| 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:
|
http://hdl.handle.net/10338.dmlcz/134092 |
| . |
| Reference:
|
[1] P.-Y. Lee: Lanzhou lectures on Henstock integration.Singapore, World Scientific, 1989. Zbl 0699.26004, MR 1050957 |
| 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 |
| Reference:
|
[5] C. Swartz: Introduction to gauge integrals.Singapore, World Scientific, 2001. Zbl 0982.26006, MR 1845270 |
| 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 |
| . |
