Abstract
We study a generalization of the classical Henstock-Kurzweil integral, known as the strong ϱ-integral, introduced by Jarník and Kurzweil. Let \((S_\varrho (E)|| \cdot ||)\) be the space of all strongly ϱ-integrable functions on a multidimensional compact interval E, equipped with the Alexiewicz norm \(|| \cdot ||.\) We show that each element in the dual space of \((S_\varrho (E)|| \cdot ||)\) can be represented as a strong ϱ-integral. Consequently, we prove that fg is strongly ϱ-integrable on E for each strongly ϱ-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.
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References
A. Alexiewicz: Linear functionals on Denjoy-integrable functions. Colloq. Math. 1 (1948), 289–293.
R. A. Gordon: The Integrals of Lebesgue. Denjoy, Perron, and Henstock, Graduate Studies in Mathematics Volume 4, AMS, 1994.
J. Jarník and Kurzweil: Perron-type integration on n-dimensional intervals and its prop-erties. Czechoslovak Math. J. 45 (120) (1995), 79–106.
J. Kurzweil: On multiplication of Perron integrable functions. Czechoslovak Math. J. 23 (98) (1973), 542–566.
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.
Lee Peng Yee: Lanzhou Lectures on Henstock integration. World Scientific, 1989.
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.
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.
Lee Tuo Yeong: Multipliers for some non-absolute integrals in the Euclidean spaces. Real Anal. Exchange 24 (1998/99), 149–160.
G. Q. Liu: The dual of the Henstock-Kurzweil space. Real Anal. Exchange 22 (1996/97), 105–121.
E. J. McShane: Integration. Princeton Univ. Press, 1944.
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; Lecture Notes Math. vol. 1419, Springer-Verlag, Berlin, Heideberg, New York, 1990, pp. 136–149.
K. M. Ostaszewski: The space of Henstock integrable functions of two variables. Internat. J. Math. Math. Sci. 11 (1988), 15–22.
S. Saks: Theory of the Integral, second edition. New York, 1964. [
W. L. C. Sargent: On the integrability of a product. J. London Math. Soc. 23 (1948), 28–34.
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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Tuo-Yeong, L. A Full Characterization of Multipliers for the Strong ϱ-Integral in the Euclidean Space. Czechoslovak Mathematical Journal 54, 657–674 (2004). https://doi.org/10.1007/s10587-004-6415-7
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DOI: https://doi.org/10.1007/s10587-004-6415-7