A Representation Theorem for $\phi$-Bounded Variation of Functions in the Sense of Riesz

In this paper we extend the well known Riesz lemma to the class of bounded $\phi$-variation functions in the sense of Riesz defined on a rectangle $I_a^b\subset {\mathbb{R}}^2$. This concept was introduced in [2], where the authors proved that the space $B{V}_{\phi }^R(I_a^b;\mathbb{R}$ of such functions is a Banach Algebra. Moreover, they characterized also the Nemytskii operator acting in this space. Thus our result creates a continuation of the paper [2].