The Dual of the Space of Regulated Functions in Several Variables - Yang, Yunyun; Estrada, Ricardo (Sarajevo Journal of Mathematics, 2024)
We consider a class of regulated functions of several variables, namely, the class of functions $\phi$ defined in an open set $U\subset\mathbb{R}^{n}$ such that at each $\mathbf{x}_{0}\in U$ the \[\phi_{\mathbf{x}_{0}}\left( \mathbf{w}\right) =\lim_{\varepsilon \rightarrow0^{+}}\phi\left(f{x}_{0}+\varepsilon\mathbf{w}\right) \,,\] exists uniformly on $\mathbf{w}\in\mathbb{S},$ the unit sphere of $\mathbb{R}^{n},$ and such that $\phi_{\mathbf{x}_{0}}$ is a continuous function of $\mathbf{w}$ at each $\mathbf{x}_{0}\in U.$ We identify the elements of the dual spaces of several Banach space of such regulated functions in several variables as signed measures with absolutely convergent sums of \textquotedblleft thick\textquotedblright\ delta functions concentrated at a countable set.