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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. Jarník 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 $$E$$.

##### MSC:
 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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