Geodesic
A Characterization of a Modulus of Smoothness in Multidimensional Setting
Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 1, pp. 79-108.

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A classical result of approximation theory states that $\lim_{\delta \to 0} \omega(f, \delta) = 0$, where $\omega$ is the modulus of smoothness of $f$ defined by means of the variation functional, if and only if $f$ is absolutely continuous. Such theorem is crucial in order to obtain results about convergence and order of approximation for linear and non-linear integral operators in BV-spaces. It was an open problem to extend the above result to the setting of $\varphi$-variation in the multidimensional frame. In this paper, working with a concept of multidimensional W-variation introduced in [3], we prove that an analogous characterization holds for the multidimensional $\varphi$-modulus of smoothness. 
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Angeloni, Laura. A Characterization of a Modulus of Smoothness in Multidimensional Setting. Bollettino della Unione matematica italiana, Série 9, Tome 4 (2011) no. 1, pp. 79-108. http://geodesic.mathdoc.fr/item/BUMI_2011_9_4_1_a4/

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