Press, 2000. MR 1802924
Reference: [4] D. W. Stroock: A concise introduction to the theory of integration.Boston, Birkhäuser, 1999. Zbl 0912.28001, MR 1658777
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Reference: [6] E. Talvila: The distributional Denjoy integral.Preprint. Zbl 1154.26011, MR 2402863
Reference: [7] E. Talvila: Estimates of Henstock-Kurzweil Poisson integrals.Canad. Math. Bull. 48 (2005), 133–146. Zbl 1073.26004, MR 2118770, 10.4153/CMB-2005-012-8


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