Diagonals of separately absolutely continuous mappings coincide with the sums of absolutely convergent series of continuous functions (Q2806125)

Let \(X\), \(Z\) be topological spaces and \(f: X^{2}\to Z\) a mapping. The diagonal of~\(f\) is the mapping \(g: X\to Z\) given by \(g(x)=f(x,x)\).NEWLINENEWLINEFor the special case \(X=Z=\mathbb{R}\), Baire proved that diagonals of separately continuous functions are exactly Baire-one functions, i.e., pointwise limits of continuous functions. Recently, it was proved that there exists a separately absolutely continuous function \(f: [0,1]^{2}\to\mathbb{R}\) with partial derivatives in \(L^{p}\) for every \(p>1\), and such that its diagonal \(g\) is not absolutely continuous.NEWLINENEWLINENEWLINESo the following problem arises: characterize functions \(g: [0,1] \to\mathbb{R}\) such that there is a separately absolutely continuous function \(f: ]0,1]^{2}\to\mathbb{R}\) with diagonal \(g\).NEWLINENEWLINELet \(X\) be a topological space and let \(Z\) be a normed space. A mapping \(f: X\to Z\) is said to be an absolute Baire-one mapping if \(f\) is the sum of an absolutely convergent series of continuous functions.NEWLINENEWLINENEWLINEIn this paper, the authors solve the problem mentioned above by means of the following characterization of diagonals of separately absolutely continuous functions:NEWLINENEWLINE-- Let \(X\subset\mathbb{R}\) be an interval, let \(Z\) be a normed space and let \(g: X\to Z\). Then, the following conditions are equivalent:NEWLINENEWLINE{\parindent=0.6cm\begin{itemize}\item[(i)] There exists a separately absolutely continuous mapping \(f: X^{2}\to Z\) with diagonal \(g\). \item[(ii)] There exists a mapping \(f: X^{2}\to Z\) with diagonal \(g\) that is continuous with respect to the first variable and is of bounded variation and continuous with respect to the second variable at every point of the diagonal \(\Delta=\{(x,x): x\in X\}\). \item[(iii)] \(g\) is an absolute Baire-one mapping. NEWLINENEWLINE\end{itemize}}