Geodesic
A full characterization of multipliers for the strong $\rho$-integral in the euclidean space
Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 657-674
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho $-integral, introduced by Jarník and Kurzweil. Let $(\mathcal S_{\rho } (E), \Vert \cdot \Vert )$ be the space of all strongly $\rho $-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\Vert \cdot \Vert $. We show that each element in the dual space of $(\mathcal S_{\rho } (E), \Vert \cdot \Vert )$ can be represented as a strong $\rho $-integral. Consequently, we prove that $fg$ is strongly $\rho $-integrable on $E$ for each strongly $\rho $-integrable function $f$ if and only if $g$ is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on $E$.
We study a generalization of the classical Henstock-Kurzweil integral, known as the strong $\rho $-integral, introduced by Jarník and Kurzweil. Let $(\mathcal S_{\rho } (E), \Vert \cdot \Vert )$ be the space of all strongly $\rho $-integrable functions on a multidimensional compact interval $E$, equipped with the Alexiewicz norm $\Vert \cdot \Vert $. We show that each element in the dual space of $(\mathcal S_{\rho } (E), \Vert \cdot \Vert )$ can be represented as a strong $\rho $-integral. Consequently, we prove that $fg$ is strongly $\rho $-integrable on $E$ for each strongly $\rho $-integrable function $f$ if and only if $g$ is almost everywhere equal to a function of bounded variation (in the sense of Hardy-Krause) on $E$.
Classification : 26A39, 46E99, 46G10
Keywords: strong $\rho $-integral; multipliers; dual space 
@article{CMJ_2004_54_3_a8,
     author = {Tuo-Yeong, Lee},
     title = {A full characterization of multipliers for the strong $\rho$-integral in the euclidean space},
     journal = {Czechoslovak Mathematical Journal},
     pages = {657--674},
     year = {2004},
     volume = {54},
     number = {3},
     mrnumber = {2086723},
     zbl = {1080.26007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a8/}
}
TY  - JOUR
AU  - Tuo-Yeong, Lee
TI  - A full characterization of multipliers for the strong $\rho$-integral in the euclidean space
JO  - Czechoslovak Mathematical Journal
PY  - 2004
SP  - 657
EP  - 674
VL  - 54
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a8/
LA  - en
ID  - CMJ_2004_54_3_a8
ER  - 
%0 Journal Article
%A Tuo-Yeong, Lee
%T A full characterization of multipliers for the strong $\rho$-integral in the euclidean space
%J Czechoslovak Mathematical Journal
%D 2004
%P 657-674
%V 54
%N 3
%U http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a8/
%G en
%F CMJ_2004_54_3_a8
Tuo-Yeong, Lee. A full characterization of multipliers for the strong $\rho$-integral in the euclidean space. Czechoslovak Mathematical Journal, Tome 54 (2004) no. 3, pp. 657-674. http://geodesic.mathdoc.fr/item/CMJ_2004_54_3_a8/

[1] A.  Alexiewicz: Linear functionals on Denjoy-integrable functions. Colloq. Math. 1 (1948), 289–293. | DOI | MR | Zbl

[2] R. A.  Gordon: The Integrals of Lebesgue. Denjoy, Perron, and Henstock, Graduate Studies in Mathematics Volume 4, AMS, 1994. | MR | Zbl

[3] J.  Jarník and Kurzweil: Perron-type integration on  $n$-dimensional intervals and its properties. Czechoslovak Math. J. 45 (120) (1995), 79–106. | MR

[4] J.  Kurzweil: On multiplication of Perron integrable functions. Czechoslovak Math. J. 23 (98) (1973), 542–566. | MR | Zbl

[5] J.  Kurzweil and J.  Jarník: Perron-type integration on  $n$-dimensional intervals as an extension of integration of stepfunctions by strong equiconvergence. Czechoslovak Math. J. 46 (121) (1996), 1–20. | MR

[6] Lee Peng Yee: Lanzhou Lectures on Henstock integration. World Scientific, 1989. | MR | Zbl

[7] Lee Peng Yee and Rudolf Výborný: The integral: An Easy Approach after Kurzweil and Henstock. Australian Mathematical Society Lecture Series 14, Cambridge University Press, 2000. | MR

[8] Lee Tuo Yeong, Chew Tuan Seng and Lee Peng Yee: Characterisation of multipliers for the double Henstock integrals. Bull. Austral. Math. Soc. 54 (1996), 441–449. | DOI | MR

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

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

[11] E. J.  McShane: Integration. Princeton Univ. Press, 1944. | MR | Zbl

[12] Piotr Mikusiński and K.  Ostaszewski: The space of Henstock integrable functions II. In: New integrals. Proc. Henstock Conf., Coleraine / Ireland, P. S. Bullen, P. Y. Lee, J. L. Mawhin, P. Muldowney and W. F. Pfeffer (eds.), 1988.

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

[14] S.  Saks: Theory of the Integral, second edition. New York, 1964 63.0183.05. | MR

[15] W. L. C.  Sargent: On the integrability of a product. J. London Math. Soc. 23 (1948), 28–34. | DOI | MR | Zbl

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