ON THE DUAL SPACE OF THE HENSTOCK-KURZWEIL INTEGRABLE FUNCTIONS IN N DIMENSIONS* Tepper
نویسنده
چکیده
The dual space of the class of Henstock-Kurzweil integrable functions is well known in the one-dimensional case and corresponds to the space of multipliers which, in turn, coincides with the class of functions of bounded essential variation. Comparable results in higher dimensions have been elusive. For cases in which the partitions defining the Henstock-Kurzweil integrals are defined on n-cells (parallelepipeds) with their sides parallel to the coordinate axes (i.e., Cartesian products of compact intervals), we prove that a function is a multiplier of the class of n-dimensional (n ≥ 2) Henstock-Kurzweil integrable functions if and only if it is of strongly bounded essential variation as defined by Kurzweil. This result was proved earlier by T.Y. Lee, T. S. Chew, and P.Y. Lee in the two-dimensional case by a different method. The sufficiency part of our proof makes use of a generalization of a method used earlier by P.Y. Lee in the one-dimensional case. *2000 Mathematics Subject Classification. Primary 26A39; Secondary 46E99
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