:= [−π, π). We give sufficient conditions in terms of the coefficients in order that >f belong to one of the multiplicative Lipschitz classes Lip (α1,..., αN) and lip (α1,..., αN) for some α1,..., αN > 0. These multiplicative Lipschitz classes of functions are defined in terms of the multiple difference operator of first
order in each variable. The conditions given by us are not only sufficient, but also necessary for a special subclass of coefficients.
Our auxiliary results on the equivalence between the order of magnitude of the rectangular partial sums and that of the rectangular
remaining sums of related N-multiple numerical series may be useful in other investigations, too.