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Article

Tuo-Yeong, Lee
A full characterization of multipliers for the strong $\rho$-integral in the euclidean space. (English). Czechoslovak Mathematical Journal, vol. 54 (2004), issue 3, pp. 657-674
MSC: 26A39, 46E99, 46G10 | MR 2086723 | Zbl 1080.26007
Keywords:
strong $\rho $-integral; multipliers; dual space
Summary:
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 $E$.
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