| Title:
|
A multidimensional integration by parts formula for the Henstock-Kurzweil integral (English) |
| Author:
|
Lee, Tuo-Yeong |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
133 |
| Issue:
|
1 |
| Year:
|
2008 |
| Pages:
|
63-74 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
It is shown that if $g$ is of bounded variation in the sense of Hardy-Krause on ${\mathop {\prod }\limits _{i=1}^{m}} [a_i, b_i]$, then $g \chi _{ _{{\mathop {\prod }\limits _{i=1}^{m}} (a_i, b_i)}}$ is of bounded variation there. As a result, we obtain a simple proof of Kurzweil’s multidimensional integration by parts formula. (English) |
| Keyword:
|
Henstock-Kurzweil integral |
| Keyword:
|
bounded variation in the sense of Hardy-Krause |
| Keyword:
|
integration by parts |
| MSC:
|
26A39 |
| idZBL:
|
Zbl 1199.26029 |
| idMR:
|
MR2400151 |
| DOI:
|
10.21136/MB.2008.133945 |
| . |
| Date available:
|
2009-09-24T22:34:29Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133945 |
| . |
| Reference:
|
[1] J. Kurzweil: On multiplication of Perron integrable functions.Czech. Math. J 23 (1973), 542–566. Zbl 0269.26007, MR 0335705 |
| Reference:
|
[2] Tuo-Yeong Lee, Tuan Seng Chew, Peng Yee Lee: Characterisation of multipliers for the double Henstock integrals.Bull. Austral. Math. Soc. 54 (1996), 441–449. MR 1419607, 10.1017/S0004972700021857 |
| Reference:
|
[3] Tuo-Yeong Lee: Multipliers for some non-absolute integrals in the Euclidean spaces.Real Anal. Exchange 24 (1998/99), 149–160. MR 1691742 |
| Reference:
|
[4] Tuo-Yeong Lee: A full descriptive definition of the Henstock-Kurzweil integral in the Euclidean space.Proc. London Math. Soc. 87 (2003), 677–700. MR 2005879 |
| Reference:
|
[5] Tuo-Yeong Lee: Every absolutely Henstock-Kurzweil integrable function is McShane integrable: an alternative proof.Rocky Mountain J. Math. 34 (2004), 1353–1365. MR 2095582, 10.1216/rmjm/1181069805 |
| Reference:
|
[6] Tuo-Yeong Lee: A full characterization of multipliers for the strong $\rho $-integral in the Euclidean space.Czech. Math. J. 54 (2004), 657–674. MR 2086723, 10.1007/s10587-004-6415-7 |
| Reference:
|
[7] Tuo-Yeong Lee: A characterisation of multipliers for the Henstock-Kurzweil integral.Math. Proc. Cambridge Philos. Soc. 138 (2005), 487–492. MR 2138575, 10.1017/S030500410500839X |
| Reference:
|
[8] Tuo-Yeong Lee: Some full descriptive characterizations of the Henstock-Kurzweil integral in the Euclidean space.Czech. Math. J. 55 (2005), 625–637. MR 2153087, 10.1007/s10587-005-0050-9 |
| Reference:
|
[9] Tuo-Yeong Lee: The Henstock variational measure, Baire functions and a problem of Henstock.Rocky Mountain J. Math. 35 (2005), 1981–1997. MR 2210644 |
| Reference:
|
[10] Tuo-Yeong Lee: On the dual space of ${\text{BV}}$-integrable functions in Euclidean space.Real Anal. Exchange 30 (2004/2005), 323–328. MR 2127537, 10.14321/realanalexch.30.1.0323 |
| Reference:
|
[11] Tuo-Yeong Lee: Product variational measures and Fubini-Tonelli type theorems for the Henstock-Kurzweil integral II.J. Math. Anal. Appl. 323 (2006), 741–745. MR 2262241, 10.1016/j.jmaa.2005.10.045 |
| Reference:
|
[12] Tuo-Yeong Lee: Multipliers for generalized Riemann integrals in the real line.Math. Bohem. 131 (2006), 161–166. MR 2242842 |
| Reference:
|
[13] Tuo-Yeong Lee: A Fubini’s theorem for generalized Riemann integrals.Preprint. |
| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
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| Reference:
|
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[19] W. H. Young: On multiple integration by parts and the second theorem of the mean.Proc. London Math. Soc. 16 (1918), 273–293. |
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