Geodesic
A multidimensional integration by parts formula for the Henstock-Kurzweil integral
Mathematica Bohemica, Tome 133 (2008) no. 1, pp. 63-74

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
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.
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.
DOI : 10.21136/MB.2008.133945
Classification : 26A39
Keywords: Henstock-Kurzweil integral; bounded variation in the sense of Hardy-Krause; integration by parts 
Lee, Tuo-Yeong. A multidimensional integration by parts formula for the Henstock-Kurzweil integral. Mathematica Bohemica, Tome 133 (2008) no. 1, pp. 63-74. doi: 10.21136/MB.2008.133945
@article{10_21136_MB_2008_133945,
     author = {Lee, Tuo-Yeong},
     title = {A multidimensional integration by parts formula for the {Henstock-Kurzweil} integral},
     journal = {Mathematica Bohemica},
     pages = {63--74},
     year = {2008},
     volume = {133},
     number = {1},
     doi = {10.21136/MB.2008.133945},
     mrnumber = {2400151},
     zbl = {1199.26029},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133945/}
}
TY  - JOUR
AU  - Lee, Tuo-Yeong
TI  - A multidimensional integration by parts formula for the Henstock-Kurzweil integral
JO  - Mathematica Bohemica
PY  - 2008
SP  - 63
EP  - 74
VL  - 133
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133945/
DO  - 10.21136/MB.2008.133945
LA  - en
ID  - 10_21136_MB_2008_133945
ER  - 
%0 Journal Article
%A Lee, Tuo-Yeong
%T A multidimensional integration by parts formula for the Henstock-Kurzweil integral
%J Mathematica Bohemica
%D 2008
%P 63-74
%V 133
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2008.133945/
%R 10.21136/MB.2008.133945
%G en
%F 10_21136_MB_2008_133945

[1] J. Kurzweil: On multiplication of Perron integrable functions. Czech. Math. J 23 (1973), 542–566. | MR | Zbl

[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. | DOI | MR

[3] Tuo-Yeong Lee: Multipliers for some non-absolute integrals in the Euclidean spaces. Real Anal. Exchange 24 (1998/99), 149–160. | MR

[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

[5] Tuo-Yeong Lee: Every absolutely Henstock-Kurzweil integrable function is McShane integrable: an alternative proof. Rocky Mountain J. Math. 34 (2004), 1353–1365. | DOI | MR

[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. | DOI | MR

[7] Tuo-Yeong Lee: A characterisation of multipliers for the Henstock-Kurzweil integral. Math. Proc. Cambridge Philos. Soc. 138 (2005), 487–492. | DOI | MR

[8] Tuo-Yeong Lee: Some full descriptive characterizations of the Henstock-Kurzweil integral in the Euclidean space. Czech. Math. J. 55 (2005), 625–637. | DOI | MR

[9] Tuo-Yeong Lee: The Henstock variational measure, Baire functions and a problem of Henstock. Rocky Mountain J. Math. 35 (2005), 1981–1997. | MR

[10] Tuo-Yeong Lee: On the dual space of ${\text{BV}}$-integrable functions in Euclidean space. Real Anal. Exchange 30 (2004/2005), 323–328. | DOI | MR

[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. | DOI | MR

[12] Tuo-Yeong Lee: Multipliers for generalized Riemann integrals in the real line. Math. Bohem. 131 (2006), 161–166. | MR

[13] Tuo-Yeong Lee: A Fubini’s theorem for generalized Riemann integrals. Preprint.

[14] G. Q. Liu: The dual of the Henstock-Kurzweil space. Real Anal. Exchange 22 (1996/97), 105–121. | MR

[15] S. Lojasiewicz: An Introduction to the Theory of Real Functions. John Wiley & Sons, Ltd., Chichester, 1988. | MR | Zbl

[16] M. S. Macphail: Functions of bounded variation in two variables. Duke Math. J. 8 (1941), 215–222. | DOI | MR | Zbl

[17] P. Mikusiński, K. Ostaszewski: The space of Henstock integrable functions II. New integrals, (P. S. Bullen, P. Y. Lee, J. L. Mawhin, P. Muldowney and W. F. Pfeffer, eds.), Lecture Notes in Math. 1419 (Springer, Berlin, Heideberg, New York, 1990), 136–149. | MR

[18] K. M. Ostaszewski: The space of Henstock integrable functions of two variables. Internat. J. Math. and Math. Sci. 11 (1988), 15–22. | DOI | MR | Zbl

[19] W. H. Young: On multiple integration by parts and the second theorem of the mean. Proc. London Math. Soc. 16 (1918), 273–293.

[20] W. H. Young, G. C. Young: On the discontinuities of monotone functions of several variables. Proc. London Math. Soc. 22 (1924), 124–142. | MR

[21] S. K. Zaremba: Some applications of multidimensional integration by parts. Ann. Pol. Math. 21 (1968), 85–96. | DOI | MR | Zbl

Cité par Sources :