On an Sz.-Nagy theorem (Q2341948)

Let \(n>1\) and let us consider the Wiener's algebra \(W_0(\mathbb R^n)\), i.e. the space of the functions \(f: \mathbb R^n\to C\) which are the transforms of the functions \(g\in L_1 (\mathbb R^n)\), endowed with the norm \(| | f | |_{W_0} = | | g | |_{L_{1}\mathbb R^n}\). Various conditions of belonging to Wiener's algebra are known. In particular, in the year 1948, \textit{B. Sz.-Nagy} [Hung. Acta Math. 1, 14--52 (1948; Zbl 0034.04401)] gave in the case \(n=1\) some conditions involving \(f\) and \(f'\) which ensure that \(f\in W_0(\mathbb R)\). In the year 2014 \textit{R. M. Trigub} [Math. Notes 96, No. 3, 454--457 (2014); translation from Mat. Zametki 96, No. 3, 473--475 (2014; Zbl 1314.42005)], still in the one-dimensional case, extended the result to some lightly more general classes of functions in such a way that the given conditions are both necessary and sufficient. In this paper, the theorem of Trigub is generalized to the \(n\)-dimensional case. In particular, the obtained result holds for the classes \(f:\mathbb{R}^n\to \mathbb{C}\) with bounded Hardy variation.