On the dual space of \({\mathcal B}{\mathcal V}\)-integrable functions in Euclidean space (Q1772948)

The \({\mathcal BV}\)-integral is an analogue, in several dimensions, of the Denjoy and Henstock-Kurzweil integrals [see \textit{W. Pfeffer}, ``Derivation and integration'' (Cam\-bridge Tracts in Mathematics 140, Cambridge University Press) (2001; Zbl 0980.26008)]. \textit{T. De Pauw} [J. Funct. Anal. 144, No.1, 190--231 (1997; Zbl 0887.46007)] introduced a norm, the Alexiewicz norm, on the space of \({\mathcal BV}\)-integrable functions. In the paper under review, the author shows that the dual of the space \({\mathcal R}(E)\) of the space of \({\mathcal BV}\)-integrable functions on \(E=\prod_{i=1}^m [a_i,b_i]\subseteq {\mathbb R}^m\) with the Alexiewicz norm is isometrically isomorphic to the space of finite signed Borel measures on \(\prod_{i=1}^m [a_i,b_i)\). He also gives an example of a Lipschitz function \(H\colon [0,1]\times [0,1]\to {\mathbb R}\) which is not contained in the dual of \({\mathcal R}(E)\): let \(h(x,y)= (1/xy) \sin (xy)\) if \(xy\neq 0\) and \(h(x,y)=0\) otherwise; if \[ H(s,t)=\int_0^s\biggl(\int_0^t h(x,y)\,dy\biggr)\,dx, \] then \(H\) is not in the dual of \({\mathcal R}([0,1]\times [0,1])\) because \(h\) is not Lebesgue-integrable on \([0,1]\times[0,1]\). This answers a question of T. De Pauw.