Convergence in variation for a homothetic modulus of smoothness in multidimensional setting

In order to get approximation results for linear and nonlinear convolution integral operators in $BV^{\phi}$-spaces, it is crucial to study convergence of the modulus of continuity. In the case of the modulus of continuity defined by means of the classical variation, it is well known that, if f is absolutely continuous, then the modulus of smoothness $\omega(f,\delta)$ converges to 0, as $\delta$ tends to 0 from the right. The purpose of this paper is to extend the above result to the frame of $BV^phi((R+_0)^N)$ for the modulus of smoothness $\omega^{\varphi}(f,\delta):= sup_{|1-t|<\delta} V^{\phi} [\tau_t f- f]$, where $\tau_t f(x) = f(xt)$ is the homothetic operator and $V^{\phi}[f]$ is the multidimensional $\phi$-variation introduced in [L. Angeloni-G. Vinti, "Convergence and rate of approximation for linear integral operators in BV^{\phi}-spaces in multidimensional setting", J. Math. Anal. Appl., 349(2) (2009), 317--334].