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A full characterization of multipliers for the strong \(\varrho \)-integral in the Euclidean space. (English) Zbl 1080.26007
Summary: We study a generalization of the classical Henstock-Kurzweil integral, known as the strong \(\rho \)-integral, introduced by J. JarnĂ­k and J. Kurzweil. Let \((\mathcal S_{\rho } (E), \| \cdot \| )\) be the space of all strongly \(\varrho \)-integrable functions on a multidimensional compact interval \(E\), equipped with the Alexiewicz norm \(\| \cdot \| \). We show that each element in the dual space of \((\mathcal S_{\rho } (E), \| \cdot \| )\) 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 \(E\).

26A39 Denjoy and Perron integrals, other special integrals
46E99 Linear function spaces and their duals
46G10 Vector-valued measures and integration
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