Previous |  Up |  Next


strong $\rho $-integral; multipliers; dual space
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 $E$.
[1] A.  Alexiewicz: Linear functionals on Denjoy-integrable functions. Colloq. Math. 1 (1948), 289–293. DOI 10.4064/cm-1-4-289-293 | MR 0030120 | Zbl 0037.32302
[2] R. A.  Gordon: The Integrals of Lebesgue. Denjoy, Perron, and Henstock, Graduate Studies in Mathematics Volume 4, AMS, 1994. MR 1288751 | Zbl 0807.26004
[3] J.  Jarník and Kurzweil: Perron-type integration on  $n$-dimensional intervals and its properties. Czechoslovak Math. J. 45 (120) (1995), 79–106. MR 1314532
[4] J.  Kurzweil: On multiplication of Perron integrable functions. Czechoslovak Math. J. 23 (98) (1973), 542–566. MR 0335705 | Zbl 0269.26007
[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 1371683
[6] Lee Peng Yee: Lanzhou Lectures on Henstock integration. World Scientific, 1989. MR 1050957 | Zbl 0699.26004
[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 1756319
[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 10.1017/S0004972700021857 | MR 1419607
[9] Lee Tuo Yeong: Multipliers for some non-absolute integrals in the Euclidean spaces. Real Anal. Exchange 24 (1998/99), 149–160. MR 1691742
[10] G. Q.  Liu: The dual of the Henstock-Kurzweil space. Real Anal. Exchange 22 (1996/97), 105–121. MR 1433600
[11] E. J.  McShane: Integration. Princeton Univ. Press, 1944. MR 0082536 | Zbl 0060.13010
[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 10.1155/S0161171288000043 | MR 0918213 | Zbl 0662.26003
[14] S.  Saks: Theory of the Integral, second edition. New York, 1964 63.0183.05. MR 0167578
[15] W. L. C.  Sargent: On the integrability of a product. J. London Math. Soc. 23 (1948), 28–34. DOI 10.1112/jlms/s1-23.1.28 | MR 0026113 | Zbl 0031.29201
[16] W. H.  Young: On multiple integration by parts and the second theorem of the mean. Proc. London Math. Soc. 16 (1918), 273–293.
Partner of
EuDML logo