Approximation in variation for nonlinear Mellin integral operators in multidimensional setting

In this paper we study convergence results for a family of nonlinear integral operators of Mellin type defined as $$ (T_w f)({\tt s}) =\int_{\R^N_+} K_w({\tt t},f({\tt st})){ \,d{\tt t} \over \langle{\tt t}\rangle} , \ {\tt s}\in\R^N_+,\ w>0; $$ here $\{K_w\}_{w>0}$ is a family of kernel functions, $\langle{\tt t}\rangle:=\prod_{i=1}^N t_i$, ${\tt t}=(t_1,\dots,t_N)\in \R^N_+$, and $f$ is a function of bounded variation on $\R^N_+$. The interest about approximation results in $BV-$spaces in the multidimensional setting of $\R^N_+$ is due, apart from a mathematical point of view, also to the important applications of such results in image reconstruction and in other fields. For this reason, in order to treat our problem, we use a multidimensional concept of variation in the sense of Tonelli, adapted from the classical definition to the present setting of $\R^N_+$, equipped with the log-Haar measure.